Scalable Gaussian process modeling of parametrized spatio-temporal fields
arXiv:2603.00290v1 Announce Type: new 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
arXiv:2603.00290v1 Announce Type: new 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.
Executive Summary
This article introduces a novel Gaussian process (GP) framework for scalable modeling of parametrized spatio-temporal fields. The proposed framework leverages deep product kernels and Kronecker matrix algebra to achieve efficient training and uncertainty quantification. The method demonstrates competitive accuracy with operator learning methods on various benchmark problems, including the one-dimensional unsteady Burgers' equation. This work presents a promising tool for data-driven surrogate modeling, particularly when uncertainty estimates are required. The scalable GP framework has the potential to accelerate simulations in fields such as climate modeling, financial modeling, and materials science. The method's ability to handle large datasets and provide uncertainty estimates makes it an attractive option for applications where model uncertainty is crucial.
Key Points
- ▸ Scalable Gaussian process framework for parametrized spatio-temporal fields
- ▸ Deep product kernels and Kronecker matrix algebra for efficient training
- ▸ Uncertainty quantification with posterior variance computation at nearly the same cost as the posterior mean
Merits
Strength in Scalability
The proposed framework demonstrates near-linear complexity scaling with the total number of spatio-temporal grid points, making it an attractive option for large-scale simulations.
Improved Uncertainty Quantification
The method efficiently computes the posterior variance, enabling scalable uncertainty quantification, which is crucial for downstream tasks such as decision-making under uncertainty.
Demerits
Assumes Regular Grids
The method relies on Kronecker matrix algebra, which assumes a regular grid structure. This limitation may hinder its applicability to unstructured grids, which are common in many fields.
Computational Requirements
The proposed framework may still require significant computational resources, particularly for large-scale simulations, which could limit its adoption in certain fields.
Expert Commentary
The proposed scalable GP framework is a significant contribution to the field of spatio-temporal modeling. The method's ability to efficiently handle large datasets and provide uncertainty estimates makes it an attractive option for applications where model uncertainty is crucial. While the method assumes regular grids and may require significant computational resources, its competitive accuracy with operator learning methods and improved uncertainty quantification make it a promising tool for data-driven surrogate modeling. Future research should focus on extending the method to unstructured grids and exploring its applications in various fields.
Recommendations
- ✓ Further research is needed to extend the method to unstructured grids and explore its applications in various fields.
- ✓ The proposed framework should be compared with other scalable methods, such as tensor train decompositions, to evaluate its performance and limitations.