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摘要: 自1982年著名的Hopfield神经网络问世以来, 神经网络的分岔动力学受到了学术界的广泛关注. 首先, 回顾四类经典神经网络的数学模型和它们在各个领域的应用. 接着, 综述近三十年来关于整数阶神经网络(Integer-order neural networks, IONNs)、分数阶神经网络(Fractional-order neural networks, FONNs)、超数域神经网络(Supernumerary-domain neural networks, SDNNs)以及反应扩散神经网络(Reaction-diffusion neural networks, RDNNs)分岔动力学的相关研究成果. 分析诸多组合因素, 包括节点规模、耦合情形、拓扑结构、系统阶次、复值、四元数、八元数、扩散、时滞、随机性、脉冲、忆阻、激活函数等对神经网络分岔动力学的影响, 并展示神经网络在多个领域的广泛应用. 最后, 对神经网络分岔动力学所面临的挑战以及未来的研究方向进行总结和展望.Abstract: Since the introduction of the renowned Hopfield neural network in 1982, the bifurcation dynamics of neural networks has garnered significant academic attention. Firstly, an overview of the mathematical models of four types of classical neural networks and their applications in various fields is provided. Subsequently, the research results on the bifurcation dynamics of integer-order neural networks (IONNs), fractional-order neural networks (FONNs), supernumerary-domain neural networks (SDNNs), and reaction-diffusion neural networks (RDNNs) in the past three decades are summarized. The effects of various combinations of factors, including node size, coupling, topology, system order, complex value, quaternion, octonion, diffusion, time delay, stochasticity, impulse, memristor, and activation function, on the bifurcation dynamics of neural networks are analyzed, and the wide applications of neural networks in various fields are also demonstrated. Finally, the challenges and potential research directions concerning neural network bifurcation dynamics are summarized and prospected.
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Key words:
- Neural networks /
- time delay /
- nonlinear dynamics /
- stability /
- bifurcation /
- periodicity /
- chaos
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要地防空作战中, 武器部署优化问题是指防御方在到达保卫目标附近后, 根据战场信息设计防区并确定武器单元的具体部署位置, 从而有效应对战场环境的高度不确定性并最大限度地发挥其作战效能. 制定陆用武器系统作战方案是一项复杂且具有系统性的研究工作[1], 而武器部署是系统级别联合防空反导作战的关键, 是战前任务规划的核心部分, 也是成功拦截弹道导弹的重要前提[2]. 因此防空武器优化部署研究具有重要的实用价值和军事意义.
武器部署问题的研究主要侧重于构建部署问题模型和设计模型求解算法两个方面. 建模方面, 其目标函数和约束条件是两个重要内容: 目标函数的设计可分为单目标和多目标两类, 前者大多以防区覆盖率为目标进行优化求解, 后者则根据具体的作战任务和条件设置多目标函数[3]; 约束条件主要是针对布防阵地的地形限制、武器总数、武器性能和弹药储备等实际因素所建立的对应[4], 旨在确保作战计划的可行性和有效性. 模型求解算法方面, 因武器部署本质是一个选址问题, 属于对武器资源的多任务分配问题, 所以大多数学者通过设计各种改进启发式搜索算法进行求解, 例如Bai等[5−6] 提出基于拍卖算法的方法, 在多任务分配问题的求解中展现较好的收敛性和鲁棒性. 而主要的改进目标集中于提高算法求得解集的质量以增强整体防区部署的防御效率.
现有的成果中, 大多数学者的研究思路是根据实际问题背景, 提取贴合实际作战的目标函数和约束条件, 从而构建针对性的问题模型, 然后根据问题的具体特点改进选取的智能算法或神经网络结构以提高对此特定问题的求解有效性, 从而达到实际应用的最终目的. 如Li等[7] 建立基于混合整数非线性规划的导弹阵地部署问题模型, 针对多层位置网络, 建立以生存能力最大化为目标的优化模型, 并提出两阶段求解方法降低算法早熟收敛的概率, 提高求解质量. 温包谦等[8] 结合扇形部署与环型部署模型应对大规模、多约束、非线性的兵力部署问题, 提出基于粒子群和遗传算法的求解方法增强求解搜索速度, 有效解决网格化防空火力单元的优化布阵问题. Sun等[9] 分析反导部队部署中多因素相互作用及环境不确定性, 结合深度学习与层次降维, 基于基本模型设计以拦截弧长为优化目标的双层嵌套优化部署模型. 李珂铭等[10] 综合考虑火力覆盖面积、有效掩护宽度和有效防御扇区等因素, 设计改进算法以组合不同解集, 实现区域和目标掩护的混合部署, 并综合使用导弹和高炮等不同武器, 从而优化编队的防御性能. 岳韶华等[11] 通过距离纵深、雷达法线偏离度和武器部署均匀度来衡量拦截窗口与火力分布均匀性, 并改进遗传蜂群算法来优化部署方案以适应重点保卫目标. 上述部署模型和求解算法为武器部署问题提供新的解决方案, 展示良好的可行性和先进性.
综合文献分析可知, 现有的武器部署研究在求解算法方面取得显著成果, 但是现有成果所考虑的部署规模有限, 适用性不足. 而粒子群优化(Particle swarm optimization, PSO)算法适用性强和计算复杂度低的特点在处理大规模场景时具有显著的优势[12−13]. 因此, 粒子群算法在求解武器部署问题时具有良好的应用潜力. 此外, 大多数学者在构建部署模型时仅从防御方的单一视角出发, 未考虑从战术层面对部署问题的模型框架进行创新改进, 这导致其难以有效抵御日益智能化的敌方空袭. 针对这些不足, 必须从全局战术高度出发, 考虑攻防双方来构建部署模型, 并针对求解算法进行贴合实际的改进. 对此, 本文提出一种基于改进粒子群优化和Stackelberg博弈的武器部署方法(Improved particle swarm optimization, IPSO), 具体创新点如下: 1)结合Stackelberg博弈构建武器部署模型以充分考虑战场环境下的攻防博弈; 2)分类设计初始化部署方案, 降低求解过程陷入局部最优的风险; 3)设计新的粒子群最优更新方法指导进化, 增强跳出局部最优的能力. 本文其余内容安排如下: 第1节从目标函数和约束条件两方面建立新型武器部署问题模型; 第2节改进粒子群求解算法; 第3节开展针对模型和算法的仿真实验; 第4节进行全文总结与展望.
1. 新型武器部署问题建模
1.1 目标函数
本文先从进攻方的角度设计其突防的目标函数, 然后以此作为参数设计对应的防御方武器部署优化的目标函数.
1) 进攻方目标函数
进攻策略优化可以抽象为规划一条从防御方防区边界到进攻投弹线的最优路径, 其中防区边界指防御方远程预警雷达覆盖区边界, 进攻投弹线指空袭武器在发射炮弹或空地导弹时与要地的最远距离线, 如图1所示.
对于常见路径规划问题, 其最大化目标函数可描述为[14]:
$$ \begin{equation} \mathrm{fitvalue} = \frac{\mathrm{collision}}{\mathrm{distance}} \end{equation} $$ (1) 式中, distance为路径总长度; collision为碰撞系数, 用于表示路径点与障碍物的碰撞关系[15], 描述为:
$$ \begin{equation} \mathrm{collision} = \begin{cases} 0,\; & \text{发生碰撞} \\ 1,\; & \text{其它情况} \end{cases} \end{equation} $$ (2) 进攻方在规划攻击路径时把防御武器位置视作障碍点, 各武器具有相应探测区和杀伤区范围. 因此从总路径长度、探测威胁规避、杀伤威胁规避三方面评价攻击路径的总体效果, 表示如下:
$$ \begin{equation} J_1 = \lambda_1J_L+\lambda_2J_D+\lambda_3J_A \end{equation} $$ (3) 式中, $ J_L $、$ J_D $和$ J_A $分别表示攻击路径总长度、探测威胁和杀伤威胁的目标函数; $ \lambda_1 $、$ \lambda_2 $和$ \lambda_3 $分别为对应目标函数的权重系数, 据决策偏好确定. 各目标函数可描述为:
$$ \begin{equation} J_L = \sum\limits_{i = 1}^{n-1}\sqrt{(x_{i+1}-x_i)^2+(y_{i+1}-y_i)^2} \end{equation} $$ (4) $$ \begin{equation} J_D = p_dL_d \end{equation} $$ (5) $$ \begin{equation} J_A = p_aL_a \end{equation} $$ (6) 式中, ($ x_i $, $ y_i $) 为第$ i $个路径点的位置坐标; $ n $为路径点的个数; $ p_d $为被探测概率; $ L_d $为总进攻路径中在探测区内的路径长度; $ p_a $为被打击概率; $ L_a $为总进攻路径中在杀伤区内的路径长度. 通过对应概率与路径长度的乘积表示探测威胁和杀伤威胁.
需要注意的是, 与所设计目标函数有关的坐标及距离均在二维环境中求解得到, 因在进行防空部署研究时均是以俯视的视角进行考虑. 若考虑三维环境, 可通过分析高程信息并划定障碍物区域, 进而在二维环境中表示三维地形对部署的影响, 随后可继续使用本文方法.
2) 防御方目标函数
防御方武器部署可以抽象为设施选址问题, 但在设计防御方目标函数时有必要考虑到进攻方的路径参数, 其表达式设计如下:
$$ \begin{equation} J_2 = \lambda_4J_C+\lambda_5J_1 \end{equation} $$ (7) 式中, $ \lambda_4 $和$ \lambda_5 $为对应目标函数的权重系数, $ J_C $代表武器对防区有效防御覆盖率的目标函数, 其表达式如下:
$$ \begin{equation} J_C = \frac{\sum S_i-\sum\left(S_i\cap S_j\right)+\cdots+(-1)^{m-1}\cap_{i = 1}^mS_i}{S} \end{equation} $$ (8) 式中, $ S_i $为防御武器$ i $的打击区面积; $ S $为远程预警雷达覆盖区面积; $ m $为武器单元个数.
由此建立从攻防双方考虑的武器部署双层优化模型, 该模型属于带约束的组合优化问题, 具体约束条件将在下文进行设计说明.
1.2 约束条件
1)防区范围约束
在进行防御武器优化部署时, 武器的部署位置具有一个总体的范围限制. 设防区边界为远程预警雷达覆盖区边界, 则对应武器部署位置约束表示如下:
$$ \begin{equation} g(x_i,\;y_i)\leq0,\;\forall i\in\{1,\;2,\;\cdots,\;m\} \end{equation} $$ (9) 式中, $ g(\cdot,\cdot) $为防区边界的函数表达式; ($ x_i $, $ y_i $)表示第$ i $个武器单元部署的位置坐标. 通过式(9)将武器的部署范围约束在防区之内, 在远程预警雷达的协助下以达到更好的防御拦截效果.
由于进攻方来袭时其进攻武器对于每个保卫目标均有投弹线的存在, 在投弹线覆盖范围内的目标均存在较大的被袭击概率, 在防御优化部署时有必要将武器单元部署于投弹线覆盖范围之外以保证防御武器自身的安全. 故设计约束条件如下:
$$ \begin{equation} h_k(x_i,\;y_i)\geq0,\;\forall i\in\{1,\;\cdots,\;m\},\;\forall k\in\{1,\;\cdots,\;s\} \end{equation} $$ (10) 式中, $ h_{k}(\cdot,\cdot) $为待保卫目标$ k $的投弹线边界函数表达式; $ s $表示防区内待保卫目标的个数. 由此将武器单元的部署位置约束在防区内.
2)武器单元部署间距约束
武器部署时各武器单元间的部署间距对火力覆盖有较大影响, 因此在优化部署中考虑将其作为约束条件. 设防御武器单元的实际最大杀伤距离[16] 即最大航路捷径为$ p $, 用于防区火力重叠的计算.
如图2所示, 当武器单元按两层火力重叠部署时, 武器之间的部署间距满足下式:
$$ \begin{equation} D = 2p-L \end{equation} $$ (11) 式中, $ L $为火力重叠区最大正面宽; $ D $为武器部署间距, 推知各武器均匀部署构成$ n $层火力重叠时, 武器部署间距满足下式条件:
$$ \begin{equation} (n-1)D = 2p-L \end{equation} $$ (12) 对于$ n $层火力重叠非均匀部署情形, 最接近的武器间的最大部署距离不超过式(12)中的$ D $.
综上可得, 在进行部署时, 考虑火力重叠层数$ n $和重叠区最大正面宽度$ L $, 最接近的武器单元之间的最大部署距离应满足:
$$ \begin{equation} D_{\max} = \begin{cases} 2p,\; & n = 1 \\ \dfrac{2p-L}{n-1},\; & n>1 \end{cases} \end{equation} $$ (13) 武器单元部署间距的约束条件表示如下:
$$ \begin{equation} D_{\min}\leq\sqrt{\left(x_i-x_j\right)^2+\left(y_i-y_j\right)^2} \end{equation} $$ (14) $$ \begin{equation} D_{\min}\leq\sqrt{\left(x_i-x_j\right)^2+\left(y_i-y_j\right)^2}\leq D_{\max} \end{equation} $$ (15) 式中, ($ x_i $, $ y_i $)、($ x_j $, $ y_j $)分为第$ i $个和第$ j $个武器单元部署的位置坐标; $ D_{\text{min}} $为武器单元之间出于电磁兼容考虑的最小部署距离. 式(14)表示对于一般的保卫目标, 只需保证武器单元部署距离满足最小间距约束; 而对于重要的保卫目标, 武器部署距离还需要同时满足构成相应火力重叠层数的最大间距约束.
综上, 通过目标函数和约束条件的设计完成对新型武器部署问题的模型构建, 后续将确定问题的求解方法. 粒子群算法作为启发式优化算法, 计算高效可靠, 且粒子位置与本模型的武器部署变量相契合, 有益于求解本文所涉及的连续变量问题. 因此, 选用粒子群算法作为武器部署问题的求解算法.
2. 改进粒子群优化算法
2.1 部署方案初始化
本节针对粒子群算法中随机生成初始种群的操作, 结合前述提出的武器部署问题优化模型具体特点进行改进, 提出新型初始化种群方法应用于求解不同规模的部署问题.
新型初始化种群方法从两个方面进行改进. 一方面, 引入混沌映射机制增强在初始化部署时的随机性; 另一方面, 结合K均值聚类与重心法, 在初始化部署位置时针对重要保卫目标的防御区域优先考虑火力重叠, 增强算法初始化时的针对性. 在实际情况中就是防御方在确定武器单元部署的预选位置时, 既从过往作战经验上确定最优部署位置可能存在的区域, 又考虑其它部署位置在作战中取得防御奇效的可能性. 基于上述思路, 下文将从初始化武器单元部署时资源紧缺与充足的两种情况阐述其实现过程, 其框架如图3所示.
1) 可用武器单元数小于需要保卫的目标数
对于此类防御资源紧缺的情况, 暂不考虑火力重叠部署及其间距约束, 根据可用武器单元数对保卫目标进行聚类操作, 然后为每一簇保卫目标部署武器单元进行防御. 参考K均值聚类算法, 直接根据可用武器数确定聚类数量, 该算法以欧氏距离作为聚类判断准则, 符合部署的实际情况, 故可设计划分保卫目标簇的目标函数如下:
$$ \begin{equation} \min J(G) = \sum\limits_{i = 1}^m\sum\limits_{j = 1}^s\varphi_{ij}\sqrt{\left(x_j-\mu_{xi}\right)^2+\left(y_j-\mu_{yi}\right)^2} \end{equation} $$ (16) $$ \begin{equation} \varphi_{ij} = \begin{cases} 1,\; & \text{目标 }\; i\; \text{在聚类}\; j\; \text{中} \\ 0,\; & \text{目标 }\; i\; \text{不在聚类}\; j\; \text{中} \end{cases} \end{equation} $$ (17) 式中, $ G = \{g_1,\;g_2,\;\cdots,\;g_m\} $为保卫目标的聚类情况且满足$ \sum\nolimits_{i = 1}^{m}|g_{i}| = s $; $ (\mu_{xi},\; \mu_{yi}) $为保卫目标簇$ g_i $的中心位置坐标; ($ x_j $, $ y_j $)为保卫目标$ j $的位置坐标.
在完成对需要保卫目标的聚类操作之后, 使用重心法为每一簇的目标配置一个武器单元进行保卫. 因为在前一步骤使用K均值聚类时只考虑到各目标之间的欧氏距离, 所以在当前步骤需对各个保卫目标的重要性加以考虑, 而重心法正好满足该需要.
根据对保卫目标进行重要性评估值的排序, 得到所有目标的重要性取值向量$ {\boldsymbol{\rho}} = \{\rho_{1},\;\rho_{2},\;\cdots,\;\rho_{s}\} $, 将该值作为每个保卫目标点的重量从而引入重心法进行计算. 在地形的二维平面表示图中, 将所有目标点视为质点, 则由所有质点对坐标轴的静力矩与重心对轴的静力矩等效, 可求得聚类目标簇$ g_j $的初始重心坐标如下:
$$ \begin{equation} x_0^j = \frac{\sum\limits_{i = 1}^{|g_j|} \rho_i x_i}{\sum\limits_{i = 1}^{|g_j|} \rho_i} \end{equation} $$ (18) $$ \begin{equation} y_0^j = \frac{\sum\limits_{i = 1}^{|g_j|} \rho_i y_i}{\sum\limits_{i = 1}^{|g_j|} \rho_i} \end{equation} $$ (19) 式中, ($ x_i $, $ y_i $)为保卫目标簇中各目标的坐标信息; $ |g_{j}| $为聚类之后保卫目标簇 $ g_j $中的目标数量; $ (x_0^j,\;\; y_0^j) $为聚类目标簇 $ g_j $中所有保卫目标的初始重心坐标. 以此为圆心, 以武器单元预部署最大机动调整距离为半径划定为最优部署位置潜在区域边界, 并定义为重点防区$ A_1 $, 剩余部分定义为一般防区$ A_2 $.
把整个防区划分为两类区域之后, 在通过粒子群算法求解部署位置时, 种群初始化操作也根据这两大区域进行: 将初始种群划分为数量相等的两个子种群, 其中一个子种群在重点布防区内随机生成部署位置, 另一个子种群在一般布防区内随机生成. 为了增强种群初始化的随机性, 采用Tent混沌映射的方法进行不同防区内的种群生成, 生成随机数的公式如下:
$$ \begin{align} r_{n+1} & = \begin{cases} \dfrac{r_n}{\alpha_t},\; & 0 < r < \alpha_t \\ \dfrac{1 - r_n}{1 - \alpha_t},\; & \alpha_t \leq r < 1 \end{cases} \end{align} $$ (20) 式中, $ \alpha_{t} $为混沌参数; $ r_n $为[0, 1]范围内的随机数. 由此得到混沌序列$ {\boldsymbol{r}} = (r_{1},\;r_{2},\;\cdots,\;r_{\text{NP}}) $, 其中NP为种群数量. 再结合防区搜索空间的远界、近界限制约束转化到种群个体的搜索空间中, 则初始化种群位置如下:
$$ \begin{equation} posx_i = lb_x+r_i(ub_x-lb_x) \end{equation} $$ (21) $$ \begin{equation} posy_i = lb_y+r_i\left(ub_y-lb_y\right) \end{equation} $$ (22) 式中, $ (posx_i,\; posy_i) $表示武器单元$ i $的初始化部署位置; $ lb $和$ ub $分别表示搜索空间近界和远界限制.
综上所述, 在武器资源紧缺、不考虑火力重叠的情况下, 根据需要保卫目标的位置信息, 本文通过结合K均值聚类算法与重心法确定出武器初始部署可能存在的最优区域, 将防区划分为两部分, 种群等分成两个子种群, 通过Tent混沌映射分别在这两个区域内进行初始化操作, 增强算法的随机性和针对性, 使得求解结果更加准确和可靠.
2) 可用武器单元数大于或等于需保卫的目标数
对于防御资源充足的情形, 不需要根据武器单元数对保卫目标进行聚类操作. 此类情况下算法的特定性体现在通过设置武器单元火力重叠部署以提高对重要目标的保卫效果上.
由前述内容已知所有待保卫目标的重要性取值向量为$ {\boldsymbol{\rho}} $, 据此设计种群初始化操作步骤如下:
1) 按向量$ {\boldsymbol{\rho}} $中重要性取值从大到小的顺序为各个保卫目标分别配备一个武器单元. 以目标为圆心, 以投弹线远界与武器单元预部署最大机动调整距离之和为半径划定部署区域.
2) 为保卫目标按重要性排序配备剩余武器单元构成双层火力重叠部署.
3) 继续部署剩余武器单元, 不断增加火力重叠层数, 直到所有武器单元预部署完成.
上述初始化部署方法在防御资源充足的情况下对重要保卫目标进行火力重叠部署, 增强算法对所提出实际问题的求解针对性. 在随机性方面, 仍采用与防御资源紧缺情况一样的Tent混沌映射方法以增强随机性. 通过将初始种群等分成两个子种群, 分别按上述初始化流程操作, 提高初始种群质量, 增强算法的针对性与随机性, 降低陷入局部最优的风险.
综上, 防御资源短缺和资源充足两种情形下的初始化部署过程如算法1所示.
算法1. 针对武器部署的种群初始化算法
1: 输入. 保卫目标总数$ s $, 保卫目标坐标信息$ x_{(1 \times s)} $和$ y_{(1 \times s)} $, 保卫目标重要性取值向量$ {\boldsymbol{\rho}} = \{\rho_1,\; \rho_2, \dots,\; \rho_s\} $, 武器单元数$ m $, 种群大小$ N_p $.
2: 输出. 初始种群位置信息$ [\bf{posx},\; \bf{posy}]. $
3: $ np = \frac{N_p}{2} $
4: if $ m < s $ then
5: 簇的数量 = $ m $;
6: $ J(G) $由式(16)得到;
7: 式(18)和式(19)更新$ x_0 $和$ y_0 $;
8: 更新$ A_1 $和$ A_2 $;
9: for $ i = 1 $ : $ np $ do
10: 式(21) ~ (22)计算子种群在$A_1 $防区的重心坐标$ subposx_i^{A_1} $和$ subposy_i^{A_1} $;
11: 式(21) ~ (22)计算子种群在$A2 $防区的重心坐标$ subposx_i^{A_2} $和$ subposy_i^{A_2} $;
12: end for
13: $ \bf{posx} = [subposx^{A_1},\; subposx^{A_2}] $
14: $ \bf{posy} = [subposy^{A_1},\; subposy^{A_2}] $
15: else
16: 3 ~ 8 更新$ A_1 $和$ A_2 $
17: for $ i = 1 $ : $ np $ do
18: 式(21)$ \sim $(22)计算$ subposx_i^{A_1} $和$ subposy_i^{A_1} $;
19: 式(21)$ \sim $(22)计算$ subposx_i^{A_2} $和$ subposy_i^{A_2} $;
20: end for
21: $ \bf{posx} = [subposx^{A_1},\; subposx^{A_2}] $
22: $ \bf{posy} = [subposy^{A_1},\; subposy^{A_2}] $
23: end if
2.2 最优粒子更新方法
PSO求解迭代过程中, 确定个体历史最优值和种群全局最优值是十分重要的步骤, 直接影响粒子进化方向和最终求解质量. 常见的最优值更新操作是在迭代中为粒子求得适应度值以进行比较, 然后根据比较结果更新个体最优值和全局最优值. 但是这种方法可能会放弃潜在的最优进化方向. 由此本文从个体最优和全局最优两个方面设计新的更新方法, 综合考虑潜在的最优进化方向, 从而降低陷入局部最优的风险.
1) 个体最优更新方法
本文结合Metropolis准则[17] 针对性地设计个体最优更新方法, 增强算法跳出局部最优的能力, 更新方法如下式:
$$ \begin{equation} \Delta = \frac{i(t)[pfit_i(t)-pbest_i]}{pbest_i} \end{equation} $$ (23) $$ \begin{equation} p_{\mathrm{update}} = \begin{cases} 1,\; & \Delta < 0 \\ \text{e}^{-\frac{\Delta}{T - t}},\; & \Delta \geq 0 \end{cases} \end{equation} $$ (24) $$ \begin{align} pbest_{i} & = \begin{cases} pfit_{i}(t),\; & p_{\mathrm{update}} > rand \\ pbest,\; & \text{其它} \end{cases} \end{align} $$ (25) 式中, $ pfit_i(t) $为粒子$ i $在第$ t $次迭代中的适应度值; $ pbest_{i} $为粒子$ i $的个体最优适应度值; $ T $为总迭代次数; $ i(t) $为迭代进程因子, 其值等于当前迭代次数除以总迭代次数; $ rand $为取值范围[0, 1)的随机数.
2) 全局最优更新方法
Stackelberg博弈模型将参与者划分为领导者和跟随者两个角色. 领导者首先做出决策, 跟随者在观察到领导者的决策后作出反应. 领导者依据跟随者反应制定相应的策略. 而跟随者则根据领导者的策略来选择自己的最佳决策. 由此可知, 此种模式与武器部署中的攻防双方决策过程相契合, 防御方在敌方进攻前优先做出决策, 而进攻方在观察到防区情况后再进行进攻路径规划. 因此, 本文将在武器部署中考虑攻防博弈模型.
由前述部署模型可知, 优化过程中每个粒子都代表一种防御方的部署策略, 对应着博弈模型中领导者的每种纯策略, 则粒子的适应度值可以作为领导者的纯策略收益. 而进攻方作为跟随者, 其策略收益以参数形式在粒子适应度值中体现. 更新全局最优时, 从种群中选取一定数量的粒子个体, 将该粒子个体的最优适应度值及其中的进攻方参数分别作为博弈中领导者和跟随者的策略收益, 通过求解Stackelberg博弈模型[18], 得到全局最优更新结果, 计算如下:
$$ \begin{split} &\operatorname*{max}_{d_{i},\;a_{j}^{k}}\sum\limits_{i\in I}\sum\limits_{j\in J}\sum\limits_{k\in K}\tau^{k}D_{ij}^{k}d_{i}a_{j}^{k} \\& \text{s.t.} \quad a^{k}\in\arg\operatorname*{max}_{r^{k}}\left\{\sum\limits_{i\in I}\sum\limits_{j\in J}A_{ij}^{k}d_{i}r_{j}^{k}\right\} \\ &r_{j}^{k} \in \{0,\;1\} \\ &\sum\limits_{j\in J}r_{j}^{k} = 1 \end{split} $$ (26) 式中, $ I $为领导者部署策略集合; $ J $为跟随者进攻策略集合; $ K $为跟随者集合; $ d_i $为领导者选择策略$ i $的概率; $ a_j^{k} $为跟随者$ k $选择策略$ j $的概率; $ A^{k} $和$ D^{k} $分别为跟随者和领导者各自的收益; $ \tau^{k} $为领导者面对跟随者$ k $的概率; $ r_{j}^{k} $用于在下层优化中表示跟随者$ k $选择策略$ j $的状态. 为简化计算, 在武器部署的攻防博弈场景中取$ |K| = 1 $, 选取适应度值前2的粒子作为对应的策略数量(即$ |I| = |J| = 2 $), 则化简之后此模型可在多项式时间内直接通过求解器求解.
式(26)求解结果为选择两个策略的概率$ d_i $, 用于考虑两个策略的重要性, 以此为权重由式(27)求得全局最优, 用于后续速度更新与惯性权重更新以指导种群的搜索进化方向, 而最终的求解结果仍是选取一个确定的全局最优粒子作为武器部署方案.
$$ \begin{equation} gbest = \sum\limits_{i\in I}d_i\times pfit_i \end{equation} $$ (27) 3. 仿真实验
3.1 算法对比
为验证算法性能, 本节将与三种算法(PSO、AGAPSO[19]、SSGWO[20])分别从进攻路径优化和防御部署优化两个方面进行对比仿真实验. 设计三种规模的作战场景, 各场景中均有5个防御武器单元, 而保卫目标数量分别为2个、5个和10个, 分别对应防御武器单元数多于、等于和小于保卫目标数的情况. 在常见的防御部署场景中, 可部署的武器单元通过5个一组执行防御任务, 基于此设计需要保卫的目标数量从而形成不同资源情况的防御场景. 优化结果及其代价迭代曲线如图4 ~ 图6所示. 结果图中红色五角星为被保卫的目标, 黑色圆圈为进攻路线的起始点, 白色圆圈为进攻路径的终止点. 整个图的背景为欧氏距离场, 颜色越深表示与火力单元的距离越小. 总体上看, 四种算法在设计的三种不同规模作战场景中均能从防区远界出发尽量避开各防御武器单元并最终到达保卫目标投弹线, 但四种算法所规划的进攻路径的质量存在差异, 分析如下:
1) 图4中有2个保卫目标, 5个防御武器单元通过构建火力重叠的形式部署在保卫目标周围, 在此情况下, 四种算法从进攻方角度开展规划, 避开防区内的武器单元, 但对火力重叠区的考虑影响整个路径代价的收敛情况. 根据所展示的代价曲线可知: 本文设计的IPSO所得路径代价最小, 随后依次是SSGWO、PSO和AGAPSO所得路径. 对比算法可知在迭代5到15步的过程中, IPSO跳出局部最优的次数与幅度均优于其它三种算法, 能够更好地找到防区中的薄弱区域并以此设计进攻路径, 这也使得IPSO最终的代价值最优, 说明本文所提算法显著增强跳出局部最优的能力.
2) 图5中共计有5个保卫目标, 相较2个保卫目标, 重叠部署形式有所变动, 火力重叠有所减弱, SSGWO、AGAPSO和IPSO规划路径较为相似, 据图中的路径代价曲线可知: PSO陷于一条局部最优路径无法跳出, 其它三种算法均能找到防区中防御更加薄弱的一条路径, 以更小的杀伤区路径长度到达投弹区域且避开火力重叠区, 但在迭代5到15步的过程中, IPSO能够以更好地跳出局部最优能力搜索到合适路径从而快于SSGWO和AGAPSO收敛.
3) 图6中有10个保卫目标, 防御部署只能对其先聚类再部署, 因资源紧缺难以形成火力重叠, 防御能力有所下降, 同时这也降低进攻路径的规划难度, 故四种算法所得路径一致, 但在收敛速度上IPSO优势较为明显.
此外, 将四种算法在三种场景中重复运行100次, 统计路径代价结果如表1所示, 其评价指标为各算法求解得到的路径最小代价值, 可用于衡量算法的求解质量. 由表1可知: IPSO算法在求解质量上相对于其它三种算法有较为明显的优势, 但由方差可知因为各种改进措施的影响, IPSO算法在求解的稳定性上要弱于传统的PSO, 但其求解的均值明显优于PSO, 故该差距对于IPSO求解结果占优影响不大.
表 1 最优进攻路径代价对比Table 1 Comparison of optimal attack path costs仿真实例 对比指标 PSO AGAPSO SSGWO IPSO 规模场景1 最大值 87.9157 90.1453 83.7577 81.1419 最小值 84.7922 85.7749 79.3922 78.6324 平均值 85.9452 87.0357 81.1712 79.4218 均方差 6.3320 6.9340 7.0318 7.8558 规模场景2 最大值 84.1190 72.7094 72.0462 71.9816 最小值 79.1262 70.2760 70.4387 70.1361 平均值 82.4984 71.6797 71.1869 70.8235 均方差 3.9597 3.6551 4.4898 3.9926 规模场景3 最大值 53.3404 52.7513 52.6991 52.6948 最小值 51.2769 51.2769 51.2769 51.2769 平均值 52.8909 51.9593 51.9407 51.9171 均方差 1.3500 2.8143 2.2543 1.9502 各算法计算时间如表2所示, 数据表明, 除规模场景1外, IPSO算法在计算时间上具有较为明显的优势. 规模场景1中, AGAPSO算法计算速度最快, IPSO算法计算速度次之, 两者差距较小. 综上, IPSO算法在计算性能方面具有较佳的表现.
表 2 各算法计算时间对比Table 2 Comparison of computation time算法 规模场景1 规模场景2 规模场景3 PSO 0.6864 0.8446 0.9469 AGAPSO 0.6314 0.8430 0.9308 SSGWO 0.6437 0.8236 0.9107 IPSO 0.6357 0.7909 0.8991 对于设计的全局最优更新策略, 通过优化进行验证. 四种算法在三种场景下的防御部署方案适应度值迭代曲线如图7所示, 具体分析如下:
图 7 三种场景下防御部署方案适应度值收敛曲线Fig. 7 Convergence curves of fitness values for defence deployment schemes in three scenarios1) 从细节上看, 在迭代5到15步的过程中, IPSO在三种场景下均能快速跳出局部最优的困境, 这得益于改进的个体最优更新和全局最优更新机制为种群进化指导方向, 降低算法迭代求解过程中长时间陷入局部最优的风险, 从而使得IPSO在收敛速度上具有较大的优势.
2) 从总体上看, 随着保卫目标数量的增加, 防御部署的难度与压力逐渐增大, 规模场景3中的敌方进攻路径优化结果较为一致, 故四种算法优化得到相同适应度值的部署方案; 但在保卫目标数量较少的前两个场景之中, 防御部署的可操作性较大, 对优化算法的求解收敛性有更高要求, 故IPSO算法的优势也更加明显.
3.2 模型对比
为验证所提部署优化模型的有效性, 本节将选取对比模型进行仿真实验. 对比模型评价指标为敌方突防概率[21], 具体计算方法不再赘述, 本文模型评价指标为部署方案适应度, 可通过式 (7) 得到. 首先将本文优化所得部署方案以对比模型方法计算评价指标并进行比较; 然后交换, 将对比模型部署方案代入本文模型计算评价指标, 每轮重复执行100次, 由此分析两种模型在双方评价指标上的差异性.
基于两种模型所生成的部署方案分别采用对比模型和本文模型评价指标的计算结果如图8所示. 由图可知, 在设计的三种规模防御场景中, 按对比模型评价指标衡量两种模型所得部署方案结果差距不大, 相反按本文模型评价指标衡量结果存在较为明显的差异, 即本文模型评价指标更能反映出不同部署方案的差异.
图 8 两种部署模型评价指标值对比Fig. 8 Comparison of the evaluation indicator values of two deployment models为了定量比较两种模型评价指标之间的差异性, 本文将采用独立样本$ t $检验对数据进行分析. 独立样本$ t $检验用于检验两组数据之间是否存在显著差异, 从而可通过适应度值的差异反映模型之间的质量差异, 具体结果如表3和表4所示.
表 3 两种模型在对比模型评价指标中的差异Table 3 Differences between two models in comparing model evaluation indicators规模 组别 评价指标均值 平均值差 t 显著性 场景1 对比模型 0.5927 0.0288 − 1.6265 0.1212 本文模型 0.6215 场景2 对比模型 0.6438 0.0088 0.3301 0.7451 本文模型 0.6350 场景3 对比模型 0.7500 0.0036 − 0.9334 0.3629 本文模型 0.7536 表 4 两种模型在本文模型评价指标中的差异Table 4 Differences between two models in the model evaluation indicators of this paper规模 组别 评价指标均值 平均值差 t 显著性 场景1 对比模型 75.5000 3.5126 − 2.4348 0.0255 *本文模型 79.0126 场景2 对比模型 73.4573 2.6694 2.2003 0.0411 *本文模型 76.1267 场景3 对比模型 63.1573 1.9786 − 1.7206 0.1025 本文模型 65.1359 由表3可知, 当对所有部署方案以对比模型的评价指标进行衡量时, 三种规模下的结果均不存在显著差异; 而由表4可知, 当对所有部署方案以本文模型的评价指标进行衡量时, 场景1和场景2下的结果在0.05的显著性水平下呈现差异($* $表示在显著性水平0.05下呈现差异). 进一步比较评价指标的均值发现本文模型所得部署方案表现优于对比模型所得方案. 两个模型求得的部署方案在对比模型评价下差异不显著, 但在本文模型评价下却存在明显差异, 且本文模型结果更优. 这说明本文模型能有效反映不同部署方案的差异, 验证了其有效性.
综合上述关于算法对比和模型对比的仿真结果, 概括如下: 1) 算法方面, 选取多种对比算法通过进攻路径规划验证了本文改进的个体最优更新策略及惯性权重的有效性, 通过攻防优化验证了设计的全局最优更新策略的有效性. 2) 模型方面, 根据选取的对比模型与本文模型进行评价指标的衡量, 结果表明本文模型更能体现不同部署方案的差异且求得的方案表现更佳, 验证了本文模型的有效性. 3) 实用性与逼真度方面, 以上模型的实现需要保卫目标的位置信息参数和来袭目标距离反馈信号作为支撑. 在实际的武器部署情景中, 保卫目标位置信息均为已知, 来袭目标的距离度量可通过雷达或光电探测等方式获取. 本文设计的求解算法为基于粒子群算法的改进, 此智能优化算法简单可靠, 计算量小[22], STM32F407、树莓派等单片机的算力可以支撑迭代求解的有效开展, 可通过嵌入式软件开发进行算法配置.
4. 结束语
本文从进攻方的角度出发, 设计基于Stackelberg博弈思想的防御武器部署模型, 可根据不断变化的战场信息与不断进行的攻防博弈为武器单元设计合理的部署方案. 通过与对比模型的独立样本$ t $检验得知所提模型对部署问题的描述更为有效, 在显著性水平0.05下更具优势. 针对求解部署模型时容易陷入局部最优的问题, 本文结合战场实际与作战经验制定初始化部署方案, 增强算法的随机性与求解具体问题的针对性. 设计新的粒子群最优更新方法指导种群进化方向. 仿真实验表明, 改进后的算法在收敛效果方面表现出更优的路径代价收敛性和更快的收敛速度, 表明相关改进增强算法跳出局部最优的能力, 进而可以提高最终部署方案的求解质量. 在后续的研究中, 可以进一步优化验证环节, 采用三维仿真环境或实物实验来更加深入地验证算法和模型的有效性.
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图 3 双向联想记忆神经网络拓扑图
Fig. 3 Topology of the bidirectional associative memory neural network
图 4 高维耦合下环型拓扑神经网络的发展历程
Fig. 4 Development of ring topology neural networks under high-dimensional coupling
图 5 高维耦合下混合型拓扑神经网络的发展历程
Fig. 5 Development of hybrid topology neural networks under high-dimensional coupling
表 1 神经网络模型分类
Table 1 Classification for neural network models
神经网络类型 具体分类 代表性文献 应用领域 特点 IONNs 少节点全连接 [20, 34−35] 嵌入式系统
实时系统
边缘计算
低功耗设备结构简单, 计算速度较快
适用于低功耗、资源受限的实时系统
灵活性不足、精度不高、适用范围受限少节点非全连接 [41, 45] 多节点Ring [70−71, 77] 多节点Star [26, 78−79] 多节点Hybrid [82−83] FONNs 少节点耦合 [98−99, 101] 信号处理
动态系统建模
时间序列预测具有记忆和遗传特性、适用于非平稳信号处理
适用于建模复杂的非线性系统和时间序列数据
计算复杂度较高、训练过程比较困难高维耦合 [75, 81, 104] SDNNs CVNNs [108, 122, 124] 信号处理
通信系统
量子计算能够更好地处理复数数据、提高数据表示能力
训练复杂度较高、需要特殊的数学处理技巧QVNNs [125, 127] OVNNs [134−135] RDNNs 少节点耦合 [150, 156−157] 模式生成
自组织系统模拟能够模拟物理世界中各类反应扩散过程
计算复杂度较高、训练过程困难高维耦合 [158−160] 
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