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Physics-augmented Multi-task Gaussian Process for Modeling Spatiotemporal Dynamics

Xizhuo Zhang, Bing Yao

TL;DR

This work tackles predicting multi-physics spatiotemporal dynamics on complex 3D geometries with multiple interrelated outputs. It introduces the physics-augmented multi-task Gaussian Process (P-M-GP), which fuses a geometry-aware, Kronecker-structured M-GP with a physics-based regularization to enforce governing PDE constraints. The spatial kernel leverages Laplacian eigenfunctions on the geometry, the temporal kernel uses a Matérn-$\tfrac{3}{2}$ form, and the task dependencies are captured via a learnable low-rank kernel $\mathcal{K}_\text{f}=LL^\top$, enabling scalable posterior predictions through Kronecker algebra and SVD. A reaction-diffusion PDE on a 3D surface provides the physics residuals used in a weighted regularization term, and the hyperparameters are learned by jointly optimizing data fit and physics fidelity. Empirical results on 3D cardiac electrodynamics show that P-M-GP outperforms pure data-driven M-GP and PINN baselines, demonstrating robustness to sparse data and noise and illustrating the method's potential for broad applications in complex spatiotemporal systems.

Abstract

Recent advances in sensing and imaging technologies have enabled the collection of high-dimensional spatiotemporal data across complex geometric domains. However, effective modeling of such data remains challenging due to irregular spatial structures, rapid temporal dynamics, and the need to jointly predict multiple interrelated physical variables. This paper presents a physics-augmented multi-task Gaussian Process (P-M-GP) framework tailored for spatiotemporal dynamic systems. Specifically, we develop a geometry-aware, multi-task Gaussian Process (M-GP) model to effectively capture intrinsic spatiotemporal structure and inter-task dependencies. To further enhance the model fidelity and robustness, we incorporate governing physical laws through a physics-based regularization scheme, thereby constraining predictions to be consistent with governing dynamical principles. We validate the proposed P-M-GP framework on a 3D cardiac electrodynamics modeling task. Numerical experiments demonstrate that our method significantly improves prediction accuracy over existing methods by effectively incorporating domain-specific physical constraints and geometric prior.

Physics-augmented Multi-task Gaussian Process for Modeling Spatiotemporal Dynamics

TL;DR

This work tackles predicting multi-physics spatiotemporal dynamics on complex 3D geometries with multiple interrelated outputs. It introduces the physics-augmented multi-task Gaussian Process (P-M-GP), which fuses a geometry-aware, Kronecker-structured M-GP with a physics-based regularization to enforce governing PDE constraints. The spatial kernel leverages Laplacian eigenfunctions on the geometry, the temporal kernel uses a Matérn- form, and the task dependencies are captured via a learnable low-rank kernel , enabling scalable posterior predictions through Kronecker algebra and SVD. A reaction-diffusion PDE on a 3D surface provides the physics residuals used in a weighted regularization term, and the hyperparameters are learned by jointly optimizing data fit and physics fidelity. Empirical results on 3D cardiac electrodynamics show that P-M-GP outperforms pure data-driven M-GP and PINN baselines, demonstrating robustness to sparse data and noise and illustrating the method's potential for broad applications in complex spatiotemporal systems.

Abstract

Recent advances in sensing and imaging technologies have enabled the collection of high-dimensional spatiotemporal data across complex geometric domains. However, effective modeling of such data remains challenging due to irregular spatial structures, rapid temporal dynamics, and the need to jointly predict multiple interrelated physical variables. This paper presents a physics-augmented multi-task Gaussian Process (P-M-GP) framework tailored for spatiotemporal dynamic systems. Specifically, we develop a geometry-aware, multi-task Gaussian Process (M-GP) model to effectively capture intrinsic spatiotemporal structure and inter-task dependencies. To further enhance the model fidelity and robustness, we incorporate governing physical laws through a physics-based regularization scheme, thereby constraining predictions to be consistent with governing dynamical principles. We validate the proposed P-M-GP framework on a 3D cardiac electrodynamics modeling task. Numerical experiments demonstrate that our method significantly improves prediction accuracy over existing methods by effectively incorporating domain-specific physical constraints and geometric prior.
Paper Structure (14 sections, 40 equations, 5 figures)

This paper contains 14 sections, 40 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Research Background
  3. Research Methodology
  4. Spatiotemporal M-GP Modeling
  5. Posterior Prediction
  6. Physics-augmented M-GP
  7. Physics-based Spatiotemporal Reaction-diffusion Model
  8. Physics-Augmented Learning
  9. Numerical Experiments
  10. Prediction under Simulation Protocol I
  11. Prediction under Simulation Protocol II
  12. Conclusions
  13. Details of Eq. \ref{['eq:dK_dt']}
  14. Details of Eq. \ref{['eq:cross-val']}

Figures (5)

  • Figure 1: Flowchart of the proposed methodology. The P-M-GP framework is developed to model complex spatiotemporal dynamics by designing effective task-spatial-temporal correlation kernels and integrating physics-based knowledge. The physics knowledge incorporation is achieved by augmenting the physics-based loss, $\mathcal{L}_{\text{phy}}$, into the data-driven loss, $\mathcal{L}_{\text{d}}$, to respect the underlying physics-based principle.
  • Figure 2: Prediction results under Simulation Protocol I: (a) Reference mapping of task-u and task-v under Protocol I at time point $t=800$. (b) Estimated mappings by M-GP, PINN ($N_\text{col}=200$ or $500$) and our P-M-GP model ($N_\text{col}=200$ or $500$) under different training dataset sizes ($|\mathcal{X}_\text{tr}| = 50, 75$) and noise levels ($\sigma_\xi = 0.01, 0.02$) at time $t=800$. (c) Bar chart comparing the aggregated $RE$ from 3 replications.
  • Figure 3: Temporal evolution of cardiac electrodynamics at spatial location $\boldsymbol{x}_{599}$ under Protocol I. The predictions from M-GP, PINN ($N_\text{col}=500$) and P-M-GP ($N_\text{col}=500$) are compared under varying training data sizes ($|\mathcal{X}_\text{tr}| = 50, 75$) and noise levels: (a) $\sigma_\xi = 0.01$, (b) $\sigma_\xi = 0.02$.
  • Figure 4: Prediction results under Protocol II: (a) Ground-truth visualization at $t=800$ under Simulation Protocol II. (b) Predicted distributions from M-GP, PINN ($N_\text{col}=200$ or $500$) and P-M-GP ($N_\text{col}=200$ or $500$) using varying training sizes ($|\mathcal{X}_\text{tr}| = 50, 75$) and noise conditions ($\sigma_\xi = 0.01, 0.02$) at time $t=800$. White dotted lines are marked for better visualization. (c) Comparison of the aggregated $RE$ between P-M-GP and M-GP averaged over 3 trials.
  • Figure 5: Comparison of temporal dynamics under Protocol II at spatial location $\boldsymbol{x}_{636}$ between M-GP, PINN ($N_\text{col}=500$) and P-M-GP ($N_\text{col}=500$) predictions across different training sample sizes ($|\mathcal{X}_\text{tr}|=50, 75$) and noise conditions: (a) $\sigma_\xi = 0.01$ and (b) $\sigma_\xi = 0.02$.