PDE-GAN for solving PDE optimal control problems more accurately and efficiently

Yuan-dong Cao, Yi-fan Dai, Chi Chiu SO, Jun-Min Wang

Keywords: Optimal control, deep learing, PINNs, GANs

TL;DR: Using a Generative Adversarial Network (GAN) to train a Physics-Informed Neural Network (PINN) to solve the problem of weight selection in PDE optimal control problems.

Abstract:

PDE optimal control (PDEOC) problems aim to optimize the performance of physical systems constrained by partial differential equations (PDEs) to achieve desired characteristics. Such problems frequently appear in scientific discoveries and are of huge engineering importance. Physics-informed neural networks (PINNs) are recently proposed to solve PDEOC problems, but it may fail to balance the different competing loss terms in such problems. Our work proposes PDE-GAN, a novel approach that puts PINNs in the framework of generative adversarial networks (GANs) “learn the loss function” to address the trade-off between the different competing loss terms effectively. We conducted detailed and comprehensive experiments to compare PDE-GANs with vanilla PINNs in solving four typical and representative PDEOC problems, namely, (1) boundary control on Laplace Equation, (2) time-dependent distributed control on Inviscous Burgers' Equation, (3) initial value control on Burgers' Equation with Viscosity, and (4) time-space-dependent distributed control on Burgers' Equation with Viscosity. Strong numerical evidence supports the PDE-GAN that it achieves the highest accuracy and shortest computation time without the need of line search which is necessary for vanilla PINNs.

Primary Area: other topics in machine learning (i.e., none of the above)

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Submission Number: 4575