Probabilistic ODE Solutions in Millions of Dimensions

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.Preprint.

Fenrir: Physics-Enhanced Regression for Initial Value Problems

“…That is, it is linear in the number of data points, in contrast to cubic complexity for standard Gaussian process regresison. Further speed-ups may be obtainable by exploiting structural simplifications for certain probabilistic solvers (Krämer et al, 2021).…”

“…A probabilistic numerics approach has also been developed for estimating time varying parameters in the context of latent force modelling (Schmidt et al, 2021). However, for the constant parameter problem, using linearised models can cause divergence in certain situations (Ljung, 1979).…”