mlsanityData to 7 Oct 2026

A unified perspective of Gaussian process approximation for differential equations

arXiv 2607.06292 ↗First posted 7 Jul 2026math.NA cs.CE cs.NAcited by 0

Not on the map. Why?

Abstract (arXiv)

The use of Gaussian processes for approximating differential equations has expanded rapidly, leading to a growing, diverse, and fragmented body of numerical methods. We present a unified Bayesian perspective that places these techniques within a common probabilistic framework, based on a derivative matching interpretation for incorporating differential equation constraints into likelihood. This unified perspective supports both parameter estimation and solution approximation, and shows how a range of existing methods can be understood within it. This work aims to consolidate current developments and provide a foundation for future research.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 17 of its 31 references that are in the map. First posted on arXiv 7 Jul 2026.

Builds on

17 of 31 references.

References · 17

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 139
  3. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  4. 2017Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationscited by 72
  5. 2023Learning Physical Models that Can Respect Conservation Lawscited by 58
  6. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 40
  7. 2018Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspectivecited by 31
  8. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  9. 2020Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processescited by 18
  10. 2023Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Priorcited by 15
  11. 2020Physics Informed Deep Kernel Learningcited by 12
  12. 2023Gaussian processes for Bayesian inverse problems associated with linear partial differential equationscited by 9
5 more
  1. 2023Gaussian process learning of nonlinear dynamicscited by 5
  2. 2025Uncertainty-Aware Diagnostics for Physics-Informed Machine Learningcited by 3
  3. 2025PDE-DKL: PDE-constrained deep kernel learning in high dimensionalitycited by 2
  4. 2023PDE-constrained Gaussian process surrogate modeling with uncertain data locationscited by 1
  5. 2025Physics-based deep kernel learning for parameter estimation in high dimensional PDEscited by 1

Built on by

Not cited yet.