  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.
