Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
Dongkun Zhang Lu Lu Ling Guo George Em Karniadakis Division of Applied Mathematics, Brown University, Providence RI, USA Department of Mathematics, Shanghai Normal University, Shanghai, China
Abstract
Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In particular, in addition to the deep neural network (DNN) for the solution, a second DNN is considered that represents the residual of the PDE. The residual is then combined with the mismatch in the given data of the solution in order to formulate the loss function. This framework is effective but is lacking uncertainty quantification of the solution due to the inherent randomness in the data or due to the approximation limitations of the DNN architecture. Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty. We first account for the parametric uncertainty when the parameter in the differential equation is represented as a stochastic process. Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors. We
中文速览
用物理信息神经网络（Physics-Informed Neural Networks, PINNs）求解随机偏微分方程时，既要处理方程参数本身的随机性（参数不确定性），又要应对神经网络近似带来的误差（近似不确定性），而现有方法对这两类不确定性的联合量化几乎是空白。为此，研究者提出了一种叫做 NN-aPC 的新框架：先用任意多项式混沌展开（arbitrary Polynomial Chaos, aPC）把随机解分解为一系列确定性模态函数，再为每个模态函数训练一个独立的物理信息神经网络来学习它；与此同时，引入 Dropout 技术在训练中随机屏蔽神经元，以此估计网络近似本身的不确定度，并据此设计了一套主动学习策略，自动决定在哪里加装新传感器以最高效地提升预测精度。在随机泊松方程（正问题）和随机椭圆方程（逆问题）上的数值实验表明，NN-aPC 能够准确重建解的统计特征，主动学习进一步以最少的新数据显著降低预测误差。这项工作首次将参数不确定性与近似不确定性统一纳入物理驱动深度学习框架，为多维随机偏微分方程的数据高效求解提供了一条实用路径。
原文 arXiv:1809.08327;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1809.08327v1