Composing Partial Differential Equations with Physics-Aware Neural Networks

Matthias Karlbauer, Timothy Praditia, Sebastian Otte, Sergey Oladyshkin, Wolfgang Nowak, Martin V. Butz

Keywords: physics-aware neural networks, partial differential equations, advection-diffusion equations, learning constituents, out-of-distribution generalization

Abstract: We introduce a compositional physics-aware 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 (Burger's, diffusion-sorption, diffusion-reaction) demonstrate FINN's superior process 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.

One-sentence Summary: A physics-aware neural network for compositional learning of constituents of partial differential equations demonstrating eminent generalization abilities.

Community Implementations: [ 2 code implementations](https://www.catalyzex.com/paper/composing-partial-differential-equations-with/code)