Physics-informed reduced order model with conditional neural fields

TL;DR

This paper introduces a physics-informed neural network framework called CNF-ROM that efficiently approximates solutions to parametrized PDEs by combining neural ODEs, coordinate-based networks, and exact boundary conditions.

Contribution

The study develops a novel physics-informed reduced-order model using conditional neural fields, integrating neural ODEs and exact boundary conditions with stability enhancements.

Findings

Effective in parameter extrapolation and interpolation

Accurate temporal extrapolation demonstrated

Outperforms traditional methods in PDE solution approximation

Abstract

This study presents the conditional neural fields for reduced-order modeling (CNF-ROM) framework to approximate solutions of parametrized partial differential equations (PDEs). The approach combines a parametric neural ODE (PNODE) for modeling latent dynamics over time with a decoder that reconstructs PDE solutions from the corresponding latent states. We introduce a physics-informed learning objective for CNF-ROM, which includes two key components. First, the framework uses coordinate-based neural networks to calculate and minimize PDE residuals by computing spatial derivatives via automatic differentiation and applying the chain rule for time derivatives. Second, exact initial and boundary conditions (IC/BC) are imposed using approximate distance functions (ADFs) [Sukumar and Srivastava, CMAME, 2022]. However, ADFs introduce a trade-off as their second- or higher-order derivatives become unstable at the joining points of boundaries. To address this, we introduce an auxiliary network inspired by [Gladstone et al., NeurIPS ML4PS workshop, 2022]. Our method is validated through parameter extrapolation and interpolation, temporal extrapolation, and comparisons with analytical solutions.