Multi-Fidelity High-Order Gaussian Processes for Physical Simulation
This addresses the problem of expensive high-fidelity PDE simulations for researchers and engineers in computational physics, offering a novel method that integrates multi-fidelity data more effectively than existing approaches.
The paper tackles the high cost of solving partial differential equations (PDEs) in physical simulation by proposing Multi-Fidelity High-Order Gaussian Process (MFHoGP), which leverages multi-fidelity examples to predict high-dimensional outputs, achieving scalability to millions of outputs without sparse approximations.
The key task of physical simulation is to solve partial differential equations (PDEs) on discretized domains, which is known to be costly. In particular, high-fidelity solutions are much more expensive than low-fidelity ones. To reduce the cost, we consider novel Gaussian process (GP) models that leverage simulation examples of different fidelities to predict high-dimensional PDE solution outputs. Existing GP methods are either not scalable to high-dimensional outputs or lack effective strategies to integrate multi-fidelity examples. To address these issues, we propose Multi-Fidelity High-Order Gaussian Process (MFHoGP) that can capture complex correlations both between the outputs and between the fidelities to enhance solution estimation, and scale to large numbers of outputs. Based on a novel nonlinear coregionalization model, MFHoGP propagates bases throughout fidelities to fuse information, and places a deep matrix GP prior over the basis weights to capture the (nonlinear) relationships across the fidelities. To improve inference efficiency and quality, we use bases decomposition to largely reduce the model parameters, and layer-wise matrix Gaussian posteriors to capture the posterior dependency and to simplify the computation. Our stochastic variational learning algorithm successfully handles millions of outputs without extra sparse approximations. We show the advantages of our method in several typical applications.