Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradients
Abstract
Deep learning based approaches like Physics-informed neural networks (PINNs) and DeepONets have shown promise on solving PDE constrained optimization (PDECO) problems. However, existing methods are insufficient to handle those PDE constraints that have a complicated or nonlinear dependency on optimization targets. In this paper, we present a novel bi-level optimization framework to resolve the challenge by decoupling the optimization of the targets and constraints. For the inner loop optimization, we adopt PINNs to solve the PDE constraints only. For the outer loop, we design a novel method by using Broyden's method based on the Implicit Function Theorem (IFT), which is efficient and accurate for approximating hypergradients. We further present theoretical explanations and error analysis of the hypergradients computation. Extensive experiments on multiple large-scale and nonlinear PDE constrained optimization problems demonstrate that our method achieves state-of-the-art results compared with strong baselines.
Cite
@article{arxiv.2209.07075,
title = {Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradients},
author = {Zhongkai Hao and Chengyang Ying and Hang Su and Jun Zhu and Jian Song and Ze Cheng},
journal= {arXiv preprint arXiv:2209.07075},
year = {2023}
}