Pick-and-Mix Information Operators for Probabilistic ODE Solvers

TL;DR

This paper introduces a flexible probabilistic approach for solving differential equations that allows incorporating diverse information types, including higher-order derivatives and physical laws, leading to more accurate and meaningful solutions.

Contribution

It proposes a novel method to include general likelihood terms in probabilistic ODE solvers, directly using second-order equations and physical constraints for improved accuracy.

Findings

Including higher-order information improves solution accuracy.

Directly providing second-order equations enhances physical relevance.

Flexible information operators enable solving differential-algebraic equations.

Abstract

Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, instead of transforming the problem to first order. Additionally, by including higher-order information or physical conservation laws in the model, solutions become more accurate and more physically meaningful. Lastly, we demonstrate the utility of flexible information operators by solving differential-algebraic equations. In conclusion, the probabilistic formulation of numerical solvers offers a flexible way to incorporate various types of information, thus improving the resulting solutions.