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Multifidelity Gaussian process regression for solving nonlinear partial differential equations

topic: othertop score: 2released: 2026-05-12first surfaced: 2026-05-12arXivPDFmethods2026-05-12

Authors: Fatima-Zahrae El-Boukkouri, Josselin Garnier, Olivier Roustant

arXiv · PDF

Summary

The authors propose using multifidelity Gaussian process regression (cokriging) to solve nonlinear partial differential equations. The key idea is to learn a good kernel from cheap low-fidelity simulations, then use that learned kernel in a high-fidelity Gaussian process framework. They fit a differentiable non-stationary kernel to an empirical kernel from low-fidelity runs, derive a high-fidelity kernel with estimated hyperparameters, and construct a high-fidelity mean using the multifidelity framework. They demonstrate the approach on Burgers' equation.

Main takeaways:

  • Kernel-method PDE solvers depend heavily on kernel choice, which is hard to specify a priori for nonlinear equations.
  • The authors use cokriging (multifidelity Gaussian processes) to learn a kernel from cheap low-fidelity simulations.
  • The learned kernel is then used in a high-fidelity Gaussian process for solving the PDE.
  • The approach leverages empirical information from multifidelity simulations to avoid manual kernel tuning.
  • Demonstration on Burgers' equation shows the method works in practice.

Relevance

Not related to my LLM persona work — this is about solving physics PDEs using multifidelity Gaussian process methods. No connection to behavioral installation, prompting, or language model steering.

Abstract

arXiv:2605.10383v1 Announce Type: new Abstract: 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.