Partial differential equations (PDEs) find applications across various scientific and engineering fields.There is a growing trend for integrating physics-aware machine learning models to solve PDEs.Among them, the Finite Volume Neural Network (FINN) has proven to be efficient in uncovering latent structures in data.This study explores the capabilities of FINN in the investigation of shallow-water equations, which simulate wave dynamics in coastal regions.Specifically, we investigate the efficacy of FINN in reconstructing underwater topography.We find that FINN excels at inferring topography solely from wave dynamics, stressing the importance of application-specific inductive bias in neural network architectures.

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.