Inferring solutions of differential equations using noisy multi-fidelity data

TL;DR

This paper introduces a probabilistic machine learning framework that infers solutions to differential equations from noisy, multi-fidelity data, providing uncertainty quantification and adaptive refinement without traditional discretization.

Contribution

It develops Gaussian process-based algorithms for linear differential equations that handle scarce, noisy, multi-fidelity data, bypassing numerical discretization and stability issues.

Findings

Provides uncertainty quantification for solutions.

Enables adaptive solution refinement via active learning.

Scalable to high-dimensional problems.

Abstract

For more than two centuries, solutions of differential equations have been obtained either analytically or numerically based on typically well-behaved forcing and boundary conditions for well-posed problems. We are changing this paradigm in a fundamental way by establishing an interface between probabilistic machine learning and differential equations. We develop data-driven algorithms for general linear equations using Gaussian process priors tailored to the corresponding integro-differential operators. The only observables are scarce noisy multi-fidelity data for the forcing and solution that are not required to reside on the domain boundary. The resulting predictive posterior distributions quantify uncertainty and naturally lead to adaptive solution refinement via active learning. This general framework circumvents the tyranny of numerical discretization as well as the consistency and stability issues of time-integration, and is scalable to high-dimensions.