为物理信息神经网络求解李雅普诺夫与霍姆顿-雅可比-贝尔曼方程提供可验证误差界。
Verifiable Error Bounds for Physics-Informed Neural Network Solutions of Lyapunov and Hamilton-Jacobi-Bellman Equations
- 基于残差构建可验证的误差上界，无需依赖真实解
- 给出最优值函数上下界及反馈策略的优化差距量化
- 即使单边残差小，近似解仍为有效李雅普诺夫函数
非线性系统分析与控制中的许多核心问题可转化为求解偏微分方程（PDE），如李雅普诺夫方程和霍姆顿-雅可比-贝尔曼（HJB）方程。物理信息神经网络（PINNs）作为一种无网格方法，在逼近其解方面展现出巨大潜力，但现有工作大多缺乏严格保证：小的PDE残差并不意味着小的解误差。本文针对李雅普诺夫与HJB方程的近似解，建立了可验证的误差界，尤其聚焦于基于PINN的近似。对于两类方程，我们证明：可验证的残差界能导出相对于真实解的相对误差界，并给出以近似解表示的可计算后验估计。对HJB方程，这进一步提供了紧致子水平集上最优值函数的认证上下界，以及诱导反馈策略的最优性差距量化。我们还证明，单边残差界已足以保证近似解本身构成有效的李雅普诺夫或控制李雅普诺夫函数。数值实验验证了理论结果。
原文摘要 · Abstract (English)
Many core problems in nonlinear systems analysis and control can be recast as solving partial differential equations (PDEs) such as Lyapunov and Hamilton-Jacobi-Bellman (HJB) equations. Physics-informed neural networks (PINNs) have emerged as a promising mesh-free approach for approximating their solutions, but in most existing works there is no rigorous guarantee that a small PDE residual implies a small solution error. This paper develops verifiable error bounds for approximate solutions of Lyapunov and HJB equations, with particular emphasis on PINN-based approximations. For both the Lyapunov and HJB PDEs, we show that a verifiable residual bound yields relative error bounds with respect to the true solutions as well as computable a posteriori estimates in terms of the approximate solutions. For the HJB equation, this also yields certified upper and lower bounds on the optimal value function on compact sublevel sets and quantifies the optimality gap of the induced feedback policy. We further show that one-sided residual bounds already imply that the approximation itself defines a valid Lyapunov or control Lyapunov function. We illustrate the results with numerical examples.
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