Multifidelity Gaussian process regression for solving nonlinear partial differential equations
For researchers solving PDEs with kernel methods, this work provides a way to leverage multifidelity data to improve kernel learning, though it is an incremental extension of existing cokriging techniques.
The authors propose a multifidelity Gaussian process regression method for solving nonlinear PDEs, using cokriging to learn kernels from low- and high-fidelity simulations. The method is demonstrated on Burgers' equation, achieving improved accuracy over single-fidelity approaches.
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.