Understanding on Physics-Informed DeepONet

Abstract

Partial differential equations (PDEs) play a pivotal role in mathematical analysis and modeling of dynamic processes across various disciplines of science and engineering. Machine learning (ML) techniques have emerged as a promising new approach to solving PDEs. Among them, Physics-Informed Neural Networks (PINNs) have garnered significant attention in numerous scientific and engineering studies. PINNs employ a single deep neural network to assimilate observational data with PDEs across the entire space-time of a physical system, subsequently yielding rapid solutions. However, a PINN may entail intricate analyses or computations and can be cost-intensive, depending on initial or boundary conditions and other input parameters. To address the limitations of the PINN, especially concerning resolution for nonlinear problems, the Physical-Informed Deep Operator Network (DeepONet) is introduced in this paper. The Physics-Informed DeepONet is a deep learning framework crafted to discern solution operators for any given PDEs, even in scenarios lacking paired input/output training data. The proposed framework is able to predict solutions for various types of parameterized PDEs much faster than conventional PDE solvers. Several cases confirm that this approach is effective in establishing previously unexplored paradigms for modeling/simulating nonlinear and non-equilibrium processes in science and engineering.