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.