基于多项式混沌不确定性传播的物理信息神经网络代理模型用于随机模型预测控制

中文摘要 AI 辅助

随机偏微分方程（PDE）支配着关键的工程和地球物理系统，但在参数不确定性下用于实时控制具有挑战性。我们提出了一个统一框架，将物理信息神经网络（PINNs）与多项式混沌展开（PCE）相结合，以构建一个快速且可微的代理模型。PCE表示能够解析地传播参数不确定性并计算相应的矩，而无需蒙特卡洛采样。我们为PINN-PCE代理模型提供了误差分解，将PCE截断误差、随机求积误差和PINN近似误差分开。将该代理模型嵌入随机模型预测控制（SMPC）方案中，能够基于解析均值和协方差预测进行有限时域控制更新。我们进一步展示了如何将代理模型近似误差纳入收紧的概率约束中。该方法在三个基准上进行了验证：Korteweg-de Vries方程、Burgers方程和二维不可压缩Navier-Stokes方程，分别代表色散、对流和对流-扩散动力学。在所有情况下，代理模型能够实现实时控制更新，同时保持规定的风险水平，并以显著较低的计算成本紧密匹配相应的高保真求解器。

英文摘要

Stochastic partial differential equations (PDEs) govern critical engineering and geophysical systems but are challenging to use for real-time control under parametric uncertainty. We present a unified framework that couples Physics-Informed Neural Networks (PINNs) with Polynomial Chaos Expansion (PCE) to construct a fast and differentiable surrogate. The PCE representation enables analytical propagation of parametric uncertainty and computation of the corresponding moments without requiring Monte Carlo sampling. We provide an error decomposition for the PINN-PCE surrogate that separates PCE truncation, stochastic quadrature, and PINN approximation errors. Embedding this surrogate into a stochastic model predictive control (SMPC) scheme enables finite-horizon control updates based on analytic mean and covariance predictions. We further show how the surrogate approximation error can be incorporated into tightened probabilistic constraints. The approach is validated on three benchmarks: the Korteweg-de Vries equation, Burgers' equation, and the two-dimensional incompressible Navier-Stokes equations, representing dispersive, convective, and convective-diffusive dynamics. Across all cases, the surrogate enables real-time control updates while maintaining prescribed risk levels and closely matching the corresponding high-fidelity solvers at substantially lower computational cost.

发表机构

• Massachusetts Institute of Technology（麻省理工学院）
• University of Washington（华盛顿大学）

机构由 AI 辅助整理，请以论文原文为准。