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Scalable Gaussian process modeling of parametrized spatio-temporal fields

Srinath Dama, Prasanth B. Nair

TL;DR

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 and achieves accuracy competitive with operator learning methods such as Fourier neural operators and deep operator networks is introduced.

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.

Scalable Gaussian process modeling of parametrized spatio-temporal fields

TL;DR

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 and achieves accuracy competitive with operator learning methods such as Fourier neural operators and deep operator networks is introduced.

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.
Paper Structure (26 sections, 2 theorems, 60 equations, 8 figures, 4 tables)

This paper contains 26 sections, 2 theorems, 60 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Background on Gaussian process regression
  5. Deep product kernels
  6. Gaussian process modeling of parametrized spatio-temporal fields on rectilinear spatial grids
  7. Rectilinear spatial grids and Kronecker-structured covariance
  8. Scalable computation of the marginal likelihood
  9. Posterior mean and variance computation
  10. Gaussian process modeling of parametrized spatio-temporal fields on general spatial grids
  11. Gappy rectilinear embedding and problem setup
  12. Scalable training and inference
  13. Efficient computation of the posterior mean
  14. Efficient computation of the posterior variance
  15. Extension to parametrized spatial domains
  16. ...and 11 more sections

Key Result

Lemma 1

The solution of $(\mathbf{K}_{\mathbf{Z}_r}+\sigma_n^2\mathbf{I})\boldsymbol{\alpha}_r=\mathbf{y}_r$ can be written as $\boldsymbol{\alpha}_r := {\mathbf{W}}\boldsymbol{\alpha}$, where $\boldsymbol{\alpha}\in\mathbb{R}^{N M N_t}$ is the solution of the Kronecker product structured system of linear a where $\mathbf{y}_r\in\mathbb{R}^{NN_rN_t}$ denotes the known observations at regular locations and

Figures (8)

  • Figure 1: Illustration of the geometric transformation: mapping a parametrized physical domain $\Omega(\boldsymbol{\mu}_{\Omega})$ (left) to a fixed reference domain $\tilde{\Omega}$ (right) to enable tensor-product grid embedding.
  • Figure 2: 1D Burgers' problem. Predicted solutions using the proposed GP framework with DPK and Matérn-$5/2$ based product structured kernel at times $t = 3.5, 7.0, 10.5,$ and $14$ are shown.
  • Figure 3: 1D Burgers' problem. For test parameter $\boldsymbol{\mu}^{(1)}_{test}=[4.3, 0.021]$, predicted solutions and confidence bounds ($\pm 2\sigma$) using the proposed GP framework with DPK (left) and Matérn-$5/2$ based product kernel (right) at times $t = 3.5, 7.0, 10.5,$ and $14$ are shown.
  • Figure 4: 1D Burgers' problem. For test parameter $\boldsymbol{\mu}^{(2)}_{test}=[5.15, 0.0285]$, predicted solutions and confidence bounds ($\pm 2\sigma$) using the proposed GP framework with DPK (left) and Matérn-$5/2$ based product kernel (right) at times $t = 3.5, 7.0, 10.5,$ and $14$ are shown.
  • Figure 5: Elasticity problem. Comparison of the stress field predicted by the proposed GP framework (DPK) with the high-fidelity solution for a test geometry in the reference domain.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof