Composing Partial Differential Equations with Physics-Aware Neural Networks

TL;DR

This paper presents FINN, a neural network that combines physical knowledge and learning to accurately model complex PDEs, outperforming existing models in accuracy, efficiency, and generalization, including real-world data scenarios.

Contribution

FINN introduces a novel compositional approach to integrate physical PDE constituents into neural networks, enhancing accuracy and generalization with fewer parameters.

Findings

FINN outperforms pure machine learning models in accuracy and parameter efficiency.

FINN demonstrates superior out-of-distribution generalization beyond initial conditions.

FINN effectively models real-world diffusion-sorption data, revealing unknown physical parameters.

Abstract

We introduce a compositional physics-aware FInite volume Neural Network (FINN) for learning spatiotemporal advection-diffusion processes. FINN implements a new way of combining the learning abilities of artificial neural networks with physical and structural knowledge from numerical simulation by modeling the constituents of partial differential equations (PDEs) in a compositional manner. Results on both one- and two-dimensional PDEs (Burgers', diffusion-sorption, diffusion-reaction, Allen--Cahn) demonstrate FINN's superior modeling accuracy and excellent out-of-distribution generalization ability beyond initial and boundary conditions. With only one tenth of the number of parameters on average, FINN outperforms pure machine learning and other state-of-the-art physics-aware models in all cases -- often even by multiple orders of magnitude. Moreover, FINN outperforms a calibrated physical model when approximating sparse real-world data in a diffusion-sorption scenario, confirming its generalization abilities and showing explanatory potential by revealing the unknown retardation factor of the observed process.