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

Abstract

Recent advances in sensing and imaging technologies have enabled the acquisition of high-dimensional spatiotemporal data from complex geometric domains. However, predictive modeling of such systems remains challenging due to irregular spatial manifolds, coupled multi-output dynamics, limited observations, and the need for reliable uncertainty quantification. This paper presents a physics-augmented, geometry-aware, multi-task Gaussian Process (P-G-MGP) framework for spatiotemporal modeling. We develop a geometry-aware multi-task GP (G-MGP) to capture spatiotemporal structures and inter-task dependencies. To enhance model fidelity and robustness, we incorporate governing physical laws through a physics-based regularization scheme, thereby constraining predictions to be consistent with governing principles. Furthermore, our framework provides closed-form estimates of posterior variance, enabling calibrated uncertainty quantification for downstream decision-making. We validate P-G-MGP on 3D cardiac electrophysiological modeling, demonstrating superior predictive performance over existing methods by effectively incorporating geometric priors, multi-task interactions, and domain-specific physical constraints.

Note to Practitioners—

This article proposes a P-G-MGP framework designed for predictive modeling of spatiotemporal systems over complex geometric domains. A central feature of the proposed approach is its ability to capture interrelated physical variables while respecting underlying dynamical laws. This framework can provide more accurate and physically consistent predictions even when available data are limited or irregularly distributed. Although our case study focuses on 3D cardiac electrodynamics, the framework is broadly applicable to other engineering and scientific domains where accurate spatiotemporal prediction is critical, such as environmental monitoring, structural integrity assessment, fluid dynamics, and advanced manufacturing. By fusing geometry priors, multi-task relationships, and physics-based regularization, our framework achieves higher predictive fidelity and greater robustness in the presence of sparse or noisy measurements. These capabilities enable reliable decision support in real-world systems where the interplay of geometry and physics is indispensable.

I. Introduction

Spatiotemporal dynamic processes are fundamental to a wide array of scientific and engineering disciplines [1], governing complex phenomena such as cardiac electrophysiology [2]–[4], thermal-fluid dynamics in additive manufacturing [5], [6], and soil moisture and salinity transport in agricultural farmlands [7]. These processes are inherently characterized by interactions among system components evolving across space and time, often unfolding over unstructured 3D geometries with complex spatial-temporal dependencies. Additionally, many real-world spatiotemporal systems are inherently multi-physics, where distinct physical phenomena interact to drive system dynamics. For example, in cardiac electrodynamics, the propagation of electrical signals is intrinsically coupled with the tissue’s conductivity, the cell’s transmembrane potential, and recovery electric current, forming a multi-physics process that spans over the heart geometry [3]. In additive manufacturing, fabricating 3D structures involves tightly coupled thermal and mechanical processes, all of which directly impact the microstructural evolution and quality of the final part [8].

Recent advances in sensing and imaging technologies have revolutionized our ability to collect spatiotemporal signals, providing unprecedented visibility into complex dynamic systems [9], [10]. For example, in cardiac electrophysiology, modern multi-lead electrocardiograms (ECGs) and intracardiac catheter mapping techniques enable real-time monitoring of electrical signals both on the torso surface and within cardiac chambers, thereby informing personalized treatment strategies for arrhythmias [11]–[14]. In AM processes, thermal and optical sensing tools such as infrared cameras and photodiodes are used to track layer-wise temperature fields, melt pool dynamics, and phase transitions, offering insights critical for real-time quality control [15]. The massive sensor signals enhance the multifaceted information visibility and can be leveraged to facilitate a better understanding of dynamic systems.

Moreover, many spatiotemporal systems are governed by physical laws, e.g., reaction-diffusion equations, advection-diffusion dynamics, or Navier-Stokes equations, which offer a rich foundation for integrating mechanistic insights into data-driven modeling [16]. In cardiac systems, the propagation of electrical potentials is described by nonlinear reaction-diffusion partial differential equations (PDEs) that couple with anatomical and electrophysiological parameters [2], [3]. In AM processes, the transient temperature fields follow the heat conduction equation, with process-induced variability affecting melt pool dynamics and part quality [17], [18]. Deviations in these physical processes often signal system anomalies or failures, such as arrhythmic events in cardiac tissue due to disrupted electrical propagation. Thus, effective system monitoring, anomaly detection, and predictive control depend on developing unified modeling frameworks that integrate data-driven learning with physics-based priors, enabling robust and accurate prediction.

However, multi-physics spatiotemporal predictive modeling poses unique challenges due to complex geometries, high-dimensional correlated signals, and the need to respect governing physical laws:

(1) High-dimensional multi-task signals on complex geometries: Many spatiotemporal systems are defined over complex geometries including asymmetric structures, non-uniform meshes, and curved manifolds [19]–[25], which pose substantial obstacles for conventional modeling approaches. Additionally, real-world systems often involve multiple inter-connected tasks, necessitating methods that can simultaneously model diverse variables [26], and further enable downstream decision-making tasks such as anomaly detection.

(2) Lack of physical fidelity in purely data-driven learning: Purely data-driven approaches often neglect fundamental physical principles governing spatiotemporal systems. This disregard can result in predictions that violate inherent physical constraints, exhibit poor generalizability beyond training distributions, or fail under sparse data conditions where prior knowledge of system dynamics is critical [27], [28].

(3) Uncertainty quantification: Deterministic surrogate modeling typically provides only point estimates and requires additional calibration for trustworthy predictive uncertainty. This limitation is particularly critical when dealing with sparse or noisy measurements, as reliable uncertainty quantification is essential for downstream decision-making and for assessing confidence in predictions made on complex, nonstationary spatiotemporal systems.

In this paper, we present a physics-augmented, geometry-aware, multi-task Gaussian Process (P-G-MGP) framework for predictive modeling of multi-physics spatiotemporal systems. The specific contributions are as follows:

(1) Geometry-aware multi-task spatiotemporal GP (G-MGP): We adapt the geometry-aware spatiotemporal GP (G-ST-GP) [29] to incorporate geometric information of 3D systems into spatiotemporal modeling, and extend it with a multi-task GP (MGP) model to capture cross-task dependencies.

(2) Physics-augmented learning to improve prediction robustness: We propose an effective physics-augmented strategy to incorporate physics prior knowledge of the dynamic systems into G-MGP for spatiotemporal predictive modeling. This hybrid approach leverages governing physics (often expressed as PDEs) to constrain the model predictions for improving both physical fidelity and predictive robustness.

(3) Uncertainty quantification with decision support capability: Our P-G-MGP framework provides posterior variance estimates in closed form. The predictive uncertainty contracts at observed locations and expands in data-sparse regions, which can enable uncertainty-aware decision-making, such as adaptive sensing for efficient data acquisition.

(4) Structured complexity reduction for scalable training and inference: We design a structured computational pipeline that converts high-dimensional covariance operations into reusable orthogonal transforms, diagonal-block multiplications, and tensor-mode contractions. By exploiting Kronecker separability and kernel eigenstructure, the complexity is significantly reduced, enabling scalable training and inference for large-scale 3D multi-physics systems.

We validate our P-G-MGP framework by applying it to spatiotemporal predictive modeling of cardiac electrodynamics in a 3D ventricular geometry. Numerical experiments highlight the superior prediction performance of our P-G-MGP model compared with the standard existing methods.

II. Research Background

The rapid advancement in sensing technologies has led to an abundance of multi-source data streams, creating new challenges in signal integration and prediction across spatiotemporal domains [30]–[34]. The existing literature relevant to this study can be categorized into three main stages: spatiotemporal GPs, multi-task extensions, and physics-informed modeling.

Spatiotemporal GPs.

GPs [35], [36] offer a probabilistic framework for spatiotemporal inference and have been deployed across a wide range of domains, including cardiac electrophysiology, robotics, and advanced manufacturing [37]–[39]. These models capture correlations across space and time through carefully designed covariance functions, enabling prediction at unobserved locations and time points with uncertainty quantification. For example, Senanayake et al. developed a GP regression method to model complex space-time dependencies and predict the spread of influenza [40]. Aftab et al. built a spatiotemporal GP to capture critical features in internal reference point data for dynamic extended object tracking [41]. More recently, Zhang and Yao developed a G-ST-GP model that effectively integrates temporal correlations and geometric manifold features for predictive modeling on irregular 3D geometries [29]. While conventional single-task spatiotemporal GPs have demonstrated success in modeling individual processes, they are inherently limited when faced with systems characterized by multiple interdependent variables. In particular, single-task GPs ignore the underlying correlations among related tasks and prevent information sharing, resulting in limited predictive accuracy and robustness.

Multi-task Extensions.

Multi-task extensions of GPs (MGPs) address this limitation by sharing information across correlated tasks through structured covariance matrices that model both intra-task and cross-task correlations [42]. This framework has enabled successful applications in robotics, environmental sensing, additive manufacturing, and biomedicine [43]–[45]. Building on these foundations, recent works have extended MGPs to spatiotemporal domains. For example, Akbari and Zhu integrated MGP with Kalman filter for dynamically tracking multiple dependent extended targets [46]. Hamelijnck et al. developed a multi-resolution MGP to capture varying sampling resolutions and noise levels for air pollution forecasting [47]. Gilanifar et al. leveraged an MGP to jointly model dynamic smart-meter readings, weather data, and traffic patterns from two residential areas in Florida, achieving improved prediction accuracy [48]. However, most existing spatiotemporal MGPs assume purely data-driven formulations and do not incorporate the physics-based principles governing the system dynamics. This limits their applicability in scenarios with sparse observations, where prior physical knowledge is essential for maintaining predictive fidelity.

Physics-informed Modeling.

To address the shortcomings of purely data-driven approaches, researchers have increasingly sought to embed mechanistic knowledge, typically expressed as algebraic equations, ordinary differential equations (ODEs), or PDEs, into statistical and machine learning frameworks [49], [50]. For example, Physics-Informed Neural Networks (PINNs) have emerged as a popular framework for embedding PDE constraints directly into the learning process of deep neural networks [51]. PINNs have been applied successfully to a range of spatiotemporal problems such as cardiac electrodynamics, fluid flow modeling, and material deformation [52]–[54]. However, PINNs are typically constructed using standard feedforward architectures defined on Euclidean domains. Extending PINNs to handle spatiotemporal data distributed over non-Euclidean geometries such as curved manifolds and non-uniform meshes remains challenging [55]. Additionally, PINNs involve optimization of a very large number of network parameters, often in the order of hundreds of thousands to millions, which exacerbates training instability, demands substantial computational resources, and makes convergence sensitive to hyperparameter tuning [56]. Moreover, PINNs inherently provide point estimates without calibrated uncertainty quantification, limiting their reliability for decision-making under data sparsity and noise.

Physics-informed GPs represent a powerful alternative approach for incorporating physical knowledge into probabilistic modeling [57]. By treating governing differential operators as linear transformations of GP priors, Raissi and Karniadakis introduced the Hidden Physics Model framework, which can solve nonlinear PDEs from limited observations [58]. Moreover, they introduced the concept of numerical GPs which embed time-stepping operators directly into the covariance, extending to general transient systems [59]. Recent advances have cast physics-informed GP regression as a probabilistic analogue of weighted-residual/Galerkin schemes, yielding rigorous discretization-error estimates [60]. Furthermore, the design of problem-specific covariance kernels has been shown to systematically encode boundary conditions directly into the GP prior, ensuring physical consistency of the inferred solutions [61]. These developments demonstrate the growing maturity of physics-informed GPs.

Despite these advances, very little work has been done to integrate these three strands into a unified framework. This gap motivates the proposed P-G-MGP framework, which aims to unify spatiotemporal GP modeling, multi-task covariance coupling, and mechanistic regularization to improve predictive accuracy, robustness, and physical consistency in data-limited regimes within complicated 3D geometries.

III. Research Methodology

Fig. 1 shows our P-G-MGP framework, which consists of two main components. First, we propose the G-MGP to model spatiotemporal dynamics through the design of an effective task-spatial-temporal correlation kernel. This kernel captures structural relationships across tasks, spatial locations, and temporal instances by incorporating Laplacian spectral analysis of the 3D geometry and carefully constructed temporal and task covariance functions. Second, a physics-based constraint mechanism is incorporated into the G-MGP to enforce physical consistency. This is achieved by augmenting the learning objective with a physics-based loss to integrate domain knowledge for enhanced predictive accuracy.

Fig. 1:

Flowchart of the proposed methodology. P-G-MGP is developed to model spatiotemporal dynamics by designing effective task-spatial-temporal correlation kernels and integrating physics knowledge. The physics incorporation is achieved by augmenting the physics-based loss, ${ℒ}_{\text{phy}}$, into the data-driven loss, ${ℒ}_{\text{d}}$, to respect the underlying physics-based principle.

A. Spatiotemporal G-MGP Modeling

The spatiotemporal G-MGP is designed to enable the integration of multiple tasks within a unified framework for high-dimensional predictive modeling. In our G-MGP framework, the variable $\mathit{q}(\mathit{x},t)\in {\mathbb{R}}^{N_{\text{f}}}$ represents the multi-variate system dynamics at a given location $\mathit{x}$ and time $t$, with $\mathit{y}(\mathit{x},t)\in {\mathbb{R}}^{N_{\text{f}}}$ denoting the corresponding measurements:

$$\mathit{y}(\mathit{x},t)=\mathit{q}(\mathit{x},t)+\mathit{ϵ}(\mathit{x},t),\mathit{q}~𝒢𝒫\left(\mathbf{0},{𝒦}_{\text{fst}}\right)$$ (1)

where $\mathit{ϵ}(\mathit{x},t)$ is a noise term or nugget effect, which follows a multi-variate Gaussian distribution: $\mathit{ϵ}(\mathit{x},t)~𝒩(\mathbf{0},\mathit{D})$, where $\mathit{D}$ is a diagonal matrix with elements defined as task-specific noise variance, i.e., $\mathit{D}=\text{diag}\left({\sigma }_{1,ϵ}^2,\dots ,{\sigma }_{N_{\text{f}},ϵ}^2\right)$. Additionally, to capture the multi-variate and spatiotemporal interactions, we design the kernel function for $\mathit{q}(\mathit{x},t)$ as ${𝒦}_{\text{fst}}≔{𝒦}_{\text{f}}\otimes {𝒦}_{\text{s}}\otimes {𝒦}_{\text{t}}$, which is a composite function that combines spatial ${𝒦}_{\text{s}}$, temporal ${𝒦}_{\text{t}}$, and task-related ${𝒦}_{\text{f}}$ kernels, each tailored to address specific aspects of correlation inherent in the spatiotemporal multi-physics dynamics. (Note that a detailed notation table is available in Supplementary Material S3.)

The training data, represented as ${\mathit{y}}_{\text{tr}}={\left[y\left(f,{\mathit{x}}_i,t_j\right)\right]}_{f\in ℱ,{\mathit{x}}_i\in {𝒳}_{\text{tr}},t_j\in {𝒯}_{\text{tr}}}\subset {\mathbb{R}}^{N_{\text{f}}{N}_{\text{s}}{N}_{\text{t}}}$, comprises observations collected at specified spatial locations ${𝒳}_{\text{tr}}=\left\{{\mathit{x}}_1,\dots ,{\mathit{x}}_{N_{\text{s}}}\right\}$ and time points ${𝒯}_{\text{tr}}=\left\{t_1,\dots ,t_{N_{\text{t}}}\right\}$ across different tasks $ℱ=\left\{f_1,\dots ,f_{N_{\text{f}}}\right\}$. Within the framework of G-MGP modeling, the marginal distribution of the training data is given by:

$${\mathit{y}}_{\text{tr}}~𝒩\left(\mathbf{0},{𝚺}_{\text{tr}}\right)$$ (2)

with ${𝚺}_{\text{tr}}={𝒦}_{\text{f}}(ℱ,ℱ)\otimes {𝒦}_{\text{s}}\left({𝒳}_{\text{tr}},{𝒳}_{\text{tr}}\right)\otimes {𝒦}_{\text{t}}\left({𝒯}_{\text{tr}},{𝒯}_{\text{tr}}\right)+D\otimes {\mathit{I}}_{N_{\text{s}}}\otimes {\mathit{I}}_{N_{\text{t}}}$ denoting the covariance of ${\mathit{y}}_{\text{tr}}$, where ${\mathit{I}}_{N_{\text{s}}}\left({\mathit{I}}_{N_{\text{t}}}\right)$ is an $N_{\text{s}}\times N_{\text{s}}\left(N_{\text{t}}\times N_{\text{t}}\right)$ identity matrix. Furthermore, given collected observations $\left[{\mathit{y}}_{\text{tr}};{𝒳}_{\text{tr}},{𝒯}_{\text{tr}}\right]$, the predictive distribution of the dynamics for the $f^{*}$-th task at an arbitrary coordinate $({\mathit{x}}^{*},t^{*})$ is:

$$\begin{gathered} q\left(f^{*},{\mathit{x}}^{*},t^{*}\right)\mid {\mathit{y}}_{\text{tr}};{𝒳}_{\text{tr}},{𝒯}_{\text{tr}}~𝒩\left({\mu }^{*},{\Sigma }^{*}\right), \\ {\mu }^{*}={𝒦}_{\text{fst}}^{*}{𝚺}_{\text{tr}}^{-1}{\mathit{y}}_{\text{tr}}, \\ {\Sigma }^{*}={𝒦}_{\text{fst}}^{**}-{𝒦}_{\text{fst}}^{*}{𝚺}_{\text{tr}}^{-1}{\left({𝒦}_{\text{fst}}^{*}\right)}^{\top } \end{gathered}$$ (3)

where ${𝒦}_{\text{fst}}^{*}={𝒦}_{\text{fst}}\left(\left(f^{*},{\mathit{x}}^{*},t^{*}\right);\left(ℱ,{𝒳}_{\text{tr}},{𝒯}_{\text{tr}}\right)\right)$ and ${𝒦}_{\text{fst}}^{**}={𝒦}_{\text{fst}}\left(\left(f^{*},{\mathit{x}}^{*},t^{*}\right);\left(f^{*},{\mathit{x}}^{*},t^{*}\right)\right)$. As such, the predictive performance fundamentally depends on the construction of effective kernels of ${𝒦}_{\text{s}},{𝒦}_{\text{t}}$ and ${𝒦}_{\text{f}}$, which will be detailed as follows.

Spatial Kernel Construction:

To capture the geometric features, we construct the spatial kernel based on the Laplacian operator of the system geometry, following the approach established in [29]. The Laplacian operator ${\Delta }_M$ defined on the discretized 3D surface $M$, which comprises $N$ vertices, encodes essential geometric properties such as vertex connectivity, edge orientation, and local curvature. Specifically, for a given vertex $i$, the discrete Laplacian is defined as [62]:

$${\left({\Delta }_M\mathit{q}\right)}_i=\frac{1}{2\left|{\Omega }_i\right|}\sum _{j\in N\left(i\right)}\left(\text{cot}{\alpha }_{ij}+\text{cot}{\beta }_{ij}\right)\left({\mathit{q}}_j-{\mathit{q}}_i\right)$$ (4)

where $\left|{\Omega }_i\right|$ is the Voronoi cell area around vertex $i,N\left(i\right)$ contains the neighboring vertices connected to $i,{\alpha }_{ij}$ and ${\beta }_{ij}$ are the angles opposite to the edge between vertices $i$ and $j$. Using the eigenfunctions ${\varphi }_j(\cdot )$ and the corresponding eigenvalues ${\lambda }_j$ of ${\Delta }_M$, we define our spatial kernel as:

$${𝒦}_{\text{s}}\left(\mathit{x},{\mathit{x}}^{\prime }\right)=\sum _{j\in {\mathbb{N}}^{*}}{\sigma }_{m}S\left(\sqrt{{\lambda }_j}\right){\varphi }_j(\mathit{x}){\varphi }_j\left({\mathit{x}}^{\prime }\right)$$ (5)

where ${\sigma }_m$ is a scaling coefficient and $S\left(\sqrt{\lambda }\right)$ is a non-increasing spectral density that modulates the contribution of each eigenmode, which is selected as the spectral density of the Matérn covariance structure in the frequency domain:

$$S(\sqrt{\lambda })≔\frac{2^d{\pi }^{d/2}\Gamma (v+d/2)(2v{)}^v}{\Gamma \left(v\right)l_{\text{s}}^{2v}}\times {\left(\frac{2v}{l_{\text{s}}^2}+4{\pi }^2\lambda \right)}^{-(v+d/2)}$$ (6)

where $d=2$ corresponds to the intrinsic dimensionality of the surface manifold, $v=3/2$ is the smoothness parameter, $l_{\text{s}}$ is the spatial length-scale, and $\Gamma (\cdot )$ denotes the Gamma function.

Temporal Kernel Construction:

Similar to the prior work [29], we adopt the Matérn kernel with the smoothness $v=3/2$ to model the temporal evolution of the dynamics:

$${𝒦}_{\text{t}}\left(t,t^{\prime }\right)={\sigma }_a\left(1+\frac{\sqrt{3}‖t-t^{\prime }‖}{l_{\text{t}}}\right)\text{exp}\left(-\frac{\sqrt{3}‖t-t^{\prime }‖}{l_{\text{t}}}\right)$$ (7)

Here, $l_{\text{t}}$ is a length scale controlling the temporal range of correlation, while ${\sigma }_a$ acts as a scaling coefficient adjusting the overall magnitude of temporal variability. This structure ensures that closer time points exhibit stronger correlations, while allowing for smooth decay over longer intervals. The Matérn kernel with $v=3/2$ is mean-square differentiable once, which is sufficient to capture the smooth temporal evolution typically observed in reaction-diffusion systems, while maintaining numerical robustness during hyperparameter optimization compared to other kernel choices. Importantly, the choice of $v=3/2$ reflects a balance between regularity and flexibility rather than a uniquely optimal selection. Other Matérn variants (e.g., $v=5/2$) or alternative kernels with comparable regularity properties may also yield competitive performance, provided their smoothness assumptions are consistent with the characteristics of the target spatiotemporal system.

Task-Correlation Kernel Construction:

We use a free-form kernel to capture the correlations between multiple tasks. Specifically, we define the task-correlation kernel as:

$${𝒦}_{\text{f}}(ℱ,ℱ)=\mathit{L}{\mathit{L}}^{\top },\mathit{L}=\left[\begin{array}{cccc}{\beta }_{11}& 0& \cdots & 0\\ {\beta }_{21}& {\beta }_{22}& 0& 0\\ ⋮& ⋮& \ddots & ⋮\\ {\beta }_{N_{\text{f}}1}& {\beta }_{N_{\text{f}}2}& \cdots & {\beta }_{N_{\text{f}}{N}_{\text{f}}}\end{array}\right]$$ (8)

where $\mathit{L}$ is a lower-triangular matrix with learnable parameters $\left\{{\beta }_{ij}\right\}$’s. This decomposition guarantees that ${𝒦}_{\text{f}}$ is symmetric and positive semi-definite, thus satisfying the requirements of a valid covariance kernel. The free-form structure of $\mathit{L}$ allows the model to flexibly learn arbitrary task relationships from data without imposing restrictive parametric assumptions.

B. Posterior Prediction

Given the kernel design and G-MGP parameters, ${\Theta }_{\text{GP}}=\left\{{\sigma }_m,l_{\text{s}},{\sigma }_a,l_{\text{t}},{\beta }_{11},\dots ,{\beta }_{N_{\text{f}}{N}_{\text{f}}},{\sigma }_{1,ϵ}^2,\dots ,{\sigma }_{N_{\text{f}},ϵ}^2\right\}$, the predicted mean of the dynamics is derived as:

$${\left.q\left(f^{*},{\mathit{x}}^{*},t^{*}\right)\right|}_{{\mathit{y}}_{\text{tr}},{𝒳}_{\text{tr}},{𝒯}_{\text{tr}};{\Theta }_{\text{GP}}}=\left({𝒦}_{\text{f}}^{*}\otimes {𝒦}_{\text{s}}^{*}\otimes {𝒦}_{\text{t}}^{*}\right){𝚺}_{\text{tr}}^{-1}{\mathit{y}}_{\text{tr}}$$ (9)

where ${𝒦}^{*}$ is the kernel matrix computed between test and training points, ${𝚺}_{\text{tr}}^{-1}$ denotes the inverse covariance matrix constructed from the training data, and ${\mathit{y}}_{\text{tr}}$ contains the observed values in the training dataset, stacked as a vector. According to the kernel design, ${𝚺}_{\text{tr}}$ can be written as ${𝚺}_{\text{tr}}={𝒦}_{\text{f}}\otimes {𝚺}_{\text{tr},\text{s}}\otimes {𝚺}_{\text{tr},\text{t}}+\mathit{D}\otimes {\mathit{I}}_{N_{\text{s}}}\otimes {\mathit{I}}_{N_{\text{t}}}$, where ${𝚺}_{\text{tr},\text{s}}≔{𝒦}_{\text{s}}\left({𝒳}_{\text{tr}},{𝒳}_{\text{tr}}\right),{𝚺}_{\text{tr},\text{t}}≔{𝒦}_{\text{t}}\left({𝒯}_{\text{tr}},{𝒯}_{\text{tr}}\right)$. In a special case of two tasks (i.e., $ℱ=\{1,2\}$): the task kernel is a 2×2 matrix, i.e., ${𝒦}_{\text{f}}(ℱ,ℱ)=\mathit{L}{\mathit{L}}^{\top }≔\left[\begin{array}{ll}{\beta }_{uu}& {\beta }_{uv}\\ {\beta }_{uv}& {\beta }_{vv}\end{array}\right]$, where we use $u$ and $v$ to denote task 1 and 2, respectively; the corresponding noise matrix is given by $\mathit{D}=\text{diag}\left({\sigma }_{u,ϵ}^2,{\sigma }_{v,ϵ}^2\right)$. To efficiently compute the posterior distribution, we need to address the computational challenges posed by the operations involving the large-scale covariance matrices ${𝚺}_{\text{tr}}$. Specifically, for a multi-task prediction scenario with $N_{\text{f}}$ tasks, $N_{\text{s}}$ spatial points, and $N_{\text{t}}$ temporal points, ${𝚺}_{\text{tr}}$ has dimensions of $\left(N_{\text{f}}{N}_{\text{s}}{N}_{\text{t}}\right)\times \left(N_{\text{f}}{N}_{\text{s}}{N}_{\text{t}}\right)$, which can be prohibitively large for direct inversion and determinant computation.

To overcome the computational barrier, we leverage the Kronecker product structure and employ Singular Value Decomposition (SVD) on the spatial and temporal covariance matrices, i.e., ${𝚺}_{\text{tr},\text{s}}={\mathit{U}}_{\text{s}}{𝚲}_{\text{s}}{\mathit{U}}_{\text{s}}^{\top }$ and ${𝚺}_{\text{tr},\text{t}}={\mathit{U}}_{\text{t}}{𝚲}_{\text{t}}{\mathit{U}}_{\text{t}}^{\top }$. This allows us to restructure and invert ${𝚺}_{\text{tr}}$ efficiently:

$${𝚺}_{\text{tr}}^{-1}={\left(\mathit{U}𝚲{\mathit{U}}^{\top }\right)}^{-1}=\mathit{U}{𝚲}^{-1}{\mathit{U}}^{\top }$$ (10)

where $\mathit{U}$ is an orthogonal matrix constructed from ${\mathit{U}}_{\text{s}}$ and ${\mathit{U}}_{\text{t}}$, and $𝚲$ is a block matrix with diagonal sub-blocks. The detailed block-wise definitions and block matrix inversion formulas are provided in Supplementary Material S2-A.

From Eq. (10), we can derive expressions for the posterior mean through a series of structured operations as follows (see the detailed derivation in Supplementary Material S2-B):

$${\left.\mathit{q}\left({\mathit{x}}^{*},t^{*}\right)\right|}_{{\mathit{y}}_{\text{tr}},{𝒳}_{\text{tr}},{𝒯}_{\text{tr}};{\Theta }_{\text{GP}}}=\text{vec}\left\{{\text{vec}}^{-1}\left(\tilde{\mathit{y}}\right){\times }_1\left({𝒦}_{\text{t}}^{*}{\mathit{U}}_{\text{t}}\right){\times }_2\left({𝒦}_{\text{s}}^{*}{\mathit{U}}_{\text{s}}\right){\times }_3\left({𝒦}_{\text{f}}^{*}\right)\right\}$$ (11)

where:

$$\tilde{\mathit{y}}≔\left[\begin{array}{c}{\widetilde{𝚲}}_{11}{\mathit{v}}_u+{\widetilde{𝚲}}_{12}{\mathit{v}}_v\\ {\widetilde{𝚲}}_{21}{\mathit{v}}_u+{\widetilde{𝚲}}_{22}{\mathit{v}}_v\end{array}\right],$$ (12)

$${\mathit{v}}_u≔{\left({\mathit{U}}_{\text{s}}\otimes {\mathit{U}}_{\text{t}}\right)}^{\top }{\mathit{y}}_u,$$ (13)

$${\mathit{v}}_v≔{\left({\mathit{U}}_{\text{s}}\otimes {\mathit{U}}_{\text{t}}\right)}^{\top }{\mathit{y}}_v.$$ (14)

The predictive variance can be efficiently evaluated by extracting its diagonal elements (see the detailed derivation in Supplementary Material S2-C):

$${\left({\hat{\mathit{\delta }}}^{*}\right)}^2\left({ℱ}^{*},{𝒳}^{*},{𝒯}^{*}\right)=\text{diag}\left({𝒦}_{\text{f}}^{**}\right)\otimes \text{diag}\left({𝒦}_{\text{s}}^{**}\right)\otimes \text{diag}\left({𝒦}_{\text{t}}^{**}\right)-\text{diag}(\mathit{G})$$ (15)

where ${𝒦}^{**}$ is the kernel for test-test pair. Computing $\text{diag}\left(\mathit{G}\right)$, which is a block matrix shaped as $\left[\begin{array}{ll}{\mathit{G}}_{11}& {\mathit{G}}_{12}\\ {\mathit{G}}_{21}& {\mathit{G}}_{22}\end{array}\right]$, only requires the computation of $\text{diag}\left({\mathit{G}}_{11}\right)$ and $\text{diag}\left({\mathit{G}}_{22}\right)$:

$$\text{diag}\left({\mathit{G}}_{ii}\right)=\text{diag}\left(\sum _{j=1}^2\sum _{k=1}^2{\mathit{H}}_{ij}{\widetilde{𝚲}}_{jk}{\mathit{H}}_{ik}^T\right),i=1,2$$ (16)

where the components of $\mathit{H}$ are defined as:

$$\mathit{H}≔\left[\begin{array}{ll}{𝒦}_{\text{f},uu}^{*}\left({𝒦}_{\text{s}}^{*}{\mathit{U}}_{\text{s}}\right)\otimes \left({𝒦}_{\text{t}}^{*}{\mathit{U}}_{\text{t}}\right)& {𝒦}_{\text{f},uv}^{*}\left({𝒦}_{\text{s}}^{*}{\mathit{U}}_{\text{s}}\right)\otimes \left({𝒦}_{\text{t}}^{*}{\mathit{U}}_{\text{t}}\right)\\ {𝒦}_{\text{f},vu}^{*}\left({𝒦}_{\text{s}}^{*}{\mathit{U}}_{\text{s}}\right)\otimes \left({𝒦}_{\text{t}}^{*}{\mathit{U}}_{\text{t}}\right)& {𝒦}_{\text{f}}^{*},vv\left({𝒦}_{\text{s}}^{*}{\mathit{U}}_{\text{s}}\right)\otimes \left({𝒦}_{\text{t}}^{*}{\mathit{U}}_{\text{t}}\right)\end{array}\right]$$ (17)

C. Physics-augmented G-MGP

1) Physics-based Spatiotemporal Reaction-diffusion Model: The present study focuses on spatiotemporal predictive modeling of a two-variable reaction-diffusion model defined on complex geometries. The evolution of the system dynamics on manifold $M$ is typically governed by the following PDEs:

$$\begin{gathered} \frac{\partial u}{\partial t}=e_1\Delta u+g_1(u,v), \\ \frac{\partial v}{\partial t}=e_2\Delta v+g_2(u,v), \\ {\left.\mathit{n}\cdot \nabla u\right|}_M=0 \\ {\left.\mathit{n}\cdot \nabla v\right|}_M=0 \end{gathered}$$ (18)

where $u$ and $v$ are the state variables representing the spatiotemporal concentrations of two interacting dynamics, $e_1$ and $e_2$ are the respective diffusion coefficients, $g_1(\cdot )$ and $g_2(\cdot )$ denote the nonlinear reactions kinetics. The boundary conditions (last two equations) enforce zero-flux/gradient (Neumann) constraints on the surface $M$, ensuring that the dynamics are confined to the manifold and there is no flow of material across its boundary.

2) Physics-Augmented Learning: To integrate prior physical knowledge into the GP modeling framework, we propose a physics-augmented parameter inference formulation that jointly leverages observational data and physical constraints. Specifically, the GP hyperparameter estimation is cast as the following regularized optimization problem:

$$\begin{gathered} {\Theta }_{\text{GP}}^{*}=\underset{{\Theta }_{\text{GP}}}{\text{arg}\text{min}}\left[{ℒ}_{\text{d}}+w{ℒ}_{\text{phy}}\right] \\ {ℒ}_{\text{d}}≔-\frac{1}{N_{\text{tr}}}\text{log}p\left({\mathit{y}}_{\text{tr}}\mid {𝒳}_{\text{tr}},{𝒯}_{\text{tr}};{\Theta }_{\text{GP}}\right) \end{gathered}$$ (19)

where ${ℒ}_{\text{d}}$ represents the data-driven negative log-likelihood that quantifies model fit to observed data, while ${ℒ}_{\text{phy}}$ encodes physics-based constraints, derived from governing equations (detailed below), which is introduced as a soft penalty modulated by a weight parameter $w$.

Data-driven G-MGP Likelihood:

Given the training dataset $\left[{\mathit{y}}_{\text{tr}};{𝒳}_{\text{tr}},{𝒯}_{\text{tr}}\right]$ and the marginal distribution in Eq. (2), the negative log-likelihood is expressed as:

$$\begin{gathered} -\text{log}p\left({\mathit{y}}_{\text{tr}}\mid {𝒳}_{\text{tr}},{𝒯}_{\text{tr}};{\Theta }_{\text{GP}}\right) \\ =\frac{1}{2}{\mathit{y}}_{\text{tr}}^{\top }{𝚺}_{\text{tr}}^{-1}{\mathit{y}}_{\text{tr}}+\frac{1}{2}\text{log}\left|{𝚺}_{\text{tr}}\right|+\frac{N_{\text{f}}{N}_{\text{s}}{N}_{\text{t}}}{2}\text{log}(2\pi ) \\ ≕\frac{1}{2}\left(f_1+f_2+f_3\right) \end{gathered}$$ (20)

where $N_{\text{f}}{N}_{\text{s}}{N}_{\text{t}}$ is the total number of observations across all tasks $\left(N_{\text{f}}\right)$, spatial locations $\left(N_{\text{s}}\right)$, and time points $\left(N_{\text{t}}\right)$. The likelihood function consists of three components: a data fit term $f_1$ that measures how well the model explains observed data, a complexity penalty $f_2$ that prevents overfitting by controlling model flexibility, and a normalization constant $f_3$. Computing terms $f_1$ and $f_2$ presents significant computational challenges due to the large-scale matrix operations involved. To address these challenges, we employ efficient matrix decomposition strategies that exploit the structural properties of ${𝚺}_{\text{tr}}$ to significantly reduce both computational complexity and memory requirements. Specifically, for $f_1$, we leverage the decomposition of ${𝚺}_{\text{tr}}^{-1}$ to obtain:

$$f_1≔{\mathit{y}}_{\text{tr}}^{\top }{𝚺}_{\text{tr}}^{-1}{\mathit{y}}_{\text{tr}}={\mathit{y}}_{\text{tr}}^{\top }\mathit{U}{𝚲}^{-1}{\mathit{U}}^{\top }{\mathit{y}}_{\text{tr}}=\left[\begin{array}{ll}{\mathit{y}}_u^{\top }& {\mathit{y}}_v^{\top }\end{array}\right]\left({\mathit{I}}_2\otimes {\mathit{U}}_{\text{s}}\otimes {\mathit{U}}_{\text{t}}\right)\widetilde{𝚲}{\left({\mathit{I}}_2\otimes {\mathit{U}}_{\text{s}}\otimes {\mathit{U}}_{\text{t}}\right)}^{\top }\left[\begin{array}{c}{\mathit{y}}_u\\ {\mathit{y}}_v\end{array}\right]$$ (21)

By defining transformed variables ${\mathit{v}}_u$ (Eq. (13)) and ${\mathit{v}}_v$ (Eq. (14), we can simplify this expression to:

$$\begin{gathered} f_1=\left[{\mathit{v}}_u^{\top },{\mathit{v}}_v^{\top }\right]\left[\begin{array}{ll}{\widetilde{𝚲}}_{11}& {\widetilde{𝚲}}_{12}\\ {\widetilde{𝚲}}_{12}& {\widetilde{𝚲}}_{22}\end{array}\right]\left[\begin{array}{c}{\mathit{v}}_u\\ {\mathit{v}}_v\end{array}\right] \\ ={\mathit{v}}_u^{\top }{\widetilde{𝚲}}_{11}{\mathit{v}}_u+{\mathit{v}}_u^{\top }{\widetilde{𝚲}}_{12}{\mathit{v}}_v+{\mathit{v}}_v^{\top }{\widetilde{𝚲}}_{12}{\mathit{v}}_u+{\mathit{v}}_v^{\top }{\widetilde{𝚲}}_{22}{\mathit{v}}_v \end{gathered}$$ (22)

Similarly, for $f_2$, we exploit the properties of orthogonal matrices and block-diagonal structures (see the detailed derivation in the Supplementary Material S2-D):

$$f_2≔\text{log}\left|{𝚺}_{\text{tr}}\right|=\sum _i\text{log}{\left({𝚲}_{11}\right)}_{ii}+\sum _i\text{log}\left({\left[{𝚲}_{22}-{𝚲}_{12}{𝚲}_{11}^{-1}{𝚲}_{12}\right]}_{ii}\right)$$ (23)

It is worth noting that the block matrices, ${𝚲}_{11},{𝚲}_{12}$, and ${𝚲}_{22}$ are all diagonal, which allows for a more efficient computation of the determinant compared to directly evaluating the original $\left(N_{\text{f}}{N}_{\text{s}}{N}_{\text{t}}\right)\times \left(N_{\text{f}}{N}_{\text{s}}{N}_{\text{t}}\right)$ matrix ${𝚺}_{\text{tr}}$.

Physics-based Loss:

To incorporate physics-based constraints into our model, we define residuals that quantify the deviation between the G-MGP predictions and the governing reaction-diffusion PDEs:

$$\begin{gathered} {\gamma }_{S_u}\left(\mathit{x},t;{\Theta }_{\text{GP}}\right)≔\frac{\partial \hat{u}}{\partial t}-e_1\Delta \hat{u}-g_1(\hat{u},\hat{v}), \\ {\gamma }_{S_v}\left(\mathit{x},t;{\Theta }_{\text{GP}}\right)≔\frac{\partial \hat{v}}{\partial t}-e_2\Delta \hat{v}-g_2(\hat{u},\hat{v}) \\ {\gamma }_{u,bc}\left(\mathit{x},t;{\Theta }_{\text{GP}}\right)≔{\left.\mathit{n}\cdot \nabla \hat{u}\right|}_M \\ {\gamma }_{v,bc}\left(\mathit{x},t;{\Theta }_{\text{GP}}\right)≔{\left.\mathit{n}\cdot \nabla \hat{v}\right|}_M \end{gathered}$$ (24)

These residuals measure how well the G-MGP predictions $\hat{u}$ and $\hat{v}$ satisfy the PDEs at any given spatial location and time point. Note that to evaluate the physics-based loss, we need to compute the derivatives that appear in the residuals. Specifically, the time derivatives can be computed as:

$${\left[\frac{\partial \hat{u}}{\partial t},\frac{\partial \hat{v}}{\partial t}\right]}^{\text{T}}=\frac{\partial \stackrel{\mathit{ˆ}}{\mathit{q}}}{\partial t}=\text{vec}\left\{{\text{vec}}^{-1}\left(\tilde{\mathit{y}}\right){\times }_1\left(\frac{\partial {𝒦}_{\text{t}}^{*}}{\partial t}{\mathit{U}}_{\text{t}}\right)\left.\times 2\left({𝒦}_{\text{s}}^{*}{\mathit{U}}_{\text{s}}\right){\times }_3\left({𝒦}_{\text{f}}^{*}{\mathit{I}}_{N_{\text{f}}}\right)\right\}\right.$$ (25)

For the Matérn temporal kernel with smoothness parameter $v=3/2$, we have:

$${\left[\frac{\partial {𝒦}_t^{*}}{\partial t}\right]}_{i=1~N_{\text{col}},j=1~N_{\text{tr}}}=-{\sigma }_a{a}^2\text{exp}(-am)m\cdot \text{sign}\left(t_i-t_j\right)$$ (26)

where $N_{\text{col}}$ is the number of collocation points used to enforce the physical constraints, $m=‖t_i-t_j‖$, and $a=\frac{\sqrt{3}}{l_t}$ (see the detailed derivation in the Supplementary Material S2-E.

The spatial derivative in the reaction-diffusion system is represented by the Laplacian operator $\Delta$, which by default operates in the ambient Euclidean space ${\mathbb{R}}^3$. However, because the dynamics are restricted to a 3D surface $M\subset {\mathbb{R}}^3$, it is essential to consider the intrinsic geometry of $M$ to derive the spatial derivative. To reconcile this, we leverage the Neumann boundary condition imposed on the state variable $u$ (or $v$) to prove that the Euclidean Laplacian $\Delta u$ restricted to the surface $M$ is equivalent to surface Laplacian ${\Delta }_{M}u$. Specifically, using the decomposition of the Euclidean gradient and divergence into tangential and normal components, we obtain:

$${\left.\Delta u\right|}_M=\nabla \cdot {\nabla }_{M}u+\nabla \cdot \left(\left({\left.\mathit{n}\cdot \nabla u\right|}_M\right)\mathit{n}\right)$$ (27)

Because ${\nabla }_{M}u$ is tangential to surface $M$ and ${\left.\mathit{n}\cdot \nabla u\right|}_M=0$ (i.e., the Neumann boundary condition), we conclude that ${\left.\Delta u\right|}_M={\nabla }_M\cdot {\nabla }_{M}u={\Delta }_{M}u$. As such, the physics-based loss function is then defined as the mean squared residual constructed by the temporal derivative and spatial derivative encoded by the surface Laplacian ${\Delta }_M$ across a set of collocation points:

$${ℒ}_{\text{phy}}=\frac{1}{N_{\text{col}}}\sum _{i=1}^{N_{\text{col}}}\left[{\left({\gamma }_{S_u,{\Delta }_M}\left({\mathit{x}}_i,t_i;{\Theta }_{\text{GP}}\right)\right)}^2\left.+{\left({\gamma }_{S_v,{\Delta }_M}\left({\mathit{x}}_i,t_i;{\Theta }_{\text{GP}}\right)\right)}^2\right]\right.$$ (28)

where ${\gamma }_{S_u,{\Delta }_M}\left(\mathit{x},t;{\Theta }_{\text{GP}}\right)≔\frac{\partial \hat{u}}{\partial t}-e_1{\Delta }_M\hat{u}-g_1(\hat{u},\hat{v})$ and ${\gamma }_{S_v,{\Delta }_M}\left(\mathit{x},t;{\Theta }_{\text{GP}}\right)≔\frac{\partial \hat{v}}{\partial t}-e_2{\Delta }_M\hat{v}-g_2(\hat{u},\hat{v})$.

Cross-Validation for Model Parameter Selection:

To select optimal model parameters under different initial guesses and evaluate prediction performance, we employ the leave-one-location-out cross-validation. The cross-validation error is defined as:

$${\hat{\tau }}_{n,\text{cv}}^2=\frac{1}{n}\sum _{f=1}^{N_{\text{f}}}\sum _{i=1}^{N_{\text{s}}}\sum _{j=1}^{N_{\text{t}}}{\left(y\left(f,{\mathit{x}}_i,t_j\right)-{\hat{q}}_{-{\mathit{x}}_i}\left(f,{\mathit{x}}_i,t_j\right)\right)}^2$$ (29)

where ${\hat{q}}_{-{\mathit{x}}_i}\left(f,{\mathit{x}}_i,t_j\right)$ is the prediction made by a model trained on all data except those from location ${\mathit{x}}_i$. Computing this error naively would require training $N_{\text{s}}$ separate models, leading to prohibitive computational costs.

Instead, we develop an efficient approach that requires only a single matrix inversion. By partitioning the full covariance matrix and leveraging block matrix inversion formulas, we can express the leave-one-location-out residuals directly in terms of the inverse covariance matrix of the full model:

$$\begin{gathered} {\left[y\left(f,{\mathit{x}}_i,t_j\right)-{\hat{q}}_{-{\mathit{x}}_i}\left(f,{\mathit{x}}_i,t_j\right)\right]}_{f\in ℱ,t_j\in {𝒯}_{\text{tr}}} \\ ={\left({\stackrel{\mathit{ˆ}}{\mathit{\delta }}}^{*}\right)}_{N_{\text{f}},1_i,N_{\text{t}}}^2\cdot {\left({𝚺}_{\text{tr}}^{-1}{\mathit{y}}_{\text{tr}}\right)}_{N_{\text{f}},\left(1_i,\cdot \right),N_{\text{t}}} \end{gathered}$$ (30)

where ${\left({\stackrel{\mathit{ˆ}}{\mathit{\delta }}}^{*}\right)}_{N_{\text{f}},1_i,N_{\text{t}}}^2$ is the estimated predictive variance at location ${\mathit{x}}_i$, and ${\left({𝚺}_{\text{tr}}^{-1}{\mathit{y}}_{\text{tr}}\right)}_{N_{\text{f}},\left(1_i,\cdot \right),N_{\text{t}}}$ is the sub-vector of ${𝚺}_{\text{tr}}^{-1}{\mathit{y}}_{\text{tr}}$ corresponding to the $i$-th spatial location. This formulation allows us to compute all cross-validation errors simultaneously from the precomputed ${𝚺}_{\text{tr}}^{-1}{\mathit{y}}_{\text{tr}}$, avoiding redundant matrix inversions. Detailed derivations for validation error calculation are provided in Supplementary Material S2-F.

Overall, the computational complexity of our framework is substantially reduced by systematically exploiting Kronecker separability and the diagonal-block eigenstructure. By utilizing orthogonal transforms in space and time, our structured pipeline converts massive dense matrix operations into highly efficient tensor mode contractions and elementwise (vector) arithmetic. For example, the computational cost of posterior mean evaluation combines the diagonal-block action (with a complexity of $𝒪\left(N_{\text{f}}^2{N}_{\text{s}}{N}_{\text{t}}\right)$), kernel precomputations (with a complexity of $𝒪\left(N_{\text{t}}^{*}{N}_{\text{t}}^2+N_{\text{s}}^{*}{N}_{\text{s}}^2\right)$), and mode-products (with a complexity of $𝒪\left(N_{\text{t}}^{*}{N}_{\text{t}}{N}_{\text{s}}{N}_{\text{f}}+N_{\text{t}}^{*}{N}_{\text{s}}^{*}{N}_{\text{s}}{N}_{\text{f}}+N_{\text{t}}^{*}{N}_{\text{s}}^{*}{N}_{\text{f}}^2\right)$, yielding a substantial reduction compared to the naive $𝒪\left({\left(N_{\text{f}}{N}_{\text{s}}{N}_{\text{t}}\right)}^3\right)$ inversion. A comprehensive, step-by-step complexity analysis for all model components, including posterior mean prediction, posterior variance prediction, data-driven likelihood evaluation, physics-based loss evaluation, and cross-validation, is detailed in Supplementary Material S1.

IV. Numerical Experiments

We assess the performance of our P-G-MGP framework in predictive modeling of cardiac electrodynamics within a 3D ventricular geometry. The geometry is discretized into 1,094 nodes and 2,184 mesh elements, constituting a refined mesh derived from the geometry data in the 2007 PhysioNet Computing in Cardiology Challenge [63]. The simulation data is generated by numerically solving the FitzHugh-Nagumo (FHN) model using finite element methods. The reaction kinetics in the FHN model are specified as follows:

$$\begin{gathered} g_1(u,v)=C_{1}u(u-\alpha )(1-u)-C_{2}uv \\ {g}_2(u,v)=b(u-dv) \end{gathered}$$ (31)

with FHN model parameters set to: $C_1=0.26,C_2=0.1,\alpha =0.13,b=0.013$, and $d=1.0$; and diffusion parameters set to: $e_1=10$ and $e_2=0$. We applied two protocols to generate the simulation data: (1) Protocol I – A regular-pacing activation source is placed at the apex of the ventricular geometry to stimulate the reaction-diffusion electrodynamics; (2) Protocol II – An additional activation source is introduced to imitate self-sustained, disorganized dynamics under fibrillation.

We denote the resulting simulation data as $\mathit{q}(\mathit{x},t)=\left[\mathit{u}\right(\mathit{x},t),\mathit{v}(\mathit{x},t\left)\right]={\left[u\left({\mathit{x}}_i,t_j\right),v\left({\mathit{x}}_i,t_j\right)\right]}_{i\in 𝒳,j\in 𝒯}$ with $\left|𝒳\right|=1094$. Note that the time series signals collected at each spatial location ${\mathit{x}}_i$ consist of 1570 data points for Protocol I (i.e., $\left|𝒯\right|=1570$), and 2920 points for Protocol II (i.e., $\left|𝒯\right|=2920$). Because measurement noise is inevitable in real-world data collection, we add different levels of noise to the simulation data to investigate the prediction performance. Specifically, the physical measurements are generated as $\mathit{y}(\mathit{x},t)=\mathit{q}(\mathit{x},t)+\mathit{\xi }(\mathit{x},t)$, where $\mathit{\xi }(\mathit{x},t)$ is the noise that follows a Gaussian distribution, $\mathit{\xi }(\mathit{x},t)~{\sigma }_{\xi }\cdot 𝒩(\mathbf{0},1)$, where ${\sigma }_{\xi }$ is the noise level coefficient. Our P-G-MGP is compared with the traditional methods based on the relative error $\left(RE\right)$:

$$RE=\frac{\Vert \stackrel{\mathit{ˆ}}{\mathit{q}}(\mathit{x},t)-\mathit{q}(\mathit{x},t)\Vert }{\Vert \mathit{q}(\mathit{x},t)\Vert }$$ (32)

where $\mathit{q}(\mathit{x},t)$ and $\stackrel{\mathit{ˆ}}{\mathit{q}}(\mathit{x},t)$ denote the reference and predicted cardiac dynamics, respectively.

To understand the contribution of each key module in our framework, we conduct a systematic ablation study. First, we compare P-G-MGP with G-MGP to assess the impact of incorporating physics-based constraints through the physics-augmented likelihood to improve model robustness to noise and data sparsity. Second, the comparison with G-ST-GP focuses on the role of multi-task learning, specifically examining how capturing the coupled dynamics between $u$ and $v$ can enhance the prediction performance. Finally, evaluating the performance difference between P-G-MGP and the physics-augmented MGP with a Euclidean kernel (P-E-MGP) is intended to clarify the importance of encoding geometric information, distinguishing the effect of replacing Euclidean distances with intrinsic geodesic kernels in capturing the spatial correlations of the cardiac structure. These experiments collectively isolate and probe the benefit of geometry awareness, multi-task interaction, and physics regularization in the proposed prediction framework.

A. Prediction under Simulation Protocol I

Fig. 2 compares the spatiotemporal prediction performance of our P-G-MGP with G-ST-GP, G-MGP, P-E-MGP, and PINN under two training set sizes $(\left|{𝒳}_{\text{tr}}\right|=50,75)$ and two noise levels $\left({\sigma }_{\xi }=\mathrm{0.01,0.02}\right)$. Panel (a) presents the ground-truth spatial distribution of cardiac electrodynamics at time-step $t=800$, while Panel (b) shows the spatial fields for $u$ and $v$ predicted by each method. Across all experimental settings, P-G-MGP consistently achieves the highest fidelity to the reference patterns shown in Fig. 2(a). In contrast, G-ST-GP (without multi-task interaction), G-MGP (without physics-based constraints), P-E-MGP (without geometry awareness), and standard PINN (without geometric manifold modeling) exhibit more pronounced deviations from the ground truth.

Fig. 2:

Simulation Protocol I: (a) Reference mapping of $u$ and $v$ at time point $t=800$. (b) Estimated mappings by our P-G-MGP and different benchmark methods under different training dataset sizes and noise levels. (c) Bar chart comparing the aggregated $RE$ from 3 replications.

Fig. 2(c) shows the quantitative performance comparison based on the aggregated $RE$ metric $\left(R{E}_{\text{total}}=\frac{1}{2}\left(R{E}_u+R{E}_v\right)\right)$, calculated from triplicate experiments with randomized seeds. The performance highlights the superior prediction accuracy achieved by our P-G-MGP. For $N_{\text{col,t}}=200$ and $\left|{𝒳}_{\text{tr}}\right|=50$, P-G-MGP yields $RE$ of 0.0465 when ${\sigma }_{\xi }=0.01$ and 0.0650 when ${\sigma }_{\xi }=0.02$, achieving an $RE$ reductions of 61.57% and 49.61% compared to the baseline G-MGP, 51.56% and 38.97% compared to P-E-MGP, and improvements of 36.73% and 12.75% over PINN, respectively. Furthermore, compared to G-ST-GP ($RE=0.2340$ and 0.2395), our method reduces the error by over 70%, demonstrating the efficacy of incorporating physics knowledge and multi-task interaction.

When the training size increases to $\left|{𝒳}_{\text{tr}}\right|=75\left(N_{\text{col},\text{t}}=200\right)$, the performance gains achieved by P-G-MGP are also prominent: $RE$ is reduced to 0.0390 for ${\sigma }_{\xi }=0.01$ and 0.0580 for ${\sigma }_{\xi }=0.02$, corresponding to reductions of 35.54% and 22.67% compared to G-MGP, 38.10% and 25.64% compared to P-E-MGP, and 46.58% and 21.62% over PINN, respectively. Notably, as the dataset size increases, P-G-MGP demonstrates a marked performance advantage over PINN. This performance gap is particularly significant given the substantial difference in model complexity: the PINN framework adopted in our comparison is a feedforward neural network with six hidden layers and 515 trainable parameters, while our P-G-MGP is highly compact with only 9 trainable parameters.

A similar pattern is observed when the number of collocation points increases to $N_{\text{col,t}}=500$: our P-G-MGP achieves $RE$ reductions of 62.40% $\left({\sigma }_{\xi }=0.01\right)$ and 51.16% $\left({\sigma }_{\xi }=0.02\right)$ at $\left|{𝒳}_{\text{tr}}\right|=50$, and 38.84% and 23.33% at $\left|{𝒳}_{\text{tr}}\right|=75$ compared to G-MGP. Compared to P-E-MGP, the $RE$ reductions are 52.11% and 40.28% at $\left|{𝒳}_{\text{tr}}\right|=50$, and 40.80% and 26.28% at $\left|{𝒳}_{\text{tr}}\right|=75$. Compared to PINN, the $RE$ reductions are 36.81% $\left({\sigma }_{\xi }=0.01,\left|{𝒳}_{\text{tr}}\right|=50\right)$, 13.10% $\left({\sigma }_{\xi }=0.02,\left|{𝒳}_{\text{tr}}\right|=50\right)$, 49.66% $\left({\sigma }_{\xi }=0.01,\left|{𝒳}_{\text{tr}}\right|=75\right)$, and 21.77% $\left({\sigma }_{\xi }=0.02,\left|{𝒳}_{\text{tr}}\right|=75\right)$, respectively. It is worth noting that increasing $N_{\text{col,t}}$ from 200 to 500 yields marginal improvements in $RE$ for P-E-MGP, PINN and P-G-MGP. This indicates a possible saturation effect, where the additional collocation points provide diminishing returns in encoding the underlying physical constraints. The limited impact is likely due to the already sufficient physical information captured by $N_{\text{col},\text{t}}=200$, suggesting that beyond a certain threshold, further increasing the density of physics-based supervision does not significantly enhance the model performance.

Fig. 3 illustrates the temporal evolution of $u$ and $v$ at the spatial location ${\mathit{x}}_{599}$. We compare our P-G-MGP against the benchmark models under two training set sizes and two noise levels. When $\left|{𝒳}_{\text{tr}}\right|=50$, P-G-MGP yields the closest agreement with the ground-truth trajectories for both tasks, capturing not only the overall waveform shape but also the timing of rapid transitions and the recovery phase. Among the baselines, P-E-MGP typically follows the true dynamics more closely than G-MGP and G-ST-GP, indicating that incorporating the physics-augmented likelihood improves temporal consistency. However, residual discrepancies remain due to the use of a Euclidean spatial kernel. In contrast, G-MGP exhibits larger phase and amplitude errors, while G-ST-GP shows the most pronounced deviations and reduced temporal stability. PINN may reproduce coarse trends but tends to deviate near sharp temporal changes, reflecting optimization/generalization sensitivity in this noisy, partially observed setting.

Fig. 3:

Comparison of temporal dynamics under Protocol I at spatial location ${\mathit{x}}_{599}$ between predictions from our P-G-MGP and benchmark methods across varying training set sizes $\left(\left|{𝒳}_{\text{tr}}\right|=50,75\right)$ and noise levels: (a) ${\sigma }_{\xi }=0.01$, (b) ${\sigma }_{\xi }=0.02$.

As $\left|{𝒳}_{\text{tr}}\right|$ increases to 75, the prediction fidelity of all methods improves to varying degrees, yet P-G-MGP remains consistently the most accurate across both tasks. This behavior aligns with the intended roles of our three key components: (i) the physics-augmented likelihood regularizes hyperparameter learning toward physically consistent dynamics; (ii) multi-task coupling leverages the intrinsic interaction between $u$ and $v$, which is absent in G-ST-GP; and (iii) the geometry-aware kernel better represents spatial correlations on the non-Euclidean cardiac manifold, which P-E-MGP or PINN lacks.

B. Prediction under Simulation Protocol II

Fig. 4 shows the comparison of G-ST-GP, G-MGP, P-E-MGP, PINN and our P-G-MGP under Simulation Protocol II. Reconstructed mappings in Panel (b) reveal that P-G-MGP continues to significantly outperform all other models across both tasks. Under the more complex dynamics of Protocol II, the performance gap between the benchmarks becomes even more apparent. Fig. 4(c) presents $R{E}_{\text{total}}$ averaged over three independent trials. G-ST-GP suffers significantly, with $R{E}_{\text{total}}$ of 0.1735 at $\left|{𝒳}_{\text{tr}}\right|=50$ and ${\sigma }_{\xi }=0.02$. G-MGP reduces this error to 0.1425 through task coupling, but fails to maintain precision under higher noise. The P-E-MGP variant ($R{E}_{\text{total}}=0.1175$ at $N_{\text{col},\text{t}}=500$) demonstrates that physics augmentation is vital, yet the Euclidean kernel approximation creates significant spatial blurring at the conduction wavefront. PINN $\left(N_{\text{col},\text{t}}=500\right)$ exhibits an $R{E}_{\text{total}}$ of 0.1255 for $\left|{𝒳}_{\text{tr}}\right|=50,{\sigma }_{\xi }=0.01$, which is more than double the error of our P-G-MGP (0.0595). Moreover, for $N_{\text{col},\text{t}}=200$ and $\left|{𝒳}_{\text{tr}}\right|=50$, our P-G-MGP attains $R{E}_{\text{total}}$ of 0.0595 for ${\sigma }_{\xi }=0.01$ and 0.0785 for ${\sigma }_{\xi }=0.02$. These results represent substantial reductions of 63.6% and 54.8% compared to the single-task G-ST-GP, 55.26% and 44.91% compared to G-MGP, 43.60% and 33.76% compared to P-E-MGP, and 53.15% and 38.91% compared to PINN. These findings collectively affirm that the integration of a geometry-aware spatial kernel, multi-task interactions, and physics-augmented learning is essential for achieving high-fidelity prediction in cardiac electrodynamics.

Fig. 4:

Simulation Protocol II: (a) Ground-truth visualization at $t=800$. (b) Predictions from our P-G-MGP and different benchmark methods using varying training sizes and noise conditions. White dotted lines are included for better visualization. (c) Comparison of the aggregated $RE$ between the models averaged over 3 trials.

Temporal dynamics at spatial node ${\mathit{x}}_{636}$ under Simulation Protocol II are illustrated in Fig. 5. Compared with Simulation Protocol I, the trajectories here are visibly more irregular, reflecting the increased dynamical complexity associated with pathological cardiac conditions. Despite this more challenging regime, the proposed P-G-MGP model provides the most accurate temporal predictions. For both task-$u$ and task-$v$, P-G-MGP closely follows the ground-truth waveforms, reproducing key features such as rapid upstrokes and the timing of irregular fluctuations under both noise levels.

Fig. 5:

Comparison of temporal dynamics under Protocol II at spatial location ${\mathit{x}}_{636}$ between predictions from our P-G-MGP and benchmark methods across different training set sizes $\left(\left|{𝒳}_{\text{tr}}\right|=50,75\right)$ and noise conditions: (a) ${\sigma }_{\xi }=0.01$ and (b) ${\sigma }_{\xi }=0.02$.

In contrast, the benchmark methods show systematic deficiencies. G-ST-GP exhibits the largest deviations, with noticeable phase mismatch and reduced stability over time. The purely data-driven G-MGP also displays clear amplitude distortions and temporal lag, indicating limited robustness when physical consistency is not enforced. Incorporating physics augmentation with a Euclidean kernel (P-E-MGP) typically reduces these distortions relative to G-MGP, supporting the value of physics augmentation. However, residual mismatch remains due to the fact that Euclidean distances cannot fully represent intrinsic spatial correlations on the cardiac manifold. Finally, PINN exhibits the most pronounced discrepancies among the baselines, particularly around sharp transitions, suggesting increased sensitivity to optimization challenges and limited generalization in the presence of noisy and partially observed data defined on complex geometries.

C. Uncertainty Quantification

A key advantage of GP models over many neural network-based approaches, such as PINNs, is their inherent ability to provide not just a pointwise prediction, but also an effective measure of predictive uncertainty. While pointwise accuracy metrics are essential for model validation, they do not fully capture the reliability of the model’s inferences. To address this, we evaluate the quality of the uncertainty estimates produced by our P-G-MGP model and analyze their spatial distribution across the whole spatial domain.

As demonstrated in Fig. 6, the model yields stable uncertainty bounds across different tasks and datasets, which underscore the robustness of the GP probabilistic framework. In particular, the regions highlighted in yellow correspond to the locations of actual observations. These regions consistently exhibit the lowest posterior predictive variance, reflecting the fundamental GP property that uncertainty contracts in the presence of data through conditioning on observations. In contrast, the regions of highest uncertainty $\left({\delta }_{\text{max}}^2\right)$ are located in areas devoid of any observational data points. This clear spatial alignment between predictive uncertainty and data density provides an important validation of the model’s probabilistic consistency. Specifically, the uncertainty map appropriately captures epistemic uncertainty by increasing in unobserved regions and decreasing where data constraints are strongest.

Fig. 6:

Spatial maps of predictive uncertainty $\left({\delta }^2\right)$ from the P-G-MGP model $\left(N_{\text{col},\text{t}}=500\right)$, shown across the cardiac domain for both tasks ($u$ and $v$), protocols I and II, and under representative noise $({\sigma }_{\xi }=0.01)$ and training set size $(\left|{𝒳}_{\text{tr}}\right|=50)$ conditions. Yellow markers denote actual observation locations.

The practical utility of this uncertainty information serves as a powerful tool for experimental design and active learning. By identifying regions with the highest predictive uncertainty, we can pinpoint where on the myocardium additional measurements would be most valuable. An active learning framework could iteratively suggest new observation points in these high-uncertainty zones to efficiently improve the global model accuracy, thereby reducing the number of required measurements for a comprehensive characterization of the cardiac dynamics. Future work will focus on designing and implementing active learning strategies within this probabilistic framework, including the development of acquisition functions tailored to spatiotemporal systems and physics-informed constraints.

V. Conclusions

This paper presents a physics-augmented, geometry-aware and multi-task Gaussian Process (P-G-MGP) framework for predictive modeling of spatiotemporal dynamic systems. The proposed methodology integrates geometric manifold awareness, task-wise dependency modeling, and physics-informed regularization to address key challenges associated with irregular spatial domains, multi-output prediction, physical fidelity and uncertainty quantification. Specifically, we extend the geometry-aware spatiotemporal GP with a multi-task model and design a structured computational pipeline through exploiting Kronecker separability and kernel eigenstructures to enable highly scalable training and inference for high-dimensional systems. Furthermore, we incorporate governing physical principles through a physics-augmented regularization strategy, improving predictive robustness and physical consistency, particularly under sparse or noisy observations. In addition, our framework provides closed-form posterior variance estimates, yielding effective uncertainty quantification to support decision-making. We validate our P-G-MGP on 3D cardiac electrophysiological modeling. Numerical results demonstrate superior predictive performance of P-G-MGP compared with geometry-agnostic, single-task, and purely data-driven baselines. By bridging statistical learning with geometric priors, multi-task interaction and domain-specific physics, our framework offers an effective solution for complex spatiotemporal modeling tasks, with broad applicability to real-world dynamic systems.

One limitation of this work is that the current evaluation relies on synthetic datasets. Clinical deployment (e.g., electroanatomical mapping or electrocardiographic imaging) may encounter discrepancies between the FHN model and true cardiac electrophysiology, alongside practical challenges such as heterogeneous noise and sparse, nonuniform sampling. However, we emphasize that our framework utilizes the physics principles solely as a structured regularizer. As demonstrated in the sensitivity analysis (Appendix A), the P-G-MGP model maintains stable predictive accuracy under ±10% parameter perturbations and consistently outperforms baseline methods. This robustness stems from the cooperative integration of physics-informed and data-driven losses, which enables the model to adaptively correct prior physical mismatches using real sensing signals. Building upon this demonstrated robustness to parametric uncertainty, an essential next step is comprehensive validation using real clinical datasets. In future work, we plan to expand access to experimental data through collaborations with healthcare institutions. Evaluating the P-G-MGP framework under these practical conditions and across diverse patient cohorts will further establish its reliability and translational applicability in clinical settings.

Appendix

A. Impact of FHN Parameters on Prediction Performance

To evaluate the sensitivity of our P-G-MGP to uncertainties in the FHN model, we perturb each key parameter by ±10% from its nominal value and assess the prediction performance in terms of the $RE$ for task-$u$ and task-$v$, as well as the aggregated $RE$. All experiments are conducted under Simulation Protocol-I, with a collocation dataset size of $N_{\text{col},\text{t}}=200$, a training set size of $\left|{𝒳}_{\text{tr}}\right|=50$, and a noise level of ${\sigma }_{\xi }=0.01$. As shown in Table I, after applying the designated ±10% variations to the FHN parameters, the overall prediction performance of P-G-MGP remains remarkably stable across all perturbations. Additionally, the fluctuations in $RE$ are minimal, ranging from 0.0496 to 0.0530 in $R{E}_u$ (a relative change of about 4% compared to the reference $R{E}_u$ of 0.0520 under the true physics specification), 0.0384 to 0.0415 for $R{E}_v$ (within approximately 6% relative change from the nominal case of 0.0410), and 0.0443 to 0.0471 for $R{E}_{\text{total}}$ (less than 4% of the reference $R{E}_{\text{total}}$ of 0.0460).

Moreover, compared with the benchmark methods, our P-G-MGP model with perturbed FHN parameters still achieves substantially lower $RE$. Specifically, the $R{E}_{\text{total}}$ values of the benchmark methods are 0.2341 for G-ST-GP, 0.1213 for G-MGP, 0.0962 for PINN, and 0.0732 for P-E-MGP, all significantly larger than the range achieved by our P-G-MGP under parameter perturbations (0.0443 – 0.0471). These results demonstrate that our P-G-MGP does not rely on perfectly specified governing parameters to maintain predictive accuracy. Instead, the physics component acts as a structured regularizer that cooperates with the data-driven likelihood, enabling the model to preserve stable performance even when the embedded physical parameters are moderately misspecified. In particular, the data-driven likelihood enables direct assimilation of real-world sensing signals, allowing the model to adaptively correct discrepancies between the assumed physical prior and observed measurements. This cooperative interaction ensures that the physics component constrains the solution space to physically plausible regimes, while the likelihood term incorporates empirical evidence to refine predictions, resulting in robust and reliable inference in practical settings where governing physics may be imperfectly specified.

TABLE I:

The $RE$ of our P-G-MGP when each FHN parameter is varied by ±10% from its reference value. Results are shown for task-$u$, task-$v$, and the average of the two tasks.

Parameter	$R{E}_u$		$R{E}_v$		$R{E}_{\text{total}}$
	−10%	+10%	−10%	+10%	−10%	+10%
Ref(0%)	0.0520		0.0410		0.0460
$a$	0.0527	0.0530	0.0414	0.0385	0.0470	0.0458
$b$	0.0509	0.0527	0.0402	0.0414	0.0456	0.0470
$C_1$	0.0500	0.0507	0.0387	0.0399	0.0443	0.0453
$C_2$	0.0526	0.0528	0.0414	0.0415	0.0470	0.0471
$d$	0.0521	0.0527	0.0384	0.0414	0.0453	0.0470
$e_1$	0.0496	0.0507	0.0400	0.0401	0.0448	0.0454
$e_2$	0.0520	0.0520	0.0410	0.0410	0.0460	0.0460