  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.
