# Multifidelity Gaussian process regression for solving nonlinear partial differential equations

arXiv: https://arxiv.org/abs/2605.10383 · HTML: https://mlsanity.com/paper/2605.10383/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 26 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 8 of 8)

- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Physics Informed Deep Kernel Learning](https://mlsanity.com/paper/2006.04976/) (2020)
- [Physics-informed kernel learning](https://mlsanity.com/paper/2409.13786/) (2024)
- [Physics-Information-Aided Kriging: Constructing Covariance Functions using Stochastic Simulation Models](https://mlsanity.com/paper/1809.03461/) (2018)
- [Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes](https://mlsanity.com/paper/2510.05568/) (2025)
- [General reproducing properties in RKHS with application to derivative and integral operators](https://mlsanity.com/paper/2503.15922/) (2025)

## Abstract (arXiv)

Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers' equation.