标题： 神经哈密顿算子 显示英文标题

标题： Neural Hamiltonian Operator

摘要： 高维随机控制问题由于维度灾难而难以解决。 传统动态规划的一种替代方法是庞特里亚金最大原理（PMP），它将问题重新表述为前向-后向随机微分方程（FBSDEs）系统。 在本文中，我们通过定义一个\textbf{神经哈密顿算子（NHO）}来引入一种形式框架，以深度学习解决此类问题。 该算子通过神经网络参数化耦合的FBSDE动力学，这些神经网络表示反馈控制和价值函数空间梯度的假设。 我们展示了如何通过训练底层网络来强制执行由PMP规定的相容性条件，从而找到最优的NHO。 通过采用这种算子理论观点，我们将深度FBSDE方法置于统计推断的严格语言中，将其视为从模拟数据中学习未知算子的问题。 这一视角使我们能够在一般的鞅驱动条件下证明NHO的通用逼近能力，并为分析此类模型固有的重大优化挑战提供了清晰的视角。 显示英文摘要

摘要： Stochastic control problems in high dimensions are notoriously difficult to solve due to the curse of dimensionality. An alternative to traditional dynamic programming is Pontryagin's Maximum Principle (PMP), which recasts the problem as a system of Forward-Backward Stochastic Differential Equations (FBSDEs). In this paper, we introduce a formal framework for solving such problems with deep learning by defining a \textbf{Neural Hamiltonian Operator (NHO)}. This operator parameterizes the coupled FBSDE dynamics via neural networks that represent the feedback control and an ansatz for the value function's spatial gradient. We show how the optimal NHO can be found by training the underlying networks to enforce the consistency conditions dictated by the PMP. By adopting this operator-theoretic view, we situate the deep FBSDE method within the rigorous language of statistical inference, framing it as a problem of learning an unknown operator from simulated data. This perspective allows us to prove the universal approximation capabilities of NHOs under general martingale drivers and provides a clear lens for analyzing the significant optimization challenges inherent to this class of models.