Geometry-aware Active Learning of Spatiotemporal Dynamic Systems

TL;DR

This work tackles predictive modeling of spatiotemporal dynamics on complex 3D geometries under data collection constraints. It introduces a geometry-aware spatiotemporal Gaussian Process (G-ST-GP) with a spatial kernel built from Laplacian eigenfunctions on the geometry and a temporal Matérn kernel, together with an adaptive active learning (A-AL) strategy that blends predictive uncertainty with geodesic space-filling, all computed efficiently via Kronecker algebra. Through a 3D cardiac electrodynamics case, the approach yields higher fidelity predictions than Euclidean-kernel baselines and demonstrates data-efficient sensor placement, achieving strong performance with only a fraction of locations. The framework has broad potential for geometry-rich, resource-constrained applications in bioengineering and other fields requiring accurate spatiotemporal prediction and efficient data acquisition.

Abstract

Rapid developments in advanced sensing and imaging have significantly enhanced information visibility, opening opportunities for predictive modeling of complex dynamic systems. However, sensing signals acquired from such complex systems are often distributed across 3D geometries and rapidly evolving over time, posing significant challenges in spatiotemporal predictive modeling. This paper proposes a geometry-aware active learning framework for modeling spatiotemporal dynamic systems. Specifically, we propose a geometry-aware spatiotemporal Gaussian Process (G-ST-GP) to effectively integrate the temporal correlations and geometric manifold features for reliable prediction of high-dimensional dynamic behaviors. In addition, we develop an adaptive active learning strategy to strategically identify informative spatial locations for data collection and further maximize the prediction accuracy. This strategy achieves the adaptive trade-off between the prediction uncertainty in the G-ST-GP model and the space-filling design guided by the geodesic distance across the 3D geometry. We implement the proposed framework to model the spatiotemporal electrodynamics in a 3D heart geometry. Numerical experiments show that our framework outperforms traditional methods lacking the mechanism of geometric information incorporation or effective data collection.

Figures (5)

• Figure 1: Flowchart of the proposed methodology. The G-ST-GP framework is first developed to model the spatiotemporal dynamics in a 3D complicated geometry. Then, an adaptive active learning method (A-AL) is designed by combining the prediction uncertainty in G-ST-GP with a space-filling design guided by geodesic distance to identify informative locations for additional signal measurements, enhancing the predictive capability of our G-ST-GP model.
• Figure 2: Illustration of angles $\alpha_{ij}$ and $\beta_{ij}$ associated with the edge connecting vertices $i$ and $j$, and area $\Omega_i$ corresponding to vertex $i$ in Eq.(\ref{['eq:angle']}).
• Figure 3: The variation of $RE$ with the number of eigenpairs.
• Figure 4: (a) Reference mapping of the electrodynamics at time point $t=1550$. (b) Estimated mappings produced by E-ST-GP and G-ST-GP under different training dataset sizes ($|\mathcal{X}_\text{tr}| = 50$ or $100$) and noise levels ($\sigma_\xi = 0.01, 0.05, 0.1$) at time $t=1550$. (c) Bar chart comparing the average $RE$'s for E-ST-GP and G-ST-GP from 5 replications across different scenarios in (b).
• Figure 5: (a) Comparison of the G-ST-GP model performance trained by signals collected from our A-AL strategy versus other methods including V-AL, S-AL, and F-AL with $\alpha_{n,1}=\alpha_{n,2}=0.5$ when $\sigma_\xi = 0.01$. (b) Comparison of A-AL and different F-AL approaches with $\gamma = \alpha_{n,1}/\alpha_{n,2}=0.5,\ 1.0$ and 1.5. (c) Performance evolution of the active learning processes provided by A-AL and R-AL (at time point $t=730$), shown alongside the initial estimation at $N^{+}=0$ and ground truth for reference.