Solving Differential Equations using Physics-Informed Deep Equilibrium Models

TL;DR

This paper proposes Physics-Informed Deep Equilibrium Models (PIDEQs) that integrate deep equilibrium models with physics-informed training to efficiently solve initial value problems of ordinary differential equations, demonstrating promising results on benchmark problems.

Contribution

The paper introduces PIDEQs, a novel approach combining DEQs and PINNs for solving IVPs, advancing the computational methods in scientific computing.

Findings

PIDEQs effectively solve the Van der Pol oscillator IVP.

The method demonstrates high efficiency and accuracy.

Hyperparameter analysis improves PIDEQ performance.

Abstract

This paper introduces Physics-Informed Deep Equilibrium Models (PIDEQs) for solving initial value problems (IVPs) of ordinary differential equations (ODEs). Leveraging recent advancements in deep equilibrium models (DEQs) and physics-informed neural networks (PINNs), PIDEQs combine the implicit output representation of DEQs with physics-informed training techniques. We validate PIDEQs using the Van der Pol oscillator as a benchmark problem, demonstrating their efficiency and effectiveness in solving IVPs. Our analysis includes key hyperparameter considerations for optimizing PIDEQ performance. By bridging deep learning and physics-based modeling, this work advances computational techniques for solving IVPs, with implications for scientific computing and engineering applications.