Fully differentiable model discovery

Model discovery aims at autonomously discovering differential equations underlying a dataset. Approaches based on Physics Informed Neural Networks (PINNs) have shown great promise, but a fully-differentiable model which explicitly learns the equation has remained elusive. In this paper we propose such an approach by combining neural network based surrogates with Sparse Bayesian Learning (SBL). We start by reinterpreting PINNs as multitask models, applying multitask learning using uncertainty, and show that this leads to a natural framework for including Bayesian regression techniques. We then construct a robust model discovery algorithm by using SBL, which we showcase on various datasets. Concurrently, the multitask approach allows the use of probabilistic approximators, and we show a proof of concept using normalizing flows to directly learn a density model from single particle data. Our work expands PINNs to various types of neural network architectures, and connects neural network-based surrogates to the rich field of Bayesian parameter inference.