Scalable Gaussian process modeling of parametrized spatio-temporal fields

TL;DR

This paper presents a scalable Gaussian process framework with deep kernels for modeling parametrized spatio-temporal fields, enabling accurate predictions and uncertainty quantification at arbitrary coordinates with nearly linear computational complexity.

Contribution

The authors introduce a novel scalable GP framework using deep product kernels and Kronecker algebra, allowing efficient learning and uncertainty quantification for complex spatio-temporal data.

Findings

Achieves accuracy comparable to Fourier neural operators and deep operator networks.

Surpasses reduced-order models in accuracy for Burgers' equation.

Provides efficient uncertainty estimates with minimal additional computational cost.

Abstract

We introduce a scalable Gaussian process (GP) framework with deep product kernels for data-driven learning of parametrized spatio-temporal fields over fixed or parameter-dependent domains. The proposed framework learns a continuous representation, enabling predictions at arbitrary spatio-temporal coordinates, independent of the training data resolution. We leverage Kronecker matrix algebra to formulate a computationally efficient training procedure with complexity that scales nearly linearly with the total number of spatio-temporal grid points. A key feature of our approach is the efficient computation of the posterior variance at essentially the same computational cost as the posterior mean (exactly for Cartesian grids and via rigorous bounds for unstructured grids), thereby enabling scalable uncertainty quantification. Numerical studies on a range of benchmark problems demonstrate that the proposed method achieves accuracy competitive with operator learning methods such as Fourier neural operators and deep operator networks. On the one-dimensional unsteady Burgers' equation, our method surpasses the accuracy of projection-based reduced-order models. These results establish the proposed framework as an effective tool for data-driven surrogate modeling, particularly when uncertainty estimates are required for downstream tasks.