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One major challenge for practical implementations is to make perimeter defense strategies scalable for large-scale problem instances. To this end, we leverage graph neural networks (GNNs) to develop an imitation learning framework that learns a mapping from defenders’ local perceptions and their communication graph to their actions. The proposed GNN-based learning network is trained by imitating a centralized expert algorithm such that the learned actions are close to that generated by the expert algorithm. We demonstrate that our proposed network performs closer to the expert algorithm and is superior to other baseline algorithms by capturing more intruders. Our GNN-based network is trained at a small scale and can be generalized to large-scale cases. We run perimeter defense games in scenarios with different team sizes and configurations to demonstrate the performance of the learned network." addthis:url="//www.thespel.com/articles/10.3389/fcteg.2023.1104745" data-event-param="Graph neural networks for decentralized multi-agent perimeter defense" title="" target="_blank" href="//www.thespel.com/journal/articles/10.3389/#"><span class="at-icon-wrapper"></span></a> </div></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_linkedin at300b" addthis:title="Graph neural networks for decentralized multi-agent perimeter defense" addthis:description="In this work, we study the problem of decentralized multi-agent perimeter defense that asks for computing actions for defenders with local perceptions and communications to maximize the capture of intruders. One major challenge for practical implementations is to make perimeter defense strategies scalable for large-scale problem instances. To this end, we leverage graph neural networks (GNNs) to develop an imitation learning framework that learns a mapping from defenders’ local perceptions and their communication graph to their actions. The proposed GNN-based learning network is trained by imitating a centralized expert algorithm such that the learned actions are close to that generated by the expert algorithm. We demonstrate that our proposed network performs closer to the expert algorithm and is superior to other baseline algorithms by capturing more intruders. Our GNN-based network is trained at a small scale and can be generalized to large-scale cases. We run perimeter defense games in scenarios with different team sizes and configurations to demonstrate the performance of the learned network." addthis:url="//www.thespel.com/articles/10.3389/fcteg.2023.1104745" data-event-param="Graph neural networks for decentralized multi-agent perimeter defense" title="" target="_blank" href="//www.thespel.com/journal/articles/10.3389/#"><span class="at-icon-wrapper"></span></a> </div></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_facebook at300b" addthis:title="Graph neural networks for decentralized multi-agent perimeter defense" addthis:description="In this work, we study the problem of decentralized multi-agent perimeter defense that asks for computing actions for defenders with local perceptions and communications to maximize the capture of intruders. One major challenge for practical implementations is to make perimeter defense strategies scalable for large-scale problem instances. To this end, we leverage graph neural networks (GNNs) to develop an imitation learning framework that learns a mapping from defenders’ local perceptions and their communication graph to their actions. The proposed GNN-based learning network is trained by imitating a centralized expert algorithm such that the learned actions are close to that generated by the expert algorithm. We demonstrate that our proposed network performs closer to the expert algorithm and is superior to other baseline algorithms by capturing more intruders. Our GNN-based network is trained at a small scale and can be generalized to large-scale cases. We run perimeter defense games in scenarios with different team sizes and configurations to demonstrate the performance of the learned network." addthis:url="//www.thespel.com/articles/10.3389/fcteg.2023.1104745" data-event-param="Graph neural networks for decentralized multi-agent perimeter defense" title="" target="_blank" href="//www.thespel.com/journal/articles/10.3389/#"><span class="at-icon-wrapper"></span></a> </div></li> </ul> <button class="dropdown-menu-mobile-close-button" aria-label="Close" data-toggle="dropdown"></button> </div></li> <li class="dropdown"><button type="button" class="navbar-toggle navbar-citations" data-event="citation-button-click" data-toggle="dropdown" data-target="//www.thespel.com/journal/articles/10.3389/#" aria-expanded="false"></button> <div class="dropdown-menu-wrapper"> <div class="dropdown-menu-mobile-title"> 出口的引用</gydF4y2Badiv> <ul class="dropdown-menu" role="menu"> <li><a data-test-id="article-endnote" href="//www.thespel.com/journal/articles/10.3389/fcteg.2023.1104745/endNote" data-event="endnotedownload-button-click">尾注</gydF4y2Baa></li> <li><a data-test-id="article-referencemanager" href="//www.thespel.com/journal/articles/10.3389/fcteg.2023.1104745/reference" data-event="referencemanagerdownload-button-click">引用管理器</gydF4y2Baa></li> <li><a data-test-id="article-simpletextfile" href="//www.thespel.com/journal/articles/10.3389/fcteg.2023.1104745/text" data-event="simpletextfiledownload-button-click">简单文本文件</gydF4y2Baa></li> <li><a data-test-id="article-bibtex" href="//www.thespel.com/journal/articles/10.3389/fcteg.2023.1104745/bibTex" data-event="bibtexdownload-button-click">助理</gydF4y2Baa></li> </ul> <button class="dropdown-menu-mobile-close-button" aria-label="Close" data-toggle="dropdown"></button> </div></li> </ul> </div> <div class="side-article-impact"> <ul class="nav"> <li class="impact-data" title="View numbers are not up-to-date. 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One major challenge for practical implementations is to make perimeter defense strategies scalable for large-scale problem instances. To this end, we leverage graph neural networks (GNNs) to develop an imitation learning framework that learns a mapping from defenders’ local perceptions and their communication graph to their actions. The proposed GNN-based learning network is trained by imitating a centralized expert algorithm such that the learned actions are close to that generated by the expert algorithm. We demonstrate that our proposed network performs closer to the expert algorithm and is superior to other baseline algorithms by capturing more intruders. Our GNN-based network is trained at a small scale and can be generalized to large-scale cases. We run perimeter defense games in scenarios with different team sizes and configurations to demonstrate the performance of the learned network." addthis:url="//www.thespel.com/articles/10.3389/fcteg.2023.1104745" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="facebook_count"></span></a></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_twitter at300b" data-event="shareicon-button-click" data-event-param="Graph neural networks for decentralized multi-agent perimeter defense" addthis:title="Graph neural networks for decentralized multi-agent perimeter defense" addthis:description="In this work, we study the problem of decentralized multi-agent perimeter defense that asks for computing actions for defenders with local perceptions and communications to maximize the capture of intruders. One major challenge for practical implementations is to make perimeter defense strategies scalable for large-scale problem instances. To this end, we leverage graph neural networks (GNNs) to develop an imitation learning framework that learns a mapping from defenders’ local perceptions and their communication graph to their actions. The proposed GNN-based learning network is trained by imitating a centralized expert algorithm such that the learned actions are close to that generated by the expert algorithm. We demonstrate that our proposed network performs closer to the expert algorithm and is superior to other baseline algorithms by capturing more intruders. Our GNN-based network is trained at a small scale and can be generalized to large-scale cases. We run perimeter defense games in scenarios with different team sizes and configurations to demonstrate the performance of the learned network." addthis:url="//www.thespel.com/articles/10.3389/fcteg.2023.1104745" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="twitter_count"></span></a></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_google at300b" data-event="shareicon-button-click" data-event-param="Graph neural networks for decentralized multi-agent perimeter defense" addthis:title="Graph neural networks for decentralized multi-agent perimeter defense" addthis:description="In this work, we study the problem of decentralized multi-agent perimeter defense that asks for computing actions for defenders with local perceptions and communications to maximize the capture of intruders. One major challenge for practical implementations is to make perimeter defense strategies scalable for large-scale problem instances. To this end, we leverage graph neural networks (GNNs) to develop an imitation learning framework that learns a mapping from defenders’ local perceptions and their communication graph to their actions. The proposed GNN-based learning network is trained by imitating a centralized expert algorithm such that the learned actions are close to that generated by the expert algorithm. We demonstrate that our proposed network performs closer to the expert algorithm and is superior to other baseline algorithms by capturing more intruders. Our GNN-based network is trained at a small scale and can be generalized to large-scale cases. We run perimeter defense games in scenarios with different team sizes and configurations to demonstrate the performance of the learned network." addthis:url="//www.thespel.com/articles/10.3389/fcteg.2023.1104745" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="twitter_count"></span></a></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_linkedin at300b" data-event="shareicon-button-click" data-event-param="Graph neural networks for decentralized multi-agent perimeter defense" addthis:title="Graph neural networks for decentralized multi-agent perimeter defense" addthis:description="In this work, we study the problem of decentralized multi-agent perimeter defense that asks for computing actions for defenders with local perceptions and communications to maximize the capture of intruders. One major challenge for practical implementations is to make perimeter defense strategies scalable for large-scale problem instances. To this end, we leverage graph neural networks (GNNs) to develop an imitation learning framework that learns a mapping from defenders’ local perceptions and their communication graph to their actions. The proposed GNN-based learning network is trained by imitating a centralized expert algorithm such that the learned actions are close to that generated by the expert algorithm. We demonstrate that our proposed network performs closer to the expert algorithm and is superior to other baseline algorithms by capturing more intruders. Our GNN-based network is trained at a small scale and can be generalized to large-scale cases. We run perimeter defense games in scenarios with different team sizes and configurations to demonstrate the performance of the learned network." addthis:url="//www.thespel.com/articles/10.3389/fcteg.2023.1104745" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="linkedin_count"></span></a></li> <li class="social_block"> <div class="atButton"> <a class="addthis_button_more at300b" data-event="shareicon-button-click" data-event-param="Graph neural networks for decentralized multi-agent perimeter defense" addthis:title="Graph neural networks for decentralized multi-agent perimeter defense" addthis:description="In this work, we study the problem of decentralized multi-agent perimeter defense that asks for computing actions for defenders with local perceptions and communications to maximize the capture of intruders. One major challenge for practical implementations is to make perimeter defense strategies scalable for large-scale problem instances. To this end, we leverage graph neural networks (GNNs) to develop an imitation learning framework that learns a mapping from defenders’ local perceptions and their communication graph to their actions. The proposed GNN-based learning network is trained by imitating a centralized expert algorithm such that the learned actions are close to that generated by the expert algorithm. We demonstrate that our proposed network performs closer to the expert algorithm and is superior to other baseline algorithms by capturing more intruders. Our GNN-based network is trained at a small scale and can be generalized to large-scale cases. We run perimeter defense games in scenarios with different team sizes and configurations to demonstrate the performance of the learned network." addthis:url="//www.thespel.com/articles/10.3389/fcteg.2023.1104745" title=""><span class="at-icon-wrapper" style=" background-color: rgb(29, 161, 242); line-height: 16px; height: 16px; width: 16px; "></span></a> </div><a><span class="total_count"></span></a></li> </ul> </div> <aside id="anchors" class="pull-left table-of-contents side-article"> <div class="side-people hidden-sm hidden-xs"> <section class="side-article-editors"> <header> <h5 class="like-h4">编辑</gydF4y2Bah5> </header> <a class="authors" href="//www.thespel.com/loop/people/1835831/overview" target="_blank"><img alt="" class="pr5 pull-left" onerror="this.src = '/Areas/Articles/Images/Frontiers/Common/Profile/default-loop-profile.jpg'" src="//www.thespel.com/loop/images/profile/1835831/24"><h6 class="author-link-aside">道格拉斯Guimarães马夏列</gydF4y2Bah6></a> <div class="clearfix"></div> <p><span class="notes">米纳斯吉拉斯州联邦大学，巴西</年代pan></p> <header> <h5 class="like-h4">看过的</gydF4y2Bah5> </header> <a class="authors" href="//www.thespel.com/loop/people/1117249/overview" target="_blank"><img alt="" class="pr5 pull-left" onerror="this.src = '/Areas/Articles/Images/Frontiers/Common/Profile/default-loop-profile.jpg'" src="//www.thespel.com/loop/images/profile/1117249/24"><h6 class="author-link-aside">紫阳孟</gydF4y2Bah6></a> <div class="clearfix"></div> <p><span class="notes">清华大学，中国</年代pan></p> <a class="authors" href="//www.thespel.com/loop/people/1118310/overview" target="_blank"><img alt="" class="pr5 pull-left" onerror="this.src = '/Areas/Articles/Images/Frontiers/Common/Profile/default-loop-profile.jpg'" src="//www.thespel.com/loop/images/profile/1118310/24"><h6 class="author-link-aside">克里斯托弗·尼尔森</gydF4y2Bah6></a> <div class="clearfix"></div> <p><span class="notes">加拿大滑铁卢大学</年代pan></p> </section> </div> <nav> <header> <h5 class="like-h4" style="padding-top: 14px; padding-left: 6px; margin-bottom: 6px;">目录</gydF4y2Bah5> </header> <ul class="nav nav-list list-unstyled contents" data-event="toc-link-click"> <li> <ul class="flyoutJournal"> <li><a href="//www.thespel.com/journal/articles/10.3389/#h1">摘要</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h2">1介绍</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h3">相关工作</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h4">3问题的制定</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h5">4种方法</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h6">5实验</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h7">6结论</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h8">数据可用性声明</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h9">作者的贡献</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h10">致谢</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h11">利益冲突</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h12">出版商的注意</gydF4y2Baa></li> <li><a href="//www.thespel.com/journal/articles/10.3389/#h13">参考文献</gydF4y2Baa></li> </ul></li> </ul> </nav> </aside> <div class="clearfix"></div> <div class="side-article-download side-export clearfix"> <ul class="list-unstyled list-inline clearfix"> <li class="dropdown text-center citation"><button data-target="//www.thespel.com/journal/articles/10.3389/#" data-test-id="citation-button" data-toggle="dropdown" role="button" aria-expanded="false"><span class="icon-label" data-event="citation-button-click">出口的引用</年代pan></button> <ul class="dropdown-menu" role="menu"> <li><a data-test-id="article-endnote" href="//www.thespel.com/journal/articles/10.3389/fcteg.2023.1104745/endNote" data-event="endnotedownload-button-click">尾注</gydF4y2Baa></li> <li><a data-test-id="article-referencemanager" href="//www.thespel.com/journal/articles/10.3389/fcteg.2023.1104745/reference" data-event="referencemanagerdownload-button-click">引用管理器</gydF4y2Baa></li> <li><a data-test-id="article-simpletextfile" href="//www.thespel.com/journal/articles/10.3389/fcteg.2023.1104745/text" data-event="simpletextfiledownload-button-click">简单文本文件</gydF4y2Baa></li> <li><a data-test-id="article-bibtex" href="//www.thespel.com/journal/articles/10.3389/fcteg.2023.1104745/bibTex" data-event="bibtexdownload-button-click">助理</gydF4y2Baa></li> </ul></li> </ul> </div> <div class="side-crossmark"> <a data-target="crossmark" class="crossmark"> <figure> <img id="crossmark-icon" style="border: 0; width:64px; height:64px;" src="https://crossmark-cdn.crossref.org/widget/v2.0/logos/CROSSMARK_BW_square_no_text.svg" title="" alt=""> </figure> <div id="crossmark-icon"> 检查更新</gydF4y2Badiv></a> </div> <article class="widget-listing people-also-looked-at side-article-related hidden"> <h5 class="like-h4" data-event="peoplelookedat-link-click">人们还研究了</gydF4y2Bah5> </article> </aside> <main class=""> <div class="article-section"> <div class="article-container" data-html="True"> <div class="abstract-container"> <div class="article-header-container">   <div class="header-bar-one"> <h2>原创研究文章</gydF4y2Bah2> </div> <div class="header-bar-three-container"> <div class="header-bar-three"> 前面。控制中。，13.January 2023<br>第2节网络控制<gydF4y2Babr> <a href="https://doi.org/10.3389/fcteg.2023.1104745">https://doi.org/10.3389/fcteg.2023.1104745</gydF4y2Baa> </div> </div> </div> <div class="JournalAbstract"> <a id="h1" name="h1"></a> <h1>分布式多智能体周界防御的图形神经网络</gydF4y2Bah1> <div class="authors"> <a href="//www.thespel.com/journal/people/u/2104035" class="user-id-2104035"><img class="pr5" src="//www.thespel.com/loop/images/profile/2104035/24" onerror="this.src='https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg'" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba">伊利亚·s·李</gydF4y2Baa> <sup>1</年代up>*，<gydF4y2Baa href="//www.thespel.com/journal/people/u/2104098" class="user-id-2104098"><img class="pr5" src="//www.thespel.com/loop/images/profile/2104098/24" onerror="this.src='https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg'" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba">和周</gydF4y2Baa> <sup>2</年代up>，<我mg class="pr5" src="https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba">亚历杭德罗里贝罗<年代up>1</年代up>而且<我mg class="pr5" src="https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba">维贾伊·库马尔<年代up>1</年代up> </div> <ul class="notes"> <li><span><sup>1</年代up></span>GRASP实验室，宾夕法尼亚大学，费城，美国</l我><l我><年代pan><sup>2</年代up></span>美国费城德雷塞尔大学电气与计算机工程系</l我></ul> <p class="mb15">在这项工作中，我们研究了分散的多智能体周界防御问题，该问题要求为具有本地感知和通信的防御者计算动作，以最大限度地捕获入侵者。实际实现的一个主要挑战是使外围防御策略可针对大规模问题实例进行扩展。为此，我们利用图神经网络(GNNs)开发了一个模仿学习框架，从防御者的局部感知和他们的通信图到他们的行动学习映射。所提出的基于gnn的学习网络通过模仿集中式专家算法来训练，使学习到的动作接近于专家算法生成的动作。我们证明，我们提出的网络性能更接近专家算法，并通过捕获更多的入侵者优于其他基线算法。我们基于gnn的网络是在小范围内训练的，可以推广到大规模的案例。我们在不同团队规模和配置的场景中运行外围防御游戏，以演示学习网络的性能。</p><gydF4y2Badiv class="clear"></div> </div> <div class="JournalFullText"> <a id="h2" name="h2"></a> <h2>1介绍</gydF4y2Bah2> <p class="mb15">外围防御游戏的问题考虑的是这样一种场景:防守方被限制沿着外围移动并试图捕获入侵者，而入侵者则试图到达外围而不被防守方捕获(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B30">Shishika和Kumar, 2020年</gydF4y2Baa>)．之前的许多作品都通过平面游戏空间(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B31">Shishika和Kumar, 2018年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B3">Chen等人，2021</gydF4y2Baa>)．然而，在现实世界中，由于玩家(例如，防守者和入侵者)可能具有执行三维运动的能力，周长可能由三维形状表示。例如，防御者旨在保护的建筑物外围可以被一个半球包围。因此，防御机器人应该能够在三维空间中移动。例如，空中机器人已经在各种环境中得到了很好的研究(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B15">Lee等人，2016</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B11">Lee等人，2020a</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B25">Nguyen等人，2019</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B5">Chen等，2020</gydF4y2Baa>)，所有这些设置都可以作为周界防御的真实用例。例如，入侵者试图攻击森林中的一个军事基地，防御者的目标是抓住入侵者。</p><pcl作为年代＝"mb15">在这项工作中，我们解决了多个代理协作完成任务的领域中的周界防御问题。多主体协作已在许多领域进行了探索，包括环境测绘(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B32">Thrun等人，2000年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B19">Liu等，2022</gydF4y2Baa>)、搜寻及救援(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B2">巴克斯特等人，2007年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B22">米勒等人，2020年</gydF4y2Baa>)，目标跟踪(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B14">Lee等人，2022b</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B7">Ge等，2022</gydF4y2Baa>)、按需无线基建(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B23">Mox等人，2020年</gydF4y2Baa>)、交通(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B24">Ng等人，2022</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B36">徐等，2022</gydF4y2Baa>)和多智能体学习(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B9">Kim等人，2021年</gydF4y2Baa>)．我们的方法是使用一个机器人团队，他们共同努力实现一个共同的目标，那就是防守外围。我们专注于为防守团队开发分散策略，原因如下:i)可以动态添加或删除队友，而不会破坏明确的层次结构;(2)集中式系统可能无法处理团队联合状态空间的高维;iii)防御者的通信范围有限，只能进行局部通信。</p><pcl作为年代＝"mb15">为此，我们的目标是开发一个框架，在这个框架中，一组防御者合作，使用基于当地感知和通信的分散策略来防御外围。具体来说，我们探索了基于学习的方法，通过模仿专家算法，如最大匹配算法(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B4">Chen et al.， 2014</gydF4y2Baa>)．最大匹配算法运行穷举搜索来找到最佳策略，在大规模上计算量很大，因为这种方法本质上是组合的，并且假设全局信息。我们利用GNN作为学习范式，并证明训练后的网络可以执行接近专家算法。gnn具有分散的通信架构，可以捕获相邻的交互和可转移性，从而可以推广到以前未见过的场景(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B28">Ruiz等人，2021年</gydF4y2Baa>)．我们证明了我们提出的基于gnn的网络可以推广到大规模解决多机器人外围防御游戏。</p><pcl作为年代＝"mb15">基于这一观点，我们在本文中做出了以下主要贡献:</p><pcl作为年代＝"mb15">使用图神经网络的分散周界防御框架。我们提出了一种新颖的学习框架，利用基于图形的表示外围防御游戏。据我们所知，我们是第一个通过学习分散策略来解决分散半球外围防御问题的<gydF4y2Baem>通过</gydF4y2Baem>图神经网络。</p><pcl作为年代＝"mb15">具有可扩展性的强大外围防御性能。我们证明了我们的方法执行接近于专家策略(即最大匹配算法)<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B4">Chen et al. (2014)</gydF4y2Baa>)，并优于其他基线算法。我们提出的网络是在小范围内训练的，可以推广到大范围。</p><gydF4y2Baa id="h3" name="h3"></a> <h2>相关工作</gydF4y2Bah2> <h3 class="pt0">2.1周界防御</gydF4y2Bah3> <p class="mb0">在外围防御游戏中，防御者的目标是通过沿着外围移动来捕获入侵者，而入侵者则试图在不被防御者捕获的情况下到达外围。我们参考(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B30">Shishika和Kumar, 2020年</gydF4y2Baa>)进行详细调查。许多之前的作品都是关于平面游戏空间(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B31">Shishika和Kumar, 2018年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B20">马沙尔等人，2020年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B1">Bajaj等人，2021年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B3">Chen等人，2021</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B8">Hsu等人，2022</gydF4y2Baa>)．例如，在一个平面游戏空间上求解了一个多人合作周防游戏。<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B31">Shishika和Kumar, 2018年</gydF4y2Baa>)．提出了一种基于入侵者到达估计的自适应分区策略(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B20">马沙尔等人，2020年</gydF4y2Baa>)．后来，周界防御问题的一个公式作为流动网络的一个实例被提出在(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B3">Chen等人，2021</gydF4y2Baa>)．此外，在(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B1">Bajaj等人，2021年</gydF4y2Baa>)，而具有异构团队的模型则在(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B8">Hsu等人，2022</gydF4y2Baa>)．</p><pcl作为年代＝"mb0">周界防御问题的高维扩展最近已在(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B12">Lee等人，2020b</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B13">Lee等人，2021年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B11">Lee等人，2022a</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B16">Lee和Bakolas, 2021年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B37">Yan等，2022</gydF4y2Baa>)．例如,<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B16">李和巴克拉斯(2021)</gydF4y2Baa>分析了在高维欧氏空间中防范封闭凸目标集攻击者的二人微分对策。<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B37">Yan等人(2022)</gydF4y2Baa>研究了一款3D多人避碰游戏，在该游戏中，多个追击者防御多个逃避者的目标区域。<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B12">Lee等人(2020b)</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B13">Lee等人，2021年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B11">Lee等人，2022a</gydF4y2Baa>)被认为是空中防御者和地面入侵者之间的游戏。</p><pcl作为年代＝"mb0">上述所有工作都专注于解决集中式外围防御问题，这些问题假设玩家对其他玩家的状态有全局了解。然而，当我们接触到大量玩家时，分散控制就变得必要了。为了解决这个问题，<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B34">Velhal等人(2022)</gydF4y2Baa>将周长防御博弈问题表述为一个凸形周长上的分散多机器人时空多任务分配问题。<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B27">保罗斯等人(2019)</gydF4y2Baa>提出的神经网络架构，用于在单位圆的周长上训练分散的代理策略，其中防御者具有简单的二进制动作空间。与上述作品不同的是，我们专注于高维的周长，专门为一个半球，具有连续的动作空间。我们用图神经网络学习分散策略来解决多智能体周界防御问题。</p><gydF4y2Bah3 class="pt0">2.2图神经网络</gydF4y2Bah3> <p class="mb0">我们利用图神经网络作为学习范式，因为它们具有去中心化架构的理想属性，可以捕捉相邻代理之间的相互作用和可转移性，从而可以推广到以前未见过的情况(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B6">Gama等人，2019年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B28">Ruiz等人，2021年</gydF4y2Baa>)．此外，gnn在各种多机器人问题上取得了巨大的成功，例如编队控制(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B33">托尔斯塔亚等人，2019年</gydF4y2Baa>)、路径规划(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B18">Li等，2021</gydF4y2Baa>)、任务分配(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B35">Wang和Gombolay, 2020年</gydF4y2Baa>)和多目标跟踪(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B40">Zhou等，2021</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B29">夏尔马等人，2022</gydF4y2Baa>)．特别是,<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B33">托尔斯塔亚等人(2019)</gydF4y2Baa>利用GNN模拟具有全局信息的集中式聚群控制器，学习移动机器人群的分散聚群行为。之后,<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B18">Li et al. (2021)</gydF4y2Baa>实现gnn，为多个机器人在障碍物丰富的环境中从起始位置到目标位置寻找无碰撞路径。他们证明了他们的去中心化路径规划器仅通过本地观察和邻近通信实现了接近专家的性能，这也可以推广到更大的机器人网络。基于gnn的方法还用于学习多机器人任务调度中的组合优化问题的解(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B35">Wang和Gombolay, 2020年</gydF4y2Baa>)和多目标追踪方案(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B40">Zhou等，2021</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B29">夏尔马等人，2022</gydF4y2Baa>)．</p><gydF4y2Baa id="h4" name="h4"></a> <h2>3问题的制定</gydF4y2Bah2> <h3 class="pt0">3.1的动力</gydF4y2Bah3> <p class="mb0">周长防御是一个相对较新的研究领域，最近已被探索。一个特别的挑战是，高维边界增加了防御者执行最佳策略的空间和算法复杂性。尽管之前的许多工作都考虑了平面游戏空间中的交战，并推导出2D运动中的最佳策略，但对于现实场景中周界防御游戏的实际应用来说，向高维空间的扩展是不可避免的。例如，防御者旨在保护的建筑物的周长可以被一个通用的形状包围，比如一个半球。由于防御者不能穿过建筑物，并且被认为随时都接近建筑物，因此他们被雇用沿着圆顶的表面移动，这导致了“半球外围防御游戏”。入侵者在半球的基本平面上移动，这意味着在移动过程中高度恒定。入侵者的移动被限制在2D范围内，因为假定入侵者可能希望保持低高度以躲避现实世界中的防御者。</p><pcl作为年代＝"mb0">值得注意的是，半球防御问题比允许两个代理在3D空间中自由移动的问题更具挑战性。在之前的作品中，防御者和入侵者都可以在三维空间中移动(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B37">Yan等，2022</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B38">严等，2019</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B39">严等，2020年</gydF4y2Baa>)．在所有情况下，即使在多机器人场景下，作者也能够明确地推导出最优解。尽管我们的问题将防御者的动力学限制在半球表面，但这些限制使得寻找最优解变得棘手和具有挑战性。</p><gydF4y2Bah3 class="pt0">3.2半球外围防守</gydF4y2Bah3> <p class="mb0">我们考虑一个半径为的半球形穹顶<gydF4y2Baem>R</gydF4y2Baem>周长。半球约束是为了防御者安全地在周边(例如，建筑)移动。在这个游戏中，考虑两组玩家:<我nline-formula id="inf1"> <math id="m1"> <mi mathvariant="bold"> D</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <msubsup> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msub> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mi> N</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>表示<gydF4y2Baem>N</gydF4y2Baem>后卫,<我nline-formula id="inf2"> <math id="m2"> <mi mathvariant="bold"> 一个</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <msubsup> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msub> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> j</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> j</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mi> N</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>表示<gydF4y2Baem>N</gydF4y2Baem>入侵者。一个后卫<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>被限制在穹顶表面活动而入侵者呢<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>被限制在地平面上运动。我们将去掉指标<gydF4y2Baem>我</gydF4y2Baem>而且<gydF4y2Baem>j</gydF4y2Baem>当他们无关紧要的时候。一个10对10周长防御的例子显示在<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F1">图1</gydF4y2Baa>．球员在球坐标系中的位置为:<gydF4y2Bab>z</gydF4y2Bab><sub><em>D</gydF4y2Baem></sub>= (<gydF4y2Baem>ψ</gydF4y2Baem><sub><em>D</gydF4y2Baem></sub>,ϕ<年代ub><em>D</gydF4y2Baem></sub>，<gydF4y2Baem>R</gydF4y2Baem>),<gydF4y2Bab>z</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>= (<gydF4y2Baem>ψ</gydF4y2Baem><sub><em>一个</gydF4y2Baem></sub>0,<gydF4y2Baem>r</gydF4y2Baem>),<gydF4y2Baem>ψ</gydF4y2Baem>而且<gydF4y2Baem>ϕ</gydF4y2Baem>为方位角和仰角，其中给出的相对位置为:<gydF4y2Bab>z</gydF4y2Bab>≜[<gydF4y2Baem>ψ</gydF4y2Baem>，<gydF4y2Baem>ϕ</gydF4y2Baem>，<gydF4y2Baem>r</gydF4y2Baem>),<gydF4y2Baem>ψ</gydF4y2Baem>≜<gydF4y2Baem>ψ</gydF4y2Baem><sub><em>一个</gydF4y2Baem></sub>−<gydF4y2Baem>ψ</gydF4y2Baem><sub><em>D</gydF4y2Baem></sub>而且<gydF4y2Baem>ϕ</gydF4y2Baem>≜ϕ<年代ub><em>D</gydF4y2Baem></sub>．玩家的位置也可以用笛卡尔坐标描述为:<gydF4y2Bab>x</gydF4y2Bab><sub><em>D</gydF4y2Baem></sub>而且<gydF4y2Bab>x</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>．所有代理都以单位速度移动，防御者通过缩小距离捕获入侵者<gydF4y2Baem>ϵ</gydF4y2Baem>，防御者和入侵者都在捕获过程中被消耗。如果入侵者到达边界(即，<gydF4y2Baem>r</gydF4y2Baem>（<gydF4y2Baem>t</gydF4y2Baem><sub><em>f</gydF4y2Baem></sub>) =<gydF4y2Baem>R</gydF4y2Baem>)在时间<gydF4y2Baem>t</gydF4y2Baem><sub><em>f</gydF4y2Baem></sub>没有被任何防御者捕获(即，<我nline-formula id="inf3"> <math id="m3"> <mo stretchy="false"> ‖</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <msub> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> </msub> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <mi> t</gydF4y2Bami> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mo> −</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <msub> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> j</gydF4y2Bami> </mrow> </msub> </mrow> </msub> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <mi> t</gydF4y2Bami> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mo stretchy="false"> ‖</gydF4y2Bamo> <mo> ></gydF4y2Bamo> <mi> ϵ</gydF4y2Bami> <mo> ，</gydF4y2Bamo> <mo> ∀</gydF4y2Bamo> <msub> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> j</gydF4y2Bami> </mrow> </msub> <mo> ∈</gydF4y2Bamo> <mi mathvariant="bold"> D</gydF4y2Bami> <mo> ，</gydF4y2Bamo> <mo> ∀</gydF4y2Bamo> <mi> t</gydF4y2Bami> <mo> <</gydF4y2Bamo> <msub> <mrow> <mi> t</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msub> </math> </inline-formula>)．防守者通过抓住入侵者或阻止入侵者无限期地得分(即，<gydF4y2Baem>ϕ</gydF4y2Baem>（<gydF4y2Baem>t</gydF4y2Baem>) =<gydF4y2Baem>ψ</gydF4y2Baem>（<gydF4y2Baem>t</gydF4y2Baem>) = 0，<gydF4y2Baem>r</gydF4y2Baem>（<gydF4y2Baem>t</gydF4y2Baem>) ><gydF4y2Baem>R</gydF4y2Baem>)．这项工作的主要兴趣是最大化防御者捕获的数量，给定一组初始配置。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图1</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g001.jpg" name="Figure1" target="_blank"><img id="F1" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g001.jpg"></a> <p><b>图1</gydF4y2Bab>．10对10外线防守实例。防御者被限制在穹顶的表面移动，而入侵者则被限制在地面上移动。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <h3 class="pt0">3.3最佳突破点</gydF4y2Bah3> <p class="mb0">鉴于<gydF4y2Bab>z</gydF4y2Bab><sub><em>D</gydF4y2Baem></sub>，<gydF4y2Bab>z</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>，我们称之为<gydF4y2Baem>breachingpoint</gydF4y2Baem>作为入侵者试图到达目标的周长上的一个点，如图所示<gydF4y2Baem>B</gydF4y2Baem>在<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F2">图2</gydF4y2Baa>．我们称形成破口点的方位角为<gydF4y2Baem>违反角</gydF4y2Baem>，表示为<gydF4y2Baem>θ</gydF4y2Baem>的夹角为(<gydF4y2Bab>z</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>−<gydF4y2Bab>z</gydF4y2Bab><sub><em>B</gydF4y2Baem></sub>)，切线为<gydF4y2Baem>B</gydF4y2Baem>作为<gydF4y2Baem>接近角</gydF4y2Baem>，表示为<gydF4y2Baem>β</gydF4y2Baem>．它是在(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B12">Lee等人，2020b</gydF4y2Baa>)，考虑到防守方目前的位置<gydF4y2Bab>z</gydF4y2Bab><sub><em>D</gydF4y2Baem></sub>和入侵者<gydF4y2Bab>z</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>作为点粒子，存在一个唯一的突破点，使得防御者和入侵者的最佳策略都是向它移动，称为<gydF4y2Baem>最佳突破点</gydF4y2Baem>．最优破口点所对应的破口角和接近角称为<gydF4y2Baem>最佳突破角度</gydF4y2Baem>，表示为<gydF4y2Baem>θ</gydF4y2Baem>*,<gydF4y2Baem>最佳进场角</gydF4y2Baem>，表示为<gydF4y2Baem>β</gydF4y2Baem>*。如载于(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B12">Lee等人，2020b</gydF4y2Baa>)，虽然没有封闭形式的解<gydF4y2Baem>θ</gydF4y2Baem>*和<gydF4y2Baem>β</gydF4y2Baem>*，它们可以通过求解两个控制方程随时计算:</p><gydF4y2Badiv class="equationImageholder"> <math id="e1"> <msup> <mrow> <mi> β</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> <mo> ＝</gydF4y2Bamo> <msup> <mrow> <mi> 因为</gydF4y2Bami> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <mi> ν</gydF4y2Bami> <mfrac> <mrow> <mi> 因为</gydF4y2Bami> <msub> <mrow> <mi> ϕ</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msub> <mo> ⁡</gydF4y2Bamo> <mi> 罪</gydF4y2Bami> <msup> <mrow> <mi> θ</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> </mrow> <mrow> <msqrt> <mrow> <mn> 1</gydF4y2Bamn> <mo> −</gydF4y2Bamo> <msup> <mrow> <mi> 因为</gydF4y2Bami> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </msup> <msub> <mrow> <mi> ϕ</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msub> <mo> ⁡</gydF4y2Bamo> <msup> <mrow> <mi> 因为</gydF4y2Bami> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </msup> <msup> <mrow> <mi> θ</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="indent"></p> <div class="DottedLine"></div> <div class="Imageheaders"> 图2</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g002.jpg" name="Figure2" target="_blank"><img id="F2" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g002.jpg"></a> <p><b>图2</gydF4y2Bab>．1对1半球防御游戏中的坐标和相关变量。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb0">而且</p><gydF4y2Badiv class="equationImageholder"> <math id="e2"> <msup> <mrow> <mi> θ</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> <mo> ＝</gydF4y2Bamo> <mi> ψ</gydF4y2Bami> <mo> −</gydF4y2Bamo> <msup> <mrow> <mi> β</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> <mo> +</gydF4y2Bamo> <msup> <mrow> <mi> 因为</gydF4y2Bami> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <mfrac> <mrow> <mi> 因为</gydF4y2Bami> <msup> <mrow> <mi> β</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> </mrow> <mrow> <mi> r</gydF4y2Bami> </mrow> </mfrac> </mrow> </mfenced> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="indent"></p> <h3 class="pt0">3.4目标时间和收益函数</gydF4y2Bah3> <p class="mb0">我们称之为<gydF4y2Baem>目标时间</gydF4y2Baem>当时间到达<gydF4y2Baem>B</gydF4y2Baem>和定义<gydF4y2Baem>τ</gydF4y2Baem><sub><em>D</gydF4y2Baem></sub>（<gydF4y2Bab>z</gydF4y2Bab><sub><em>D</gydF4y2Baem></sub>，<gydF4y2Bab>z</gydF4y2Bab><sub><em>B</gydF4y2Baem></sub>)作为<gydF4y2Baem>防守者目标时间</gydF4y2Baem>，<gydF4y2Baem>τ</gydF4y2Baem><sub><em>一个</gydF4y2Baem></sub>（<gydF4y2Bab>z</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>，<gydF4y2Bab>z</gydF4y2Bab><sub><em>B</gydF4y2Baem></sub>)作为<gydF4y2Baem>入侵者目标时间</gydF4y2Baem>，以下为<gydF4y2Baem>回报</gydF4y2Baem>功能:</p><gydF4y2Badiv class="equationImageholder"> <math id="e3"> <mi> p</gydF4y2Bami> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> B</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mo> ＝</gydF4y2Bamo> <msub> <mrow> <mi> τ</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msub> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> B</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mo> −</gydF4y2Bamo> <msub> <mrow> <mi> τ</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msub> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> B</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 3.</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb0">防守者伸手<gydF4y2Baem>B</gydF4y2Baem>如果快<gydF4y2Baem>p</gydF4y2Baem>< 0则入侵者到达<gydF4y2Baem>B</gydF4y2Baem>如果快<gydF4y2Baem>p</gydF4y2Baem>> 0。因此，防守者的目标是最小化<gydF4y2Baem>p</gydF4y2Baem>而入侵者的目标是最大化它。</p><gydF4y2Bah3 class="pt0">3.5最优策略与纳什均衡</gydF4y2Bah3> <p class="mb0">它已在(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B12">Lee等人，2020b</gydF4y2Baa>)，即防御者和入侵者的最佳策略是在任何时候都以最大速度向最佳突破点移动。设Ω和Γ为连续<gydF4y2Baem>v</gydF4y2Baem><sub><em>D</gydF4y2Baem></sub>而且<gydF4y2Baem>v</gydF4y2Baem><sub><em>一个</gydF4y2Baem></sub>这导致了<gydF4y2Baem>B</gydF4y2Baem>这<gydF4y2Baem>τ</gydF4y2Baem><sub><em>D</gydF4y2Baem></sub>（<gydF4y2Bab>z</gydF4y2Bab><sub><em>D</gydF4y2Baem></sub>， Ω)≜<gydF4y2Baem>τ</gydF4y2Baem><sub><em>D</gydF4y2Baem></sub>（<gydF4y2Bab>z</gydF4y2Bab><sub><em>D</gydF4y2Baem></sub>，<gydF4y2Bab>z</gydF4y2Bab><sub><em>B</gydF4y2Baem></sub>),<gydF4y2Baem>τ</gydF4y2Baem><sub><em>一个</gydF4y2Baem></sub>（<gydF4y2Bab>z</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>， Γ)≜<gydF4y2Baem>τ</gydF4y2Baem><sub><em>一个</gydF4y2Baem></sub>（<gydF4y2Bab>z</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>，<gydF4y2Bab>z</gydF4y2Bab><sub><em>B</gydF4y2Baem></sub>)，设Ω*和Γ*为最小化的最优策略<gydF4y2Baem>τ</gydF4y2Baem><sub><em>D</gydF4y2Baem></sub>（<gydF4y2Bab>z</gydF4y2Bab><sub><em>D</gydF4y2Baem></sub>， Ω)及<gydF4y2Baem>τ</gydF4y2Baem><sub><em>一个</gydF4y2Baem></sub>（<gydF4y2Bab>z</gydF4y2Bab><sub><em>一个</gydF4y2Baem></sub>， Γ)，则博弈中的最优性以纳什均衡的形式给出:</p><gydF4y2Badiv class="equationImageholder"> <math id="e4"> <mi> p</gydF4y2Bami> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="normal"> Ω</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> <mo> ，</gydF4y2Bamo> <mi mathvariant="normal"> Γ</gydF4y2Bami> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mo> ≤</gydF4y2Bamo> <mi> p</gydF4y2Bami> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="normal"> Ω</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> <mo> ，</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="normal"> Γ</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mo> ≤</gydF4y2Bamo> <mi> p</gydF4y2Bami> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <mi mathvariant="normal"> Ω</gydF4y2Bami> <mo> ，</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="normal"> Γ</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 4</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="indent"></p> <h3 class="pt0">3.6问题定义</gydF4y2Bah3> <p class="mb0">最大化捕获的数量<gydF4y2Baem>N</gydF4y2Baem>vs。<gydF4y2Baem>N</gydF4y2Baem>防守，我们首先回顾1对1外围防守游戏的动态。它已在(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B12">Lee等人，2020b</gydF4y2Baa>)，在一对一的比赛中，防守者的最佳动作是向球门移动<gydF4y2Baem>最佳突破点</gydF4y2Baem>(定义见章节3.3)。防御者比入侵者更快地到达最佳突破点<gydF4y2Baem>回报p</gydF4y2Baem>(在3.4节中定义)为负值，则入侵者到达更快<gydF4y2Baem>p</gydF4y2Baem>> 0。由此，我们推断最大化捕获的数量<gydF4y2Baem>N</gydF4y2Baem>vs。<gydF4y2Baem>N</gydF4y2Baem>防御等同于在防御者和入侵者之间找到匹配，从而使分配的负收益对的数量最大化。在最优匹配中，负收益的数量在整个游戏中保持不变，因为防守者-入侵者对在每个游戏中的最优性是a<gydF4y2Baem>纳什均衡</gydF4y2Baem>(见第3.5节)。</p><pcl作为年代＝"mb0">专家分配政策是一个<gydF4y2Baem>最大匹配</gydF4y2Baem>（<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B4">Chen et al.， 2014</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B31">Shishika和Kumar, 2018年</gydF4y2Baa>)．为了执行这个算法，我们生成一个二部图<gydF4y2Bab>D</gydF4y2Bab>而且<gydF4y2Bab>一个</gydF4y2Bab>作为两组节点(即，<我nline-formula id="inf4"> <math id="m8"> <mi mathvariant="script"> V</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <mn> 1、2</gydF4y2Bamn> <mo> ，</gydF4y2Bamo> <mo> ．</gydF4y2Bamo> <mo> ．</gydF4y2Bamo> <mo> ，</gydF4y2Bamo> <mi> N</gydF4y2Bami> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </math> </inline-formula>)，并将防御者与入侵者之间的潜在分配定义为边。对于每个防御者/节点<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>在<gydF4y2Bab>D</gydF4y2Bab>，我们找到所有的入侵者/节点<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>在<gydF4y2Bab>一个</gydF4y2Bab>防御方是明智的，并计算出相应的收益<gydF4y2Baem>p</gydF4y2Baem><sub><em>ij</gydF4y2Baem></sub>对于所有的对。我们说<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>是<gydF4y2Baem>强烈的分配</gydF4y2Baem>来<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>如果<gydF4y2Baem>p</gydF4y2Baem><sub><em>ij</gydF4y2Baem></sub>< 0。使用边集<我nline-formula id="inf5"> <math id="m9"> <mi mathvariant="script"> E</gydF4y2Bami> </math> </inline-formula>通过最大匹配，我们可以最大化强赋值对的数量。为了唯一性，我们选择一个匹配最小化<gydF4y2Baem>游戏的价值</gydF4y2Baem>，定义为</p><gydF4y2Badiv class="equationImageholder"> <math id="e5"> <mi> V</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <munder> <mrow> <mo> ∑</gydF4y2Bamo> </mrow> <mrow> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> j</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> <mo> ∈</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="script"> E</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> </mrow> </munder> <msub> <mrow> <mi> p</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mi> j</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 5</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf6"> <math id="m11"> <msup> <mrow> <mi mathvariant="script"> E</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> </math> </inline-formula>的子集<我nline-formula id="inf7"> <math id="m12"> <mi mathvariant="script"> E</gydF4y2Bami> </math> </inline-formula>负收益(即，<我nline-formula id="inf8"> <math id="m13"> <msup> <mrow> <mi mathvariant="script"> E</gydF4y2Bami> </mrow> <mrow> <mo> ＊</gydF4y2Bamo> </mrow> </msup> <mo> ＝</gydF4y2Bamo> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <msub> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> j</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> <mo> ∈</gydF4y2Bamo> <mi mathvariant="script"> E</gydF4y2Bami> <mo stretchy="false"> |</gydF4y2Bamo> <msub> <mrow> <mi> p</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mi> j</gydF4y2Bami> </mrow> </msub> <mo> <</gydF4y2Bamo> <mn> 0</gydF4y2Bamn> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </math> </inline-formula>)．这种独特的分配可确保尽可能早地最大化捕获的数量。然而，随着团队规模的增加，使用最大匹配算法运行穷举搜索的成本会非常高。这种方法在本质上是组合的，并假设集中的信息具有充分的可观察性。相反，我们的目标是找到利用当地认知的分散策略<我nline-formula id="inf9"> <math id="m14"> <msub> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mi mathvariant="script"> V</gydF4y2Bami> </mrow> </msub> </math> </inline-formula>(见章节4.1)。为此，我们将本文的主要问题形式化如下。</p><pcl作为年代＝"mb0">问题1(图神经网络的分散周长防御)。<gydF4y2Baem>设计一个基于gnn的学习框架来学习映射</gydF4y2Baem> <inline-formula id="inf10"> <math id="m15"> <mi mathvariant="script"> 米</gydF4y2Bami> </math> </inline-formula><em>从防御者在当地的认知</gydF4y2Baem> <inline-formula id="inf11"> <math id="m16"> <msub> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mi mathvariant="script"> V</gydF4y2Bami> </mrow> </msub> </math> </inline-formula><em>以及他们的交流图</gydF4y2Baem> <inline-formula id="inf12"> <math id="m17"> <mi mathvariant="script"> G</gydF4y2Bami> </math> </inline-formula><em>他们的行为</gydF4y2Baem> <inline-formula id="inf13"> <math id="m18"> <mi mathvariant="script"> U</gydF4y2Bami> </math> </inline-formula><em>,也就是说,</gydF4y2Baem> <inline-formula id="inf14"> <math id="m19"> <mi mathvariant="script"> U</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mi mathvariant="script"> 米</gydF4y2Bami> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mi mathvariant="script"> V</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <mi mathvariant="script"> G</gydF4y2Bami> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> </math> </inline-formula><em>，以致于</gydF4y2Baem> <inline-formula id="inf15"> <math id="m20"> <mi mathvariant="script"> U</gydF4y2Bami> </math> </inline-formula><em>是否尽可能接近行动设定</gydF4y2Baem> <inline-formula id="inf16"> <math id="m21"> <msup> <mrow> <mi mathvariant="script"> U</gydF4y2Bami> </mrow> <mrow> <mi mathvariant="monospace"> g</gydF4y2Bami> </mrow> </msup> </math> </inline-formula><em>由集中专家算法选择。</gydF4y2Baem></p> <p class="mb0">在下一节中，我们将详细描述解决问题1的学习架构。</p><gydF4y2Baa id="h5" name="h5"></a> <h2>4种方法</gydF4y2Bah2> <p class="mb15">在本文中，我们使用图神经网络学习外围防御的分散策略。该方法的推理是实时的，可扩展到大量的代理。我们使用专家分配政策来训练一个通过沟通渠道分享信息的防守团队。在第4.1节中，我们介绍了用于处理输入到GNN的特征的感知模块。通过GNN学习去中心化算法和为防御者规划候选匹配将在4.2节中讨论。防守队员的控制详见4.3节，训练过程详见4.4节。整体框架显示在<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F3">图3</gydF4y2Baa>．对于体系结构的选择，我们将控制模块从学习框架中解耦，因为直接学习动作是不必要的。学习代理之间的分配就足够了，并且可以通过最优策略计算出最佳操作(第3.5节)。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图3</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g003.jpg" name="Figure3" target="_blank"><img id="F3" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g003.jpg"></a> <p><b>图3</gydF4y2Bab>．总体框架。感知模块收集本地信息。学习与计划模块通过GNN处理收集到的信息<gydF4y2Baem>K</gydF4y2Baem>-hop邻居通信。控制模块计算出最优策略，并执行控制器实现闭环。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <h3 class="pt0">4.1感知</gydF4y2Bah3> <p class="mb0">在本节中，我们假设<gydF4y2Baem>N</gydF4y2Baem>防空员和<gydF4y2Baem>N</gydF4y2Baem>地面入侵者。每一个后卫<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>是否配备了传感器，并面向外围的视野<gydF4y2Baem>视场</gydF4y2Baem>．防守者的水平视野<gydF4y2Baem>视场</gydF4y2Baem>被选为<gydF4y2Baem>π</gydF4y2Baem>假设有一个鱼眼式相机。</p><gydF4y2Bah4>4.1.1入侵者特性</gydF4y2Bah4> <p class="mb0">为每一个<gydF4y2Baem>我</gydF4y2Baem>，防御者观察这群入侵者<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>，和两者在球坐标中的相对位置<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>而且<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>表示为<我nline-formula id="inf17"> <math id="m22"> <msubsup> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msubsup> <mo> ＝</gydF4y2Bamo> <msub> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msubsup> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mi> j</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msubsup> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> j</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </mrow> </msub> </math> </inline-formula>在哪里<我nline-formula id="inf18"> <math id="m23"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>是入侵者特性的数量。输入特征的数量<我nline-formula id="inf19"> <math id="m24"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>而且<我nline-formula id="inf20"> <math id="m25"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>分别选取固定数目的最近被检测agent和相邻agent。尽管防御者可以在传感范围内检测到任意数量的入侵者，但要选择固定数量的检测，以使系统具有可伸缩性。在权力分散的环境中，防御者应该能够根据当地的看法来决定其行动。我们在实验中选择固定的数字为10，因为专家算法(即最大匹配)总是会在10个最近的入侵者中为机器人分配一个防御者。</p><gydF4y2Bah4>4.1.2 Defender特性</gydF4y2Bah4> <p class="mb0">为了使系统可扩展，我们与固定数量的最接近的防御者建立通信。每一个后卫<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>与附近的防守者沟通<gydF4y2Baem>D</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>在其通讯范围内<gydF4y2Baem>r</gydF4y2Baem><sub><em>c</gydF4y2Baem></sub>．为每一个<gydF4y2Baem>我</gydF4y2Baem>，之间的相对位置<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>而且<gydF4y2Baem>D</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>表示为<我nline-formula id="inf21"> <math id="m26"> <msubsup> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msubsup> <mo> ＝</gydF4y2Bamo> <msub> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msubsup> <mrow> <mi mathvariant="bold"> z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mi> j</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msubsup> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> j</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </mrow> </msub> </math> </inline-formula>在哪里<我nline-formula id="inf22"> <math id="m27"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>是防御者特征的数量。选择的数字是3，因为与许多其他机器人通信可以让每个防御者都拥有环境的全部信息(即集中)，如果我们假设机器人分散，3是机器人在每个方向上收集信息的最小数字。如果检测到的入侵者少于10个，或者相邻的防御者少于3个，我们就会交出虚拟值来填充感知输入矩阵。保持输入特征不变是很重要的，因为神经网络无法处理变化的特征大小。</p><gydF4y2Bah4>4.1.3特征提取</gydF4y2Bah4> <p class="mb0">特征提取是通过将观察到的入侵者和通信的防御者的相对位置连接起来，形成局部感知<我nline-formula id="inf23"> <math id="m28"> <msub> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> <mo> ＝</gydF4y2Bamo> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msubsup> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> </msubsup> <mo> ，</gydF4y2Bamo> <msubsup> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> </msubsup> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </math> </inline-formula>．将提取的特征输入多层感知器(MLP)，生成后处理的特征向量<gydF4y2Bab>x</gydF4y2Bab><sub><em>我</gydF4y2Baem></sub>，这些信息将通过通信在邻居之间交换。</p><gydF4y2Bah3 class="pt0">4.2学习与规划</gydF4y2Bah3> <p class="mb0">我们使用图神经网络<gydF4y2Baem>K</gydF4y2Baem>跳的通信。防御者与邻近的机器人交流他们感知到的特征。沟通图<我nline-formula id="inf24"> <math id="m29"> <mi mathvariant="script"> G</gydF4y2Bami> </math> </inline-formula>是通过在通讯范围内连接附近的防守者形成的吗<gydF4y2Baem>r</gydF4y2Baem><sub><em>c</gydF4y2Baem></sub>，得到邻接矩阵<gydF4y2Bab>年代</gydF4y2Bab>给出了图神经网络。</p><gydF4y2Bah4>4.2.1图移位操作</gydF4y2Bah4> <p class="mb0">我们认为每个防守者<我nline-formula id="inf25"> <math id="m30"> <mi> 我</gydF4y2Bami> <mo> ，</gydF4y2Bamo> <mi> 我</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mi mathvariant="script"> V</gydF4y2Bami> </math> </inline-formula>有一个特征向量<我nline-formula id="inf26"> <math id="m31"> <msub> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> <mo> ∈</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="double-struck"> R</gydF4y2Bami> </mrow> <mrow> <mi> F</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>，表示来自的后处理信息<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>．通过收集特征向量<gydF4y2Bab>x</gydF4y2Bab><sub><em>我</gydF4y2Baem></sub>从所有防守队员中，我们得到了防守队员的特征矩阵<gydF4y2Bab>D</gydF4y2Bab>为:</p><gydF4y2Badiv class="equationImageholder"> <math id="e6"> <mi mathvariant="bold"> X</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mfenced open="[" close="]"> <mrow> <mtable class="matrix"> <mtr> <mtd columnalign="center"> <msubsup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mi mathvariant="sans-serif"> T</gydF4y2Bami> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd columnalign="center"> <mo> ⋮</gydF4y2Bamo> </mtd> </mtr> <mtr> <mtd columnalign="center"> <msubsup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi mathvariant="sans-serif"> T</gydF4y2Bami> </mrow> </msubsup> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo> ＝</gydF4y2Bamo> <mfenced open="[" close="]"> <mrow> <msup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mo> ，</gydF4y2Bamo> <mo> .．.</gydF4y2Bamo> <mo> ，</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> F</gydF4y2Bami> </mrow> </msup> </mrow> </mfenced> <mo> ∈</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="double-struck"> R</gydF4y2Bami> </mrow> <mrow> <mi> N</gydF4y2Bami> <mo> ×</gydF4y2Bamo> <mi> F</gydF4y2Bami> </mrow> </msup> <mo> ，</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 6</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf27"> <math id="m33"> <msup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msup> <mo> ∈</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="double-struck"> R</gydF4y2Bami> </mrow> <mrow> <mi> N</gydF4y2Bami> </mrow> </msup> <mo> ，</gydF4y2Bamo> <mi> f</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mrow> <mo stretchy="false"> ［</gydF4y2Bamo> <mrow> <mn> 1</gydF4y2Bamn> <mo> ，</gydF4y2Bamo> <mo> .．.</gydF4y2Bamo> <mo> ，</gydF4y2Bamo> <mi> F</gydF4y2Bami> </mrow> <mo stretchy="false"> ］</gydF4y2Bamo> </mrow> </math> </inline-formula>是集合的特征吗<gydF4y2Baem>f</gydF4y2Baem>越过所有的防守者;也就是说,<我nline-formula id="inf28"> <math id="m34"> <msup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msup> <mo> ＝</gydF4y2Bamo> <msup> <mrow> <mrow> <mo stretchy="false"> ［</gydF4y2Bamo> <mrow> <msubsup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> <mo> ，</gydF4y2Bamo> <mo> .．.</gydF4y2Bamo> <mo> ，</gydF4y2Bamo> <msubsup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </mrow> <mo stretchy="false"> ］</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi mathvariant="sans-serif"> T</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>与<我nline-formula id="inf29"> <math id="m35"> <msubsup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>表示特征<gydF4y2Baem>f</gydF4y2Baem>的<我nline-formula id="inf30"> <math id="m36"> <msub> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <mi> 我</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mi mathvariant="script"> V</gydF4y2Bami> </math> </inline-formula>．我们的行为<gydF4y2Baem>图移操作</gydF4y2Baem>为每一个<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>通过其相邻特征的线性组合，即，<我nline-formula id="inf31"> <math id="m37"> <msub> <mrow> <mo movablelimits="false" form="prefix"> ∑</gydF4y2Bamo> </mrow> <mrow> <mi> j</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="script"> N</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> </msub> <msub> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> j</gydF4y2Bami> </mrow> </msub> </math> </inline-formula>．因此，对于所有的防守者<gydF4y2Bab>D</gydF4y2Bab>与图<我nline-formula id="inf32"> <math id="m38"> <mi mathvariant="script"> G</gydF4y2Bami> </math> </inline-formula>，特征矩阵<gydF4y2Bab>X</gydF4y2Bab>移位操作后变成<gydF4y2Bab>SX</gydF4y2Bab>:</p><gydF4y2Badiv class="equationImageholder"> <math id="e7"> <msub> <mrow> <mfenced open="[" close="]"> <mrow> <mi mathvariant="bold"> 年代</gydF4y2Bami> <mi mathvariant="bold"> X</gydF4y2Bami> </mrow> </mfenced> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mi> f</gydF4y2Bami> </mrow> </msub> <mo> ＝</gydF4y2Bamo> <munderover accentunder="false" accent="true"> <mrow> <mo> ∑</gydF4y2Bamo> </mrow> <mrow> <mi> j</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mi> N</gydF4y2Bami> </mrow> </munderover> <msub> <mrow> <mfenced open="[" close="]"> <mrow> <mi mathvariant="bold"> 年代</gydF4y2Bami> </mrow> </mfenced> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mi> j</gydF4y2Bami> </mrow> </msub> <msubsup> <mrow> <mfenced open="[" close="]"> <mrow> <mi mathvariant="bold"> X</gydF4y2Bami> </mrow> </mfenced> </mrow> <mrow> <mi> j</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> <mo> ＝</gydF4y2Bamo> <munder> <mrow> <mo> ∑</gydF4y2Bamo> </mrow> <mrow> <mi> j</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="script"> N</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> </munder> <msub> <mrow> <mi> 年代</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mi> j</gydF4y2Bami> </mrow> </msub> <msubsup> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> <mrow> <mi> j</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> <mo> ，</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 7</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="indent">这里是邻接矩阵<gydF4y2Bab>年代</gydF4y2Bab>叫做<gydF4y2Baem>图移位算子</gydF4y2Baem>(GSO) (<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B6">Gama等人，2019年</gydF4y2Baa>)．</p><gydF4y2Bah4>4.2.2图卷积</gydF4y2Bah4> <p class="mb0">使用shift运算，我们定义<gydF4y2Baem>图卷积</gydF4y2Baem>通过线性组合<gydF4y2Baem>转移特性</gydF4y2Baem>在图<我nline-formula id="inf33"> <math id="m40"> <mi mathvariant="script"> G</gydF4y2Bami> </math> </inline-formula><em>通过K</gydF4y2Baem>-跳通信交换(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B6">Gama等人，2019年</gydF4y2Baa>；<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B17">李等，2020年</gydF4y2Baa>)：</p><gydF4y2Badiv class="equationImageholder"> <math id="e8"> <mi mathvariant="script"> H</gydF4y2Bami> <mfenced open="(" close=")"> <mrow> <mi mathvariant="bold"> X</gydF4y2Bami> <mo> ；</gydF4y2Bamo> <mi mathvariant="bold"> 年代</gydF4y2Bami> </mrow> </mfenced> <mo> ＝</gydF4y2Bamo> <munderover accentunder="false" accent="true"> <mrow> <mo> ∑</gydF4y2Bamo> </mrow> <mrow> <mi> k</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mn> 0</gydF4y2Bamn> </mrow> <mrow> <mi> K</gydF4y2Bami> </mrow> </munderover> <msup> <mrow> <mi mathvariant="bold"> 年代</gydF4y2Bami> </mrow> <mrow> <mi> k</gydF4y2Bami> </mrow> </msup> <mi mathvariant="bold"> X</gydF4y2Bami> <msub> <mrow> <mi mathvariant="bold"> H</gydF4y2Bami> </mrow> <mrow> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 8</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf34"> <math id="m42"> <msub> <mrow> <mi mathvariant="bold"> H</gydF4y2Bami> </mrow> <mrow> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ∈</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="double-struck"> R</gydF4y2Bami> </mrow> <mrow> <mi> F</gydF4y2Bami> <mo> ×</gydF4y2Bamo> <mi> G</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>表示组合的系数<gydF4y2Baem>F</gydF4y2Baem>位移特征矩阵中防御者的特征<gydF4y2Bab>年代</gydF4y2Bab><sup><em>k</gydF4y2Baem></sup><b>X</gydF4y2Bab>,<gydF4y2Baem>F</gydF4y2Baem>而且<gydF4y2Baem>G</gydF4y2Baem>表示图卷积的输入和输出维数。注意,<gydF4y2Bab>年代</gydF4y2Bab><sup><em>k</gydF4y2Baem></sup><b>X</gydF4y2Bab>＝<gydF4y2Bab>年代</gydF4y2Bab>（<gydF4y2Bab>年代</gydF4y2Bab><sup><em>k</gydF4y2Baem>−1</年代up><b>X</gydF4y2Bab>)的计算方法为<gydF4y2Baem>k</gydF4y2Baem>与1跳邻居的通信交流。</p><gydF4y2Bah4>4.2.3图神经网络</gydF4y2Bah4> <p class="mb0">应用逐点非线性<我nline-formula id="inf35"> <math id="m43"> <mi> σ</gydF4y2Bami> <mo> ：</gydF4y2Bamo> <mi mathvariant="double-struck"> R</gydF4y2Bami> <mo> →</gydF4y2Bamo> <mi mathvariant="double-struck"> R</gydF4y2Bami> </math> </inline-formula>作为图卷积的激活函数(Eq。<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#e8">8</gydF4y2Baa>)，我们定义<gydF4y2Baem>图感知</gydF4y2Baem>为:</p><gydF4y2Badiv class="equationImageholder"> <math id="e9"> <mi mathvariant="script"> H</gydF4y2Bami> <mfenced open="(" close=")"> <mrow> <mi mathvariant="bold"> X</gydF4y2Bami> <mo> ；</gydF4y2Bamo> <mi mathvariant="bold"> 年代</gydF4y2Bami> </mrow> </mfenced> <mo> ＝</gydF4y2Bamo> <mi> σ</gydF4y2Bami> <mfenced open="(" close=")"> <mrow> <munderover accentunder="false" accent="true"> <mrow> <mo> ∑</gydF4y2Bamo> </mrow> <mrow> <mi> k</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mn> 0</gydF4y2Bamn> </mrow> <mrow> <mi> K</gydF4y2Bami> </mrow> </munderover> <msup> <mrow> <mi mathvariant="bold"> 年代</gydF4y2Bami> </mrow> <mrow> <mi> k</gydF4y2Bami> </mrow> </msup> <mi mathvariant="bold"> X</gydF4y2Bami> <msub> <mrow> <mi mathvariant="bold"> H</gydF4y2Bami> </mrow> <mrow> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> <mo> ．</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 9</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb0">然后，通过级联定义GNN模块<gydF4y2Baem>l</gydF4y2Baem>层的图形感知(Eq。<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#e9">9</gydF4y2Baa>)：</p><gydF4y2Badiv class="equationImageholder"> <math id="e10"> <msup> <mrow> <mi mathvariant="bold"> X</gydF4y2Bami> </mrow> <mrow> <mi> ℓ</gydF4y2Bami> </mrow> </msup> <mo> ＝</gydF4y2Bamo> <mi> σ</gydF4y2Bami> <mfenced open="[" close="]"> <mrow> <msup> <mrow> <mi mathvariant="script"> H</gydF4y2Bami> </mrow> <mrow> <mi> ℓ</gydF4y2Bami> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi mathvariant="bold"> X</gydF4y2Bami> </mrow> <mrow> <mi> ℓ</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mo> ；</gydF4y2Bamo> <mi mathvariant="bold"> 年代</gydF4y2Bami> </mrow> </mfenced> </mrow> </mfenced> <mspace width="1em"></mspace> <mtext> 为</gydF4y2Bamtext> <mspace width="1em"></mspace> <mi> ℓ</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> <mo> ，</gydF4y2Bamo> <mo> .．.</gydF4y2Bamo> <mo> ，</gydF4y2Bamo> <mi> l</gydF4y2Bami> <mo> ，</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> （</gydF4y2Bamo> <mn> 10</gydF4y2Bamn> <mo stretchy="false"> ）</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">上一层的输出特征在哪里<gydF4y2Baem>ℓ</gydF4y2Baem>−1<我nline-formula id="inf36"> <math id="m46"> <msup> <mrow> <mi mathvariant="bold"> X</gydF4y2Bami> </mrow> <mrow> <mi> ℓ</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mo> ∈</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="double-struck"> R</gydF4y2Bami> </mrow> <mrow> <mi> N</gydF4y2Bami> <mo> ×</gydF4y2Bamo> <msup> <mrow> <mi> F</gydF4y2Bami> </mrow> <mrow> <mi> ℓ</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> </mrow> </msup> </math> </inline-formula>，作为当前层的输入<gydF4y2Baem>ℓ</gydF4y2Baem>生成层的输出特征<gydF4y2Baem>l</gydF4y2Baem>，<gydF4y2Bab>X</gydF4y2Bab><sup><em>ℓ</gydF4y2Baem></sup>．回想一下，第一层的输入是<gydF4y2Bab>X</gydF4y2Bab><sup>0</年代up>＝<gydF4y2Bab>X</gydF4y2Bab>(Eq。<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#e6">6</gydF4y2Baa>)．最后一层的输出特征<我nline-formula id="inf37"> <math id="m47"> <msup> <mrow> <mi mathvariant="bold"> X</gydF4y2Bami> </mrow> <mrow> <mi> l</gydF4y2Bami> </mrow> </msup> <mo> ∈</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="double-struck"> R</gydF4y2Bami> </mrow> <mrow> <mi> N</gydF4y2Bami> <mo> ×</gydF4y2Bamo> <mi> G</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>,得到了<gydF4y2Baem>通过K</gydF4y2Baem>-hop通信，表示防御方交换和融合的信息<gydF4y2Bab>D</gydF4y2Bab>．</p><gydF4y2Bah4>4.2.4候选匹配</gydF4y2Bah4> <p class="mb0">GNN的输出，它表示融合的信息从<gydF4y2Baem>K</gydF4y2Baem>-hop通信，然后与另一个MLP进行处理，为每个防御者提供一个候选匹配。<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F3">图3</gydF4y2Baa>显示一个候选匹配实例if<我nline-formula id="inf38"> <math id="m48"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> <mo> ＝</gydF4y2Bamo> <mn> 6</gydF4y2Bamn> </math> </inline-formula>．给一个防守者<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>，我们发现<我nline-formula id="inf39"> <math id="m49"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>最近的入侵者，从1到编号<我nline-formula id="inf40"> <math id="m50"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>顺时针方向旋转。顺时针为附近的入侵者编号的主要原因是解释来自我们的网络的特征输出，以确定哪些入侵者将与哪些防御者匹配。我们可以逆时针或者任意顺序编号。由于每个防御者都学习去中心化策略，因此它需要根据其本地感知指定要捕获的入侵者。入侵者没有全局id，所以在不失去通用性的情况下，我们简单地顺时针分配id。多层感知器的输出是一个赋值似然<我nline-formula id="inf41"> <math id="m51"> <mi mathvariant="script"> l</gydF4y2Bami> </math> </inline-formula>的概率<我nline-formula id="inf42"> <math id="m52"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>入侵者候选人与给定防御者匹配的可能性。例如，专家分配可能性<我nline-formula id="inf43"> <math id="m53"> <msubsup> <mrow> <mi> l</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> <mrow> <mi> g</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>为<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>在<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F3">图3</gydF4y2Baa>如果第三个入侵者(即，<gydF4y2Baem>一个</gydF4y2Baem><sub>3.</年代ub>)与<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>通过专家策略(即最大匹配)。计划模块选择入侵者候选<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>这样匹配的一对(<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>，<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>)将极有可能与专家政策相似。值得注意的是，我们的方法给出了一个去中心化的分配策略，只交换相邻的信息。</p><gydF4y2Bah4>4.2.5排列等价</gydF4y2Bah4> <p class="mb0">值得注意的是，由于排列等价，我们提出的基于gnn的学习方法是可扩展的。这意味着给定一个去中心化的防御者，它应该能够根据由任意数量的无编号入侵者组成的局部感知来决定行动。以一个周界防御游戏为例说明了这一性质<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F4">图4</gydF4y2Baa>．故事情节集中在一个防御者身上，随着时间的推移，入侵者逐渐接近边界。相同的入侵者在不同的时间戳上被涂成相同的颜色。注意一个新的浅蓝色入侵者进入防守者的视野<gydF4y2Baem>t</gydF4y2Baem>= 2时，一个紫色的入侵者开始出现<gydF4y2Baem>t</gydF4y2Baem>= 3。虽然每次检测到的入侵者数量是任意的，但我们的系统将入侵者的id显示为中蓝色数字<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F4">图4</gydF4y2Baa>．我们顺时针给它们编号，但在任何排列中(例如逆时针)都可以做不同的事情，因为图神经网络执行与标签无关的处理。编号的原因是指定哪些入侵者将与来自网络输出的哪些防御者匹配。在不丧失通用性的情况下，我们顺时针分配id，但我们注意到这些id是任意的，因为id可以在不同的戳记上更改。例如，黄色入侵者ID是2 at<gydF4y2Baem>t</gydF4y2Baem>= 1，但变成3<gydF4y2Baem>t</gydF4y2Baem>= 2,3。类似地，红色入侵者ID为3 at<gydF4y2Baem>t</gydF4y2Baem>= 1，但改为4<gydF4y2Baem>t</gydF4y2Baem>= 2和5在<gydF4y2Baem>t</gydF4y2Baem>= 3。这样，我们可以容纳任意数量的入侵者，因此我们的系统是置换等效的。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图4</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g004.jpg" name="Figure4" target="_blank"><img id="F4" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g004.jpg"></a> <p><b>图4</gydF4y2Bab>．不同时间点的外围防御游戏实例。情节集中在一个单一的防御者和当地的看法。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <h3 class="pt0">4.3控制</gydF4y2Bah3> <p class="mb0">第4.2节的输出被输入到中的防御策略模块<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F3">图3</gydF4y2Baa>．这个模块处理所有匹配的对(<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>，<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>)，并计算每个一对一半球外围防御游戏的最佳突破点(见章节3.3)。防御者策略模块集体输出位置指令，方向为最佳突破点方向。SO(3)命令(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B21">梅林格和库马尔，2011年</gydF4y2Baa>)，由推力和力矩组成，在低水平控制机器人，然后传递给防守队<gydF4y2Bab>D</gydF4y2Bab>为控制。防御者-入侵者对的状态动态在(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B12">Lee等人，2020b</gydF4y2Baa>)．防御者根据命令移动以关闭感知-动作循环。值得注意的是，专家分配可能性<我nline-formula id="inf44"> <math id="m54"> <msup> <mrow> <mi mathvariant="script"> l</gydF4y2Bami> </mrow> <mrow> <mi> g</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>会导致专家行动集吗<我nline-formula id="inf45"> <math id="m55"> <msup> <mrow> <mi mathvariant="script"> U</gydF4y2Bami> </mrow> <mrow> <mi> g</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>(在问题1中定义)。</p><gydF4y2Bah3 class="pt0">4.4培训流程</gydF4y2Bah3> <p class="mb0">为了训练我们提出的网络，我们使用模仿学习来模仿最大匹配给出的专家策略(在第3节中解释)，它提供了最优分配似然<我nline-formula id="inf46"> <math id="m56"> <msup> <mrow> <mi mathvariant="script"> l</gydF4y2Bami> </mrow> <mrow> <mi> g</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>(见第4.2节)<我nline-formula id="inf47"> <math id="m57"> <msub> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mi mathvariant="script"> V</gydF4y2Bami> </mrow> </msub> </math> </inline-formula>以及沟通图<我nline-formula id="inf48"> <math id="m58"> <mi mathvariant="script"> G</gydF4y2Bami> </math> </inline-formula>．训练集<我nline-formula id="inf49"> <math id="m59"> <mi mathvariant="script"> D</gydF4y2Bami> </math> </inline-formula>作为以下数据的集合生成:<我nline-formula id="inf50"> <math id="m60"> <mi mathvariant="script"> D</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <mrow> <mo stretchy="false"> （</gydF4y2Bamo> <mrow> <msub> <mrow> <mrow> <mo stretchy="false"> ｛</gydF4y2Bamo> <mrow> <msub> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </mrow> <mrow> <mi> 我</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mi mathvariant="script"> V</gydF4y2Bami> </mrow> </msub> <mo> ，</gydF4y2Bamo> <mi mathvariant="script"> G</gydF4y2Bami> <mo> ，</gydF4y2Bamo> <msup> <mrow> <mi mathvariant="script"> l</gydF4y2Bami> </mrow> <mrow> <mi> g</gydF4y2Bami> </mrow> </msup> </mrow> <mo stretchy="false"> ）</gydF4y2Bamo> </mrow> </mrow> <mo stretchy="false"> ｝</gydF4y2Bamo> </mrow> </math> </inline-formula>．我们训练映射<我nline-formula id="inf51"> <math id="m61"> <mi mathvariant="script"> 米</gydF4y2Bami> </math> </inline-formula>(在问题1中定义)来最小化之间的交叉熵损失<我nline-formula id="inf52"> <math id="m62"> <msup> <mrow> <mi mathvariant="script"> l</gydF4y2Bami> </mrow> <mrow> <mi> g</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>而且<我nline-formula id="inf53"> <math id="m63"> <mi mathvariant="script"> l</gydF4y2Bami> </math> </inline-formula>．我们证明受过训练的人<我nline-formula id="inf54"> <math id="m64"> <mi mathvariant="script"> 米</gydF4y2Bami> </math> </inline-formula>提供了<我nline-formula id="inf55"> <math id="m65"> <mi mathvariant="script"> U</gydF4y2Bami> </math> </inline-formula>这很接近<我nline-formula id="inf56"> <math id="m66"> <msup> <mrow> <mi mathvariant="script"> U</gydF4y2Bami> </mrow> <mrow> <mi> g</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>．我们的网络中可学习参数的数量与团队规模的数量无关<gydF4y2Baem>N</gydF4y2Baem>．因此，我们可以在小范围内训练我们的网络，并将我们的模型推广到大范围，因为任何规模的防御者都可以根据固定数量的代理的局部感知来学习分散策略。</p><gydF4y2Bah4>4.4.1模型架构</gydF4y2Bah4> <p class="mb0">我们的模型架构由2层MLP组成，其中16层和8层隐藏层用于生成后处理的特征向量<gydF4y2Bab>x</gydF4y2Bab><sub><em>我</gydF4y2Baem></sub>2层GNN(包含32层和128层隐藏层)用于交换防御者收集的信息，以及单层MLP用于生成分配似然<我nline-formula id="inf57"> <math id="m67"> <mi mathvariant="script"> l</gydF4y2Bami> </math> </inline-formula>．在MLP和GNN的层之后是ReLU。</p><gydF4y2Bah4>4.4.2图神经网络细节</gydF4y2Bah4> <p class="mb0">在实现图神经网络时，我们通过连接通信范围内的防御者来构造一个1跳连接图<gydF4y2Baem>r</gydF4y2Baem><sub><em>c</gydF4y2Baem></sub>= 1。假设默认半径为<gydF4y2Baem>R</gydF4y2Baem>= 1时，我们可以预见1跳内的三个相邻agent将为防御者提供一个广泛的传感区域。因此，我们假设通信是实时发生的<我nline-formula id="inf58"> <math id="m68"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> <mo> ＝</gydF4y2Bamo> <mn> 3.</gydF4y2Bamn> </math> </inline-formula>．每个防御者收集信息作为输入特征，包括<我nline-formula id="inf59"> <math id="m69"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> <mo> ＝</gydF4y2Bamo> <mn> 10</gydF4y2Bamn> </math> </inline-formula>最近的入侵者位置和<我nline-formula id="inf60"> <math id="m70"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> D</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> <mo> ＝</gydF4y2Bamo> <mn> 3.</gydF4y2Bamn> </math> </inline-formula>最近的防守位置。所使用的参数总结在<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#T1">表1</gydF4y2Baa>．</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 表1</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-t001.jpg" name="Table1" target="_blank"><img id="T1" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-t001.jpg"></a> <p><b>表1</gydF4y2Bab>．图神经网络实现中的参数设置。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <h4>4.4.3实现细节</gydF4y2Bah4> <p class="mb0">实验使用12核3.50 GHz i9-9920X CPU和Nvidia GeForce RTX 2080 Ti GPU进行。我们使用PyTorch v1.10.1 (<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B26">Paszke等人，2019年</gydF4y2Baa>)加速Cuda v10.2 api。我们使用动量为0.5的亚当优化器。学习率计划从5 × 10衰减<年代up>3</年代up>到10<年代up>6</年代up>在1500个周期内，批大小为64，使用余弦退火。我们选择这些超参数以获得最佳性能。</p><gydF4y2Baa id="h6" name="h6"></a> <h2>5实验</gydF4y2Bah2> <h3 class="pt0">5.1数据集</gydF4y2Bah3> <p class="mb0">我们使用模仿学习来评估我们的分散网络，其中专家分配策略是最大匹配。周长是一个有半径的半球<gydF4y2Baem>R</gydF4y2Baem>定义为<我nline-formula id="inf63"> <math id="m73"> <mi> R</gydF4y2Bami> <mo> ＝</gydF4y2Bamo> <msqrt> <mrow> <mi> N</gydF4y2Bami> <mo> /</gydF4y2Bamo> <msub> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi mathvariant="italic"> def</gydF4y2Bami> </mrow> </msub> </mrow> </msqrt> </math> </inline-formula>在哪里<gydF4y2Baem>N</gydF4y2Baem>是团队规模和<gydF4y2Baem>N</gydF4y2Baem><sub><em>def</gydF4y2Baem></sub>是默认的团队大小。由于运行最大匹配在大规模(例如，<gydF4y2Baem>N</gydF4y2Baem>> 10)，我们设置默认的团队规模<gydF4y2Baem>N</gydF4y2Baem><sub><em>def</gydF4y2Baem></sub>= 10。这样，<gydF4y2Baem>R</gydF4y2Baem>也代表了游戏的规模;例如，当<gydF4y2Baem>N</gydF4y2Baem>= 40,<gydF4y2Baem>R</gydF4y2Baem>变为2，这表明问题设置的规模比设置的规模翻了一番<gydF4y2Baem>R</gydF4y2Baem>= 1。考虑到团队规模<gydF4y2Baem>N</gydF4y2Baem>= 10时，实验场地尺寸为10 × 10 × 1 m。在线下，我们随机抽取了1000万个防御者的局部感知样本<我nline-formula id="inf64"> <math id="m74"> <msub> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </math> </inline-formula>找到相应的<我nline-formula id="inf65"> <math id="m75"> <mi mathvariant="script"> G</gydF4y2Bami> </math> </inline-formula>而且<我nline-formula id="inf66"> <math id="m76"> <msup> <mrow> <mi mathvariant="script"> l</gydF4y2Bami> </mrow> <mrow> <mi> g</gydF4y2Bami> </mrow> </msup> </math> </inline-formula>准备数据集，数据集分为训练集(60%)、验证集(20%)和测试集(20%)。</p><gydF4y2Bah3 class="pt0">5.2指标</gydF4y2Bah3> <p class="mb0">我们主要感兴趣的是被捕获的入侵者的百分比(即，捕获的数量/入侵者的总数)。在小范围内(例如，<gydF4y2Baem>N</gydF4y2Baem>≤10)，则可以运行专家策略(即最大匹配)，并且可以将专家策略与我们的策略进行直接比较。在大尺度上(例如，<gydF4y2Baem>N</gydF4y2Baem>> 10)，最大匹配太昂贵而无法运行。因此，我们将我们的算法与其他基线方法进行比较:<gydF4y2Baem>贪婪的</gydF4y2Baem>，<gydF4y2Baem>随机</gydF4y2Baem>,<gydF4y2Baem>中长期规划</gydF4y2Baem>，这将在第5.3节中解释。为了观察小尺度和大尺度上的可伸缩性，我们为每个尺度总共运行了五种不同的算法:<gydF4y2Baem>专家</gydF4y2Baem>，<gydF4y2Baem>gnn</gydF4y2Baem>，<gydF4y2Baem>贪婪的</gydF4y2Baem>，<gydF4y2Baem>随机</gydF4y2Baem>,<gydF4y2Baem>中长期规划</gydF4y2Baem>．在所有情况下，我们都计算<gydF4y2Baem>绝对精度</gydF4y2Baem>，它由捕获数量除以团队规模定义，以验证我们的网络是否可以推广到任何团队规模。此外，我们还计算了<gydF4y2Baem>比较准确</gydF4y2Baem>，定义为捕获次数的比值<gydF4y2Baem>gnn</gydF4y2Baem>以另一种算法捕获的数量，观察比较结果。</p><gydF4y2Bah3 class="pt0">5.3比较算法</gydF4y2Bah3> <p class="mb0">在基线算法中，防御者不会传达他们的“意图”，即哪些入侵者将被哪些相邻的防御者捕获，以便进行公平的比较，因为GNN也不共享此类信息。对于GNN框架，每个防御者都能感知到附近的入侵者，而被感知到的入侵者的相对位置，而不是“意图”，由GNN通过通信共享。gnn的力量在于隐式地学习这些“意图”<gydF4y2Baem>通过</gydF4y2Baem>K-hop通信。这样，去中心化决策(即GNN和基线)可能允许多个防御者瞄准捕获同一个入侵者，而集中计划者知道所有防御者的“意图”，并将避免这样的场景。</p><gydF4y2Bah4>5.3.1贪婪</gydF4y2Bah4> <p class="mb0">贪婪算法可以在多项式时间内运行，因此成为与我们使用GNN的方法进行比较的一个很好的候选算法。为了公平的比较，我们运行了一个基于局部感知的去中心化贪婪算法<我nline-formula id="inf67"> <math id="m77"> <msub> <mrow> <mi mathvariant="script"> Z</gydF4y2Bami> </mrow> <mrow> <mi> 我</gydF4y2Bami> </mrow> </msub> </math> </inline-formula>的<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>．我们使<gydF4y2Baem>K</gydF4y2Baem>-hop邻居通信，使防御者的感知区域扩大，就像GNN的网络通道是活跃的一样。这名后卫<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>计算收益<gydF4y2Baem>p</gydF4y2Baem><sub><em>ij</gydF4y2Baem></sub>(见3.4节)基于任何明智的入侵者<gydF4y2Baem>一个</gydF4y2Baem><sub><em>j</gydF4y2Baem></sub>然后贪婪地选择一个收益最小的任务<gydF4y2Baem>p</gydF4y2Baem><sub><em>ij</gydF4y2Baem></sub>．</p><gydF4y2Bah4>5.3.2随机</gydF4y2Bah4> <p class="mb0">随机算法与贪婪算法的相似之处在于<gydF4y2Baem>K</gydF4y2Baem>为扩展感知启用-hop邻居通信。在明智的入侵者中，有防御者<gydF4y2Baem>D</gydF4y2Baem><sub><em>我</gydF4y2Baem></sub>随机挑选一个入侵者来决定任务。</p><gydF4y2Bah4>5.3.3延时</gydF4y2Bah4> <p class="mb0">对于MLP算法，我们仅通过排除GNN模块来隔离训练我们提出的框架的当前MLP。通过比较我们的GNN框架和这个算法，我们可以观察到GNN是否有任何改进。</p><gydF4y2Bah3 class="pt0">5.4结果</gydF4y2Bah3> <p class="mb0">我们在不同的场景下，以不同的团队规模和初始配置运行外围防御游戏，以评估学习网络的性能。特别地，我们在小型(<gydF4y2Baem>N</gydF4y2Baem>≤10)和大(<gydF4y2Baem>N</gydF4y2Baem>> 10)刻度。在顶视图中，我们针对不同团队规模提出的网络模拟外围防御游戏的快照如下所示<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F5">图5</gydF4y2Baa>．周长、防御者状态、入侵者状态和突破点分别用绿色、蓝色、红色和黄色标记。我们观察到入侵者试图到达外围。给定防御者-入侵者匹配，入侵者执行各自的最优策略，向最优突破点移动(见章节3.5)。如果入侵者成功到达而没有被任何防御者捕获，入侵者将被消耗并留下一个称为“入侵”的标记。如果入侵者失败并被防御者拦截，两个代理都被消耗，并留下一个称为“捕获”的标记。入侵者瞄准的周边点被标记为“突破点”。在所有比赛中，比赛结束于<gydF4y2Baem>终点时间T</gydF4y2Baem><sub><em>f</gydF4y2Baem></sub>当所有入侵者都被消耗掉时。查看补充视频了解更多结果。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图5</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g005.jpg" name="Figure5" target="_blank"><img id="F5" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g005.jpg"></a> <p><b>图5</gydF4y2Bab>．<gydF4y2Bab>(两者)</gydF4y2Bab>利用所提出的方法获得了模拟周界防御的俯视图快照<gydF4y2Baem>gnn</gydF4y2Baem>针对三种不同的团队规模。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb0">如5.1节所述，我们运行这五种算法<gydF4y2Baem>专家</gydF4y2Baem>，<gydF4y2Baem>gnn</gydF4y2Baem>，<gydF4y2Baem>贪婪的</gydF4y2Baem>，<gydF4y2Baem>随机</gydF4y2Baem>,<gydF4y2Baem>中长期规划</gydF4y2Baem>在小范围内，并运行<gydF4y2Baem>gnn</gydF4y2Baem>，<gydF4y2Baem>贪婪的</gydF4y2Baem>，<gydF4y2Baem>随机</gydF4y2Baem>,<gydF4y2Baem>中长期规划</gydF4y2Baem>在大范围内。以模拟20对20外线防守比赛为例，给出了终端时刻俯视图的快照<gydF4y2Baem>T</gydF4y2Baem><sub><em>f</gydF4y2Baem></sub>四种算法的使用方法显示在<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F6">图6</gydF4y2Baa>．四个子图(a)-(d)表明，尽管游戏在所有情况下初始配置相同，但这些算法表现出不同的性能。这些算法捕获的数量<gydF4y2Baem>gnn</gydF4y2Baem>，<gydF4y2Baem>贪婪的</gydF4y2Baem>，<gydF4y2Baem>随机</gydF4y2Baem>,<gydF4y2Baem>中长期规划</gydF4y2Baem>分别是12 11 10 7。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图6</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g006.jpg" name="Figure6" target="_blank"><img id="F6" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g006.jpg"></a> <p><b>图6</gydF4y2Bab>．<gydF4y2Bab>(模拟)</gydF4y2Bab>模拟20对20外线防守游戏的快照，在终端时间的俯视图<gydF4y2Baem>T</gydF4y2Baem><sub><em>f</gydF4y2Baem></sub>使用这四种算法<gydF4y2Baem>gnn</gydF4y2Baem>，<gydF4y2Baem>贪婪的</gydF4y2Baem>，<gydF4y2Baem>随机</gydF4y2Baem>,<gydF4y2Baem>中长期规划</gydF4y2Baem>．使用这些算法的捕获次数分别为12、11、10和7。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb0">中描述了每种方法捕获的入侵者百分比的总体结果<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F7">图7</gydF4y2Baa>．据观察，<gydF4y2Baem>gnn</gydF4y2Baem>在所有情况下都优于其他基线，并且性能接近<gydF4y2Baem>专家</gydF4y2Baem>在小尺度上。特别是，考虑到我们默认的团队规模<gydF4y2Baem>N</gydF4y2Baem><sub><em>def</gydF4y2Baem></sub>10，我们提出的算法的性能是否与专家策略的性能保持竞争<gydF4y2Baem>N</gydF4y2Baem>= 10。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图7</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g007.jpg" name="Figure7" target="_blank"><img id="F7" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g007.jpg"></a> <p><b>图7</gydF4y2Bab>．不同算法在小范围(<gydF4y2Baem>N</gydF4y2Baem>≤10)和大(<gydF4y2Baem>N</gydF4y2Baem>> 10)刻度。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb0">在大尺度上，捕获的百分比为<gydF4y2Baem>gnn</gydF4y2Baem>保持不变，这表明即使在小范围内进行训练，训练后的网络也能很好地推广到大范围。捕获的百分比<gydF4y2Baem>贪婪的</gydF4y2Baem>也似乎不变，但表现要差得多<gydF4y2Baem>gnn</gydF4y2Baem>随着团队规模的扩大。在小范围内，只有少数组合是可用的匹配防御者-入侵者对，因此<gydF4y2Baem>贪婪的</gydF4y2Baem>算法的表现与专家算法相似。随着代理数量的增加，可能匹配的数量呈指数增长，因此<gydF4y2Baem>贪婪的</gydF4y2Baem>由于问题的复杂度越来越高，算法的性能越来越差。的<gydF4y2Baem>随机</gydF4y2Baem>方法在小尺度上的性能比其他算法差<gydF4y2Baem>中长期规划</gydF4y2Baem>开始表现得比<gydF4y2Baem>随机</gydF4y2Baem>当团队规模超过40人时。这种趋势表明，仅使用MLP训练的策略无法大规模扩展。由于训练是用10个agent完成的，因此它是最优的<gydF4y2Baem>N</gydF4y2Baem>= 10，但是<gydF4y2Baem>中长期规划</gydF4y2Baem>不能在更大的范围内工作，甚至表现不如<gydF4y2Baem>随机</gydF4y2Baem>算法。结果表明，在MLP中加入GNN后，性能得到了显著提高。总的来说，与其他算法相比，<gydF4y2Baem>gnn</gydF4y2Baem>在大规模中比在小规模中表现得更好，这证实了GNN有助于网络变得可扩展。</p><pcl作为年代＝"mb0">为了定量评估所提出的方法，我们报告了<gydF4y2Baem>绝对精度</gydF4y2Baem>而且<gydF4y2Baem>比较准确</gydF4y2Baem>(定义见第5.2节)<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#T2">表2</gydF4y2Baa>而且<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#T3">表3</gydF4y2Baa>．正如预期的那样，当团队规模接近时，绝对精度达到最大值<gydF4y2Baem>N</gydF4y2Baem>= 10。除以下情况外，绝对精度的总体值基本一致<gydF4y2Baem>N</gydF4y2Baem>= 2。我们推测，两个防御者之间可能没有太多共享的信息，根据初始配置，可能根本没有可感知的入侵者。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 表2</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-t002.jpg" name="Table2" target="_blank"><img id="T2" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-t002.jpg"></a> <p><b>表2</gydF4y2Bab>．小尺度精度。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <div class="Imageheaders"> 表3</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-t003.jpg" name="Table3" target="_blank"><img id="T3" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-t003.jpg"></a> <p><b>表3</gydF4y2Bab>．大尺度精度。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <p class="mb0">两者之间的比较准确性<gydF4y2Baem>gnn</gydF4y2Baem>而且<gydF4y2Baem>专家</gydF4y2Baem>这表明我们训练过的政策更接近于专家政策<gydF4y2Baem>N</gydF4y2Baem>趋于10时，我们期望的性能为<gydF4y2Baem>gnn</gydF4y2Baem>接近:接近于…的<gydF4y2Baem>专家</gydF4y2Baem>即使是在大尺度上。两者之间的比较准确性<gydF4y2Baem>gnn</gydF4y2Baem>其他基线表明，我们训练过的网络在大规模上比基线算法表现得更好(<gydF4y2Baem>N</gydF4y2Baem>≥40)，平均可多捕获1.5倍。两者之间的比较准确性<gydF4y2Baem>gnn</gydF4y2Baem>而且<gydF4y2Baem>随机</gydF4y2Baem>由于随机性的性质，在整个团队规模中有点嘈杂，但我们观察到，我们的策略在小范围和大范围的平均捕获次数比随机策略多1.8倍。我们观察到<gydF4y2Baem>中长期规划</gydF4y2Baem>在大规模的情况下，性能比其他算法差得多。</p><pcl作为年代＝"mb0">基于比较，我们证明了我们提出的网络，在小范围内训练，可以推广到大范围。直觉上，人们可能会这样想<gydF4y2Baem>贪婪的</gydF4y2Baem>会在去中心化的环境中表现最好，因为每个防守者都尽力将风险最小化<gydF4y2Baem>游戏的价值</gydF4y2Baem>(定义于Eq.<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#e5">5</gydF4y2Baa>)．然而，我们可以推断<gydF4y2Baem>贪婪的</gydF4y2Baem>不知道附近防御者的意图(例如，要捕获哪些入侵者)，因此无法实现接近集中专家算法的性能。我们的方法实现了图神经网络来交换附近防守方的信息，这些信息感知到防守方的局部特征，从而规划防守方的最终行动;因此，附近的防御者可能移动的隐式信息被传输到每个相邻的防御者。由于集中式专家策略知道防御者的所有意图，我们基于gnn的策略通过通信渠道学习意图。防守团队之间的合作是我们的关键<gydF4y2Baem>gnn</gydF4y2Baem>表现<gydF4y2Baem>贪婪的</gydF4y2Baem>的方法。这些结果验证了实现的gnn对于我们的问题来说是理想的，它具有分散通信的属性，可以捕获相邻的交互，并且允许泛化到未见的场景。</p><gydF4y2Bah3 class="pt0">5.5进一步分析</gydF4y2Bah3> <h4>5.5.1性能<gydF4y2Baem>vs</gydF4y2Baem>．专家演示次数</gydF4y2Bah4> <p class="mb0">为了分析算法性能，我们用不同数量的专家演示(例如，1000万、100万、100万和10 k)训练了基于gnn的架构。在团队规模10≤的情况下，捕获的入侵者百分比(10次试验的平均值和标准差)<gydF4y2Baem>N</gydF4y2Baem>≤50表示在<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F8">图8</gydF4y2Baa>．该图验证了我们所提出的网络通过更多的演示可以更好地学习。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图8</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g008.jpg" name="Figure8" target="_blank"><img id="F8" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g008.jpg"></a> <p><b>图8</gydF4y2Bab>．不同数量专家演示的样本效率。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <h4>5.5.2性能<gydF4y2Baem>vs</gydF4y2Baem>．周边半径</gydF4y2Bah4> <p class="mb0">我们用不同的周长半径对基于gnn的方法进行了测试。直观地说，在agent数量固定的情况下，增加半径可能会导致防御系统的失败。我们将防御者的默认团队规模设置为40，并增加周长半径，直到抓到的入侵者的百分比收敛为零。如<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F9">图9</gydF4y2Baa>时，随着半径从100 m到800 m的变化，百分比逐渐减小，收敛于0。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图9</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g009.jpg" name="Figure9" target="_blank"><img id="F9" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g009.jpg"></a> <p><b>图9</gydF4y2Bab>．在不同半径范围内抓到的入侵者的百分比。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <h4>5.5.3性能<gydF4y2Baem>vs</gydF4y2Baem>．察觉到入侵者的数量</gydF4y2Bah4> <p class="mb0">我们基于gnn的方法在不同数量的入侵者(例如，<我nline-formula id="inf68"> <math id="m78"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>)的感觉显示在<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#F10">图10</gydF4y2Baa>．我们已经做了实验<我nline-formula id="inf69"> <math id="m79"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>作为1、3、5和10，因为没有地面真相专家策略可用于生成大于10的数字的训练数据。我们观察到，感受到的入侵者特征越多，表现得越好。此外，随着团队规模的增大，性能差异趋于减小。对于某些团队规模(例如，40人)，则更高<我nline-formula id="inf70"> <math id="m80"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>表现更好，但这是基于游戏初始配置的预期。例如，如果初始配置非常稀疏，防御者将从更高的配置中受益<我nline-formula id="inf71"> <math id="m81"> <msubsup> <mrow> <mi> N</gydF4y2Bami> </mrow> <mrow> <mi> 一个</gydF4y2Bami> </mrow> <mrow> <mi> f</gydF4y2Bami> </mrow> </msubsup> </math> </inline-formula>，被抓到的入侵者的比例会更高。</p><gydF4y2Badiv class="DottedLine"></div> <div class="Imageheaders"> 图10</gydF4y2Badiv> <div class="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g010.jpg" name="Figure10" target="_blank"><img id="F10" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1104745/fcteg-04-1104745-HTML/image_m/fcteg-04-1104745-g010.jpg"></a> <p><b>图10</gydF4y2Bab>．察觉到不同数量的入侵者被抓获的百分比。</p></gydF4y2Badiv> <div class="clear"></div> <div class="DottedLine"></div> <h3 class="pt0">5.6限制</gydF4y2Bah3> <p class="mb0">由于外围防御是一个相对较新的研究领域，这项工作有潜在的限制假设。在问题的表述中，我们假设机器人是点粒子。因此，我们假设最优轨迹服从一阶假设。有一项初步工作(<gydF4y2Baa href="//www.thespel.com/journal/articles/10.3389/#B13">Lee等人，2021年</gydF4y2Baa>)来弥补点粒子假设和三维机器人之间的差距，用于一对一的半球周长防御，我们希望在未来将这项工作的思想扩展到我们的多智能体周长防御问题。另一个限制是没有可用的专家策略，可以与我们提出的方法进行大规模比较。在大规模运行最大匹配算法是非常昂贵的，因此我们将基于gnn的算法与其他基线方法进行了比较。尽管经过测试的算法在不同尺度上的一致性能证实了我们训练的网络可以推广到大规模，但我们希望探索另一种算法，可以在大规模上用作专家策略，以取代最大匹配。一个考虑因素是利用强化学习，因为算法在大规模上的性能是可用的。</p><gydF4y2Baa id="h7" name="h7"></a> <h2>6结论</gydF4y2Bah2> <p class="mb15">提出了一种利用图神经网络解决分散多智能体周界防御问题的新框架。我们的学习框架将防御者的本地感知和通信图作为输入，并返回动作，以最大限度地增加防御者团队的捕获次数。我们通过基于最大匹配算法的专家策略来训练深度网络。为了验证所提出的方法，我们使用五种不同的算法在不同的团队规模中运行外围防御游戏:<gydF4y2Baem>专家</gydF4y2Baem>，<gydF4y2Baem>gnn</gydF4y2Baem>，<gydF4y2Baem>贪婪的</gydF4y2Baem>，<gydF4y2Baem>随机</gydF4y2Baem>,<gydF4y2Baem>中长期规划</gydF4y2Baem>．我们证明了我们的基于gnn的策略在小尺度上更接近专家策略，训练过的网络可以推广到大尺度。</p><pcl作为年代＝"mb15">未来的工作是在感知模块中实现基于视觉的局部感知，这将放松完美状态估计的假设。在防守者内部实现基于视觉的感知和通信的多智能体外围防御将是最终目标。未来的另一个研究方向是利用强化学习在多机器人系统中找到一个集中的专家策略。</p><gydF4y2Baa id="h8" name="h8"></a> <h2>数据可用性声明</gydF4y2Bah2> <p class="mb15">支持本文结论的原始数据将由作者提供，毫无保留地提供。</p><gydF4y2Baa id="h9" name="h9"></a> <h2>作者的贡献</gydF4y2Bah2> <p class="mb0">EL、LZ和VK对研究的概念和设计做出了贡献。EL和AR进行统计分析。EL写了手稿的初稿。所有作者均参与了稿件修改，阅读并批准了所提交的版本。</p><gydF4y2Baa id="h10" name="h10"></a> <h2>致谢</gydF4y2Bah2> <p class="mb03">我们感谢来自ARL DCIST CRA赠款W911NF-17-2-0181, NSF赠款CCR-2112665, CNS-1446592和ec -1941529, ONR赠款N00014-20-1-2822和N00014-20-S-B001，高通研究，NVIDIA，洛克希德马丁公司，C-BRIC，由DARPA共同赞助的半导体研究公司联合大学微电子计划的支持。</p><gydF4y2Baa id="h11" name="h11"></a> <h2>利益冲突</gydF4y2Bah2> <p class="mb0">作者声明，这项研究是在没有任何商业或财务关系的情况下进行的，这些关系可能被解释为潜在的利益冲突。</p><gydF4y2Baa id="h12" name="h12"></a> <h2>出版商的注意</gydF4y2Bah2> <p class="mb15">本文中所表达的所有主张仅代表作者，并不代表他们的附属组织，也不代表出版商、编辑和审稿人。任何可能在本文中评估的产品，或可能由其制造商提出的声明，都不得到出版商的保证或认可。</p><gydF4y2Baa id="h13" name="h13"></a> <h2>参考文献</gydF4y2Bah2> <div class="References"> <p class="ReferencesCopy1"><a name="B1" id="B1"></a>Bajaj, S.， Torng, E.， Bopardikar, S. 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