  Forecasting complex dynamical phenomena in settings where only partial
knowledge of their dynamics is available is a prevalent problem across various
scientific fields. While purely data-driven approaches are arguably
insufficient in this context, standard physical modeling based approaches tend
to be over-simplistic, inducing non-negligible errors. In this work, we
introduce the APHYNITY framework, a principled approach for augmenting
incomplete physical dynamics described by differential equations with deep
data-driven models. It consists in decomposing the dynamics into two
components: a physical component accounting for the dynamics for which we have
some prior knowledge, and a data-driven component accounting for errors of the
physical model. The learning problem is carefully formulated such that the
physical model explains as much of the data as possible, while the data-driven
component only describes information that cannot be captured by the physical
model, no more, no less. This not only provides the existence and uniqueness
for this decomposition, but also ensures interpretability and benefits
generalization. Experiments made on three important use cases, each
representative of a different family of phenomena, i.e. reaction-diffusion
equations, wave equations and the non-linear damped pendulum, show that
APHYNITY can efficiently leverage approximate physical models to accurately
forecast the evolution of the system and correctly identify relevant physical
parameters. Code is available at https://github.com/yuan-yin/APHYNITY .
