CONFIDE: Contextual Finite Differences Modelling of PDEs

We introduce a method for inferring an explicit PDE from a data sample generated by previously unseen dynamics, based on a learned context. The training phase integrates knowledge of the form of the equation with a differential scheme, while the inference phase yields a PDE that fits the data sample and enables both signal prediction and data explanation. We include results of extensive experimentation, comparing our method to SOTA approaches, together with ablation studies that examine different flavors of our solution in terms of prediction error and explainability. Many scientific fields use the language of Partial Differential Equations (PDEs; Evans, 2010) to describe the physical laws governing observed natural phenomena with spatio-temporal dynamics. Typically, a PDE system is derived from first principles and a mechanistic understanding of the problem after experimentation and data collection by domain experts of the field. Well-known examples for such systems include Navier-Stokes and Burgers' equations in fluid dynamics, Maxwell's equations for electromagnetic theory, and Schrödinger's equations for quantum mechanics. Solving a PDE model could provide users with crucial information on how a signal evolves over time and space, and could be used for both prediction and control tasks. While creating PDE-based models holds great value, it is still a difficult task in many cases. For many complex real-world phenomena, we might only know some of the dynamics of the system. For example, an expert might tell us that a heat equation PDE has a specific functional form but we do not know the values of the diffusion and drift coefficient functions.