Physics-Informed Variational State-Space Gaussian Processes

TL;DR

This work addresses the challenge of incorporating mechanistic physical knowledge into probabilistic models for spatio-temporal data by introducing physics-informed state-space Gaussian processes (physs-gp). It unifies linear and nonlinear PDE/ODE constraints within a state-space GP framework and derives a variational lower bound (physs-vgp/physs-eks) that preserves linear-time inference in time while enabling efficient handling of spatial complexity. Three scalable approximations—spatio-temporal inducing points, structured variational posteriors, and spatial mini-batching—reduce cubic spatial costs, enabling application to large-scale problems. Empirical results on synthetic and real-world datasets (pendulum dynamics, curl-free magnetic fields, diffusion-reaction systems, and ocean currents) demonstrate improved predictive performance and substantial speedups over AutoIP and Helmholtz-GP, with code released for reproducibility. Overall, the approach advances uncertainty-aware physics-informed modelling by providing a scalable, end-to-end probabilistic framework for spatio-temporal physics with practical impact in scientific and engineering domains.

Abstract

Differential equations are important mechanistic models that are integral to many scientific and engineering applications. With the abundance of available data there has been a growing interest in data-driven physics-informed models. Gaussian processes (GPs) are particularly suited to this task as they can model complex, non-linear phenomena whilst incorporating prior knowledge and quantifying uncertainty. Current approaches have found some success but are limited as they either achieve poor computational scalings or focus only on the temporal setting. This work addresses these issues by introducing a variational spatio-temporal state-space GP that handles linear and non-linear physical constraints while achieving efficient linear-in-time computation costs. We demonstrate our methods in a range of synthetic and real-world settings and outperform the current state-of-the-art in both predictive and computational performance.

Figures (6)

• Figure 1: The state-space formalism allows for linear-time inference in the temporal dimension.
• Figure 2: Curl free synthetic example. The left panel displays the learnt scalar potential functions by physs-gp with $N_{\mathrm{s}}=20$, and the right panel illustrates the associated vector field.
• Figure 3: Results on the diffusion reaction system. The top row denotes the predictive mean, and the bottom the $95\%$ confidence intervals. The white line denotes where the training data ends. Only physs-eks captures the sharp boundaries, due to the iwp kernel. physs-$\textsc{svgp}\xspace_\textsc{h}\xspace$ recovers a similar solution to autoip but at half the computational cost.
• Figure 4: Predicted ocean currents by physs-$\textsc{svgp}\xspace_\textsc{h}\xspace$. True observations are in grey, and predictions in green. The thickness of the line represents uncertainty and is computed by the L2 norm of the standard deviations across both outputs.
• Figure 5: Predictive distributions of gp and physs-gp on the monotonic function in \ref{['sec:app_experimental_details']}. The gp cannot incorporate monotonicity information and fits the data.

Theorems & Definitions (5)

• Example 3.1: eks Prior and pmm
• Example 3.2: helmholtz-gp and Curl and Divergence-Free Vector Fields in $2$D
• Theorem 3.1
• Theorem A.1
• proof