发表机构
Institute of Science Tokyo（东京科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对无界域上 Hamilton--Jacobi--Bellman 方程，提出核配点法与物理信息神经网络的收敛性分析，通过经验残差控制推导出一致收敛误差界，并给出数值验证。
AI 中文摘要
我们针对无界空间域上 Hamilton--Jacobi--Bellman 方程的两种非单调、无网格逼近方法，从经验平方残差导出了误差界：一种是在本征空间范数约束下使用 Wendland 核的配点法，另一种是在 Sobolev 范数约束下使用光滑激活函数的物理信息神经网络。主要步骤是从经验损失恢复黏性解误差估计所需的逐点残差控制，同时量化离散化或采样误差。对于核方法，我们在经典与本征空间正则性条件下证明了紧集上的一致收敛，并推导出唯一有界连续黏性解的后验定量误差界。对于神经网络方法，我们证明了紧集上依概率的一致收敛，并在经典 Sobolev 正则性条件下给出了定量高概率误差界。这些界由核的填充距离以及神经网络的网络规模和样本规模所控制。证明结合了经验到总体的估计、一个将 $L^2$ 残差转化为上确界范数界的插值不等式，以及一个用于域截断所致误差的受控扩散估计。从逼近性质获得收敛性所用的正则性假设是对黏性适定性的补充。一维和二维空间中的数值实验验证了这些方法和估计。
英文摘要
We derive error bounds from empirical squared residuals for two non-monotone, mesh-free approximations of Hamilton--Jacobi--Bellman equations on unbounded spatial domains: collocation with Wendland kernels under a native space norm constraint, and physics-informed neural networks with smooth activations under a Sobolev-norm constraint. The main step is to recover the pointwise residual control needed for viscosity-solution error estimates from the empirical loss, while quantifying the discretization or sampling error. For the kernel method, we prove uniform convergence on compact sets under classical and native space regularity and derive an a posteriori quantitative error bound for the unique bounded continuous viscosity solution. For the neural-network method, we prove uniform convergence on compact sets in probability and give a quantitative high-probability error bound under classical Sobolev regularity. The bounds are governed by the fill distance for kernels and by the network and sample sizes for neural networks. The proofs combine empirical-to-population estimates, an interpolation inequality that turns an $L^2$ residual into a sup norm bound, and a controlled diffusion estimate for the error caused by truncating the domain. The regularity assumptions used to obtain convergence from approximation properties are additional to viscosity wellposedness. Numerical experiments in one and two space dimensions illustrate the methods and the estimates.
