Composing Partial Differential Equations with Physics-Aware Neural Networks
This work addresses the challenge of accurately modeling partial differential equations for researchers in computational physics and machine learning, offering a novel hybrid approach with significant performance gains.
The paper tackled the problem of learning spatiotemporal advection-diffusion processes by introducing a compositional physics-aware neural network (FINN), which achieved superior modeling accuracy and out-of-distribution generalization, outperforming state-of-the-art models with only one tenth of the parameters on average and even revealing unknown physical factors in real-world data.
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.