Probabilistic Exponential Integrators

TL;DR

This paper introduces probabilistic exponential integrators that improve stability and efficiency for stiff differential equations by incorporating linear dynamics into the prior, providing a probabilistic error estimate.

Contribution

The paper develops probabilistic exponential integrators that are L-stable and extend to non-linear systems via Jacobian-based semi-linearity, enhancing probabilistic numerics for stiff problems.

Findings

Demonstrated improved stability over existing probabilistic solvers.

Achieved better efficiency in solving stiff differential equations.

Provided probabilistic error estimates alongside solutions.

Abstract

Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator -- with the added benefit of providing a probabilistic account of the numerical error. The method is also generalized to arbitrary non-linear systems by imposing piece-wise semi-linearity on the prior via Jacobians of the vector field at the previous estimates, resulting in probabilistic exponential Rosenbrock methods. We evaluate the proposed methods on multiple stiff differential equations and demonstrate their improved stability and efficiency over established probabilistic solvers. The present contribution thus expands the range of problems that can be effectively tackled within probabilistic numerics.