论文标题
使用神经网络求解高维汉密尔顿 - 雅各比 - 贝尔曼PDE：从路径空间中受控扩散和措施的观点
Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
论文作者
论文摘要
扩散过程的最佳控制与解决某些汉密尔顿 - 雅各比 - 贝尔曼方程的问题密切相关。在最近的机器学习启发方法的基础上，我们研究了$ \ textIt {迭代扩散优化} $技术的潜力，特别是考虑到重要性采样和罕见的事件模拟的应用，并专注于没有扩散控制的问题，并具有线性控制的漂移和跑步成本，这些成本取决于控制控制。更一般而言，我们的方法适用于具有一定偏移不变的非线性抛物线PDE。在算法设计中，适当的损失函数的选择是基于路径测量之间的差异，涵盖各种现有方法的差异。通过与前向SDE的连接的动机，我们提出并研究了新颖的$ \ textit {log-variance} $ divergence，显示了相应的蒙特卡洛估计量的有利属性。开发方法的承诺由一系列高维和亚稳态的数值示例举例说明。
Optimal control of diffusion processes is intimately connected to the problem of solving certain Hamilton-Jacobi-Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the potential of $\textit{iterative diffusion optimisation}$ techniques, in particular considering applications in importance sampling and rare event simulation, and focusing on problems without diffusion control, with linearly controlled drift and running costs that depend quadratically on the control. More generally, our methods apply to nonlinear parabolic PDEs with a certain shift invariance. The choice of an appropriate loss function being a central element in the algorithmic design, we develop a principled framework based on divergences between path measures, encompassing various existing methods. Motivated by connections to forward-backward SDEs, we propose and study the novel $\textit{log-variance}$ divergence, showing favourable properties of corresponding Monte Carlo estimators. The promise of the developed approach is exemplified by a range of high-dimensional and metastable numerical examples.
