Scalable Gaussian process modeling of parametrized spatio-temporal fields

Srinath Dama; Prasanth B. Nair

cs.LGFeb 27, 2026

Scalable Gaussian process modeling of parametrized spatio-temporal fields

Authors: Srinath Dama, Prasanth B. Nair

Organizations: University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, M3H 5T6, Ontario, Canada

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.

Figures & tables

Panel of Figure 1 — Figure 1: Illustration of the geometric transformation: mapping a parametrized physical domain $\Omega(\bm{\mu}_{\Omega})$ (left) to a fixed reference domain $\tilde{\Omega}$ (right) to enable tensor-product grid embedding.

Panel of Figure 2 — 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.

Method	Relative error
	$\bm{\mu}^{(1)}_{test}$	$\bm{\mu}^{(2)}_{test}$
Deep-LSPG ( $p=20$ )	0.0009	0.001
Proposed GP (DPK-Matérn- $5/2$ )	0.0033	0.0029
Deep-Galerkin ( $p=20$ )	0.015	0.013
Proposed GP (Matérn- $5/2$ )	0.0211	0.0078
POD-Galerkin ( $p=20$ )	0.032	0.03

Table 1: 1D Burgers’ problem. Relative test errors for the test snapshots corresponding to the parameters

\bm{\mu}^{(1)}_{test}=(4.3,0.021)

and

\bm{\mu}^{(2)}_{test}=(5.15,0.0285)

are presented. Errors for the physics-based methods (POD-Galerkin, POD-LSPG, Deep-Galerkin, and Deep-LSPG) for reduced dimension

p=20

are obtained from Figure 3 in [ 25 ] .

Panel of Figure 3 — Figure 3: 1D Burgers’ problem. For test parameter $\bm{\mu}^{(1)}_{test}=[4.3,0.021]$ , 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.

Panel of Figure 4 — Figure 4: 1D Burgers’ problem. For test parameter $\bm{\mu}^{(2)}_{test}=[5.15,0.0285]$ , 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.

Method	Relative error
Method	training	testing
Proposed GP (DPK)	0.0181	0.0326
Geo-FNO (O-mesh)	0.0344	0.0363
FNO interpolation	0.0314	0.0508
DeepONet	0.0528	0.0965

Table 2: Elasticity problem. Relative errors for the proposed GP framework compared with ML-based operator methods, including Geo-FNO [ 26 ] and DeepONet [ 30 ] . Best testing results are highlighted in bold.

Panel of Figure 5 — 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.

Method	Airfoil		Pipe
Method	training	testing	training	testing
Geo-FNO	0.0134	0.0138	0.0047	0.0067
Proposed GP (DPK)	0.0051	0.0142	0.0062	0.0131
Proposed GP (Matérn- $5/2$ )	0.0065	0.0316	0.0067	0.0171
FNO interpolation	0.0287	0.0421	0.0083	0.0151
UNet interpolation	0.0039	0.0519	0.0109	0.0182

Table 3: Airfoil and pipe problems. Relative errors comparing the proposed GP framework with ML-based operator methods, including Geo-FNO [ 26 ] and FNO [ 27 ] . Best testing results are highlighted in bold.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Problem	Learning rate	Maximum training iterations
1D unsteady Burgers’	$10^{-2}$	1000
Hyper-elastic problem	$1.5\times 10^{-2}$	450
2D Transonic flow	${10^{-2}}^{\dagger}$	368
Navier-Stokes Equation (Pipe)	${10^{-2}}^{\dagger}$	900

Appendix

Table 4: Learning rate and maximum number of training iterations for all the problems. Learning rate with

(\cdot)^{\dagger}

denotes the initial learning rate. For the transonic flow and pipe flow test problems, we employed a step learning rate decay scheduler with a step size of 100 and a decay factor of 0.8.

Panel of Figure 6 — Figure 6: The subdomains and the boundary conditions for the linear elastic block problem [ 15 ] are shown.

Panel of Figure 7 — Figure 7: Elastic block problem. Convergence plot comparing the mean relative test errors corresponding to 1000 test snapshots for the proposed GP framework (using the deep product kernel (DPK)) with POD-GPR and physics-based POD-Galerkin with reduced dimension $p=20$ .

Panel of Figure 8 — Figure 8: Elastic block problem. Box plot of relative test errors on 1000 test parameters for models trained on an increasing number of snapshots ( $N=50,100,200,500,$ $1000,2000,5000$ ).

Gaussian Processes

Gaussian Processes are probabilistic models of functions that use kernels to make predictions with explicit uncertainty.

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Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA 92617

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