Universal Differential Equations for Scientific Machine Learning

TL;DR

This paper introduces universal differential equations (UDEs) within the SciML ecosystem, unifying physical laws and data-driven models to efficiently solve complex scientific problems with advanced machine learning techniques.

Contribution

It presents UDEs as a versatile framework that integrates scientific models with machine learning, enabling broad applications and efficient computation in scientific machine learning.

Findings

UDEs can model biological mechanisms and Hamilton-Jacobi-Bellman equations.

The software handles stochasticity, delays, and implicit constraints.

The toolkit is optimized for stability, parallelism, and GPU acceleration.

Abstract

In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." In this manuscript we introduce the SciML software ecosystem as a tool for mixing the information of physical laws and scientific models with data-driven machine learning approaches. We describe a mathematical object, which we denote universal differential equations (UDEs), as the unifying framework connecting the ecosystem. We show how a wide variety of applications, from automatically discovering biological mechanisms to solving high-dimensional Hamilton-Jacobi-Bellman equations, can be phrased and efficiently handled through the UDE formalism and its tooling. We demonstrate the generality of the software tooling to handle stochasticity, delays, and implicit constraints. This funnels the wide variety of SciML applications into a core set of training mechanisms which are highly optimized, stabilized for stiff equations, and compatible with distributed parallelism and GPU accelerators.