Fully differentiable model discovery

TL;DR

This paper introduces a fully differentiable approach for discovering differential equations from data by combining neural network surrogates with Sparse Bayesian Learning, and extends PINNs to new architectures including a novel Physics Informed Normalizing Flow.

Contribution

It presents a new fully differentiable model discovery method integrating neural networks with Bayesian learning, and introduces PINF for density modeling from particle data.

Findings

Robust model discovery algorithm demonstrated on various datasets.

Connection established between model discovery and multitask learning.

Proof-of-concept of PINF for density estimation from single particle data.

Abstract

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 integrating neural network-based surrogates with Sparse Bayesian Learning (SBL). This combination yields a robust model discovery algorithm, which we showcase on various datasets. We then identify a connection with multitask learning, and build on it to construct a Physics Informed Normalizing Flow (PINF). We present a proof-of-concept using a PINF 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.