Multifidelity Gaussian process regression for solving nonlinear partial differential equations
Fatima-Zahrae El-Boukkouri, Josselin Garnier, Olivier Roustant
TLDR
This paper proposes a multifidelity Gaussian process regression method using cokriging for learning optimal kernels to solve nonlinear PDEs efficiently.
Key contributions
- Proposes a multifidelity kernel learning approach for PDEs using cokriging.
- Fits a differentiable non-stationary kernel to empirical low-fidelity simulation data.
- Derives a high-fidelity kernel and mean from the multifidelity framework.
- Integrates learned kernels into a Gaussian process for solving nonlinear PDEs.
Why it matters
Kernel methods for PDEs are powerful but sensitive to kernel choice. This work offers a novel multifidelity approach to learn optimal kernels, improving accuracy and efficiency. It provides a robust framework for solving complex nonlinear PDEs.
Original 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.
📬 Weekly AI Paper Digest
Get the top 10 AI/ML arXiv papers from the week — summarized, scored, and delivered to your inbox every Monday.