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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 thosePDE constraints that have a complicated or nonlinear dependency on optimizationtargets. In this paper, we present a novel bi-level optimization framework toresolve the challenge by decoupling the optimization of the targets andconstraints. For the inner loop optimization, we adopt PINNs to solve the PDEconstraints only. For the outer loop, we design a novel method by usingBroyden's method based on the Implicit Function Theorem (IFT), which isefficient and accurate for approximating hypergradients. We further presenttheoretical explanations and error analysis of the hypergradients computation.Extensive experiments on multiple large-scale and nonlinear PDE constrainedoptimization problems demonstrate that our method achieves state-of-the-artresults compared with strong baselines.

Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradients