Parameterized Physics-informed Neural Networks for Parameterized PDEs

TL;DR

This paper introduces parameterized physics-informed neural networks (P2INNs) that efficiently model solutions to parameterized PDEs by encoding PDE parameters, improving accuracy and robustness over existing methods.

Contribution

The paper proposes P2INNs, a novel extension of PINNs that explicitly encode PDE parameters, enabling efficient and accurate solutions for parameterized PDEs.

Findings

P2INNs outperform baseline methods in accuracy.

P2INNs are more parameter-efficient.

P2INNs overcome known failure modes of PINNs.

Abstract

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Reynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P$^2$INNs). P$^2$INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P$^2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known "failure modes".