Learning the solution operator of a nonlinear parabolic equation using physics informed deep operator network

TL;DR

This paper introduces a physics-informed deep operator network (PI-DeepONet) that efficiently learns the solution operator of nonlinear parabolic equations, reducing the need for retraining across different parameters.

Contribution

It presents a novel physics-informed neural network approach that directly approximates the solution operator of complex PDEs, enhancing flexibility and efficiency.

Findings

Successfully approximates solution operators for nonlinear parabolic equations.

Reduces computational cost compared to traditional methods.

Demonstrates robustness across varying parameters.

Abstract

This study focuses on addressing the challenges of solving analytically intractable differential equations that arise in scientific and engineering fields such as Hamilton-Jacobi-Bellman. Traditional numerical methods and neural network approaches for solving such equations often require independent simulation or retraining when the underlying parameters change. To overcome this, this study employs a physics-informed DeepONet (PI-DeepONet) to approximate the solution operator of a nonlinear parabolic equation. PI-DeepONet integrates known physics into a deep neural network, which learns the solution of the PDE.