Probabilistic ODE Solutions in Millions of Dimensions

TL;DR

This paper introduces a probabilistic numerical method for solving high-dimensional ODEs efficiently, enabling uncertainty quantification in complex scientific problems involving millions of dimensions.

Contribution

It presents a novel approach to solving high-dimensional ODEs probabilistically, overcoming previous computational limitations by exploiting independence or Kronecker structure.

Findings

Efficient probabilistic solutions for ODEs with millions of dimensions.

Demonstrated applicability to discretized partial differential equations.

Achieved significant computational efficiency improvements.

Abstract

Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving {high-dimensional} ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems -- most importantly, the solution of discretised {partial} differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions.