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 of 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 benefit 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. The code is available at https://github.com/yuan-yin/APHYNITY.

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Cardiac electrophysiology (EP) models achieved good progress in simulating cardiac electrical activity. However numerical issues and computational times hamper clinical applicability of such models. Moreover, personalisation can still be challenging and model errors can be difficult to overcome. On the other hand, deep learning methods achieved impressive results but suffer from robustness issues in healthcare due to their lack of physiological knowledge. We propose a novel approach which is based on deep learning in order to replace numerical integration of partial differential equations. This has the advantage to directly learn spatio-temporal correlations, which increases stability. Moreover, once trained, solutions are very fast to compute. We present first results in state estimation based on few measurements and evaluate the forecasting power of the trained network. The proposed method performed very well on this preliminary evaluation. It opens up possibilities towards data-driven personalisation, to overcome model error by learning from the data.