Scalable Gaussian process modeling of parametrized spatio-temporal fields

2026-02-27

AI Summary

This work addresses the challenge of data-driven modeling of parametric spatiotemporal fields over fixed or parameter-dependent domains, where conventional Gaussian processes struggle to scale to large grids and lack efficient uncertainty quantification. The authors propose a scalable Gaussian process framework based on deep product kernels, leveraging Kronecker matrix algebra to achieve near-linear training complexity. The method enables continuous predictions at arbitrary spatiotemporal coordinates and efficient computation of posterior variances at the same cost as the posterior mean, on both structured and unstructured grids. This significantly overcomes the scalability limitations of traditional Gaussian processes. Experiments demonstrate that the approach matches the accuracy of Fourier Neural Operators and Deep Operator Networks, and outperforms projection-based reduced-order models on the one-dimensional unsteady Burgers equation, confirming its effectiveness as an efficient surrogate model with built-in uncertainty quantification.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Gaussian process
• spatio-temporal fields
• scalable modeling
• uncertainty quantification
• parametrized systems

Innovation

• Scalable Gaussian Process
• Deep Product Kernel
• Spatio-temporal Modeling
• Uncertainty Quantification
• Kronecker Algebra