PhysNODE: Fusion of data and expert knowledge for modeling dynamical systems

Building a representative model of a complex system remains a highly challenging problem. While by now there is basic understanding of most physical domains, model design is often hindered by lack of detail, for example concerning model dimensions or its relevant constraints. Here we present a novel model-building approach -- physNODE -- augmenting basic system descriptions, based on expert knowledge in the form of ordinary differential equations, with continuous adjoint sensitivity analysis related to artificial neural network principles, based on observable data. With this we have created a general tool, that can be applied to any physical system described by ordinary differential equations. PhysNODE allows validating or extending the initial description, for example with different variables and constraints. This way one arrives at a better-optimised, representative low-dimensional model, which can fit existing data and predict novel experimental outcomes. We validate our method on five, quite different problem domains. (1) Kolmogorov model: Lotka Volterra model where we show the application of physNODE to continuous-time Markov processes and the performance of physNODE when working with noisy data. (2) Particle model: Interactive N-body, is a demonstration of the scalability of physNODE and that even interactive system with a high number of elements can be reconstructed with high precision. (3) Excitable media (heart dynamics) by the Bueno\-Orovio\-Cherry\-Fenton model: PhysNODE can reconstruct the parameters of a high-dimensional model and fields for diffusion driven system for chaotic behaviour. (4) Fluid dynamics: Rayleigh-B\'enard Convection where a complete unknown field, the temperature, can be extracted from only velocity data. (5) New experimental data of Zebrafish embryogenesis: This is a case where we extend existing models with new variables.