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.

翻译：基于深度学习的物理信息神经网络（PINNs）和DeepONets等方法在求解偏微分方程约束优化（PDECO）问题上展现出潜力。然而，现有方法难以处理那些对优化目标具有复杂或非线性依赖关系的偏微分方程约束。本文提出一种新颖的双层优化框架，通过解耦目标与约束的优化来应对该挑战。内层优化中，我们采用PINNs仅求解偏微分方程约束；外层优化中，我们基于隐函数定理（IFT）设计了一种高效且精确的超梯度近似方法——布罗伊登法。我们进一步提供了超梯度计算的理论解释与误差分析。在多个大规模非线性偏微分方程约束优化问题上的广泛实验表明，与强基线方法相比，我们的方法取得了最先进的性能。