Physics-Informed Variational State-Space Gaussian Processes (2409.13876v2)

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.

• The paper presents PHYSS-GP, a novel physics-informed state-space GP model that reduces spatial complexity from cubic to linear.
• It employs structured variational inference and spatial mini-batching techniques to efficiently manage spatio-temporal derivatives and large datasets.
• Empirical evaluations on systems like diffusion-reaction and ocean currents demonstrate improved predictive accuracy and faster computations compared to existing methods.

Review of "Physics-Informed Variational State-Space Gaussian Processes"

The paper "Physics-Informed Variational State-Space Gaussian Processes" by Oliver Hamelijnck, Arno Solin, and Theodoros Damoulas introduces an innovative framework integrating physics-based constraints and data-driven modeling using Gaussian Processes (GPs). The principal objective of the research is to address limitations in existing models which either suffer from poor computational scaling or focus exclusively on temporal domains.

Introduction to Physics-Informed Models

Integrating physical constraints through differential equations (ODEs and PDEs) into GP models, the authors situate their work within the broader context of physics-informed machine learning. Leveraging GPs for quantifying uncertainty and incorporating intricate physical relationships assures better predictive performance and computational efficiency. Prior models like PMM, HELMHOLTZ-GP, and AUTOIP attempted similar integrations but incurred high cubic computational complexity due to full covariance matrix computations.

Contributions and Innovations

The authors propose the PHYSS-GP, a physics-informed state-space GP capable of handling both spatio-temporal scenarios with linear computational complexity in the temporal dimension. Key contributions of the paper are:

1. State-Space GP Handling Spatio-Temporal Derivatives: The GP framework developed maintains linear-time computational complexity, significantly improving efficiency.
2. Unifying Framework with Variational Inference: The derived algorithm enables integrating both linear and non-linear PDEs effectively while ensuring computational efficiency comparable to state-of-the-art methods.
3. Approximations for Spatial Efficiency: The framework incorporates structured variational posterior, spatial sparsity, and spatial mini-batching techniques, reducing the spatial computational complexity from cubic to linear order.

Methodological Insights

The methodology hinges on exploiting state-space representations and Kalman smoothing for efficient computations. By utilizing Kronecker structures and Markov properties of temporal kernels, the proposed model ensures scalable inference while embedding physical laws as soft constraints or exact solutions for particular properties like curl or divergence-free fields.

Three specific approximations further enhance computational tractability:

• Spatio-Temporal Inducing Points (PHYSS-SVGP): By augmenting the model with inducing points defined on a spatio-temporal grid, cubic costs in spatial operations are alleviated.
• Structured Variational Inference (PHYSS-SVGPH): This reduces state size by defining approximate posteriors on temporal derivatives, simplifying computations significantly.
• Spatial Mini-Batching: For large-scale data, spatial mini-batching is employed to maintain linear cost relative to the spatial dimension, facilitating application to big datasets.

Empirical Evaluation

The paper presents comprehensive experiments on synthetic and real-world datasets demonstrating both computational efficiency and predictive performance enhancements:

• Non-linear Damped Pendulum: PHYSS-GP outperformed AUTOIP significantly in terms of computational time while maintaining superior predictive accuracy with increasing collocation points.
• Curl-Free Magnetic Field Strength: Demonstrated equivalent performance to HELMHOLTZ-GP but with markedly faster computational times, particularly at larger spatial scales.
• Diffusion-Reaction System: The EKS prior in PHYSS-GP exhibited remarkable performance, capturing sharp boundaries and requiring fewer epochs than competing methods.
• Ocean Currents: PHYSS-SVGPH efficiently handled large-scale data, showing excellent agreement with test data in a computationally feasible manner.

Implications and Future Directions

The theoretical advancements and empirical validations underscore the potential of PHYSS-GP to combine physical knowledge and observational data effectively. Practically, this integration enhances predictive modeling in fields like oceanography, climate modeling, and beyond, where physical laws are pivotal.

Future research could focus on expanding the framework to deep Gaussian processes for handling more complex non-linearities and exploring methods beyond the collocation approach for imposing differential equation constraints.

Conclusion

The research delineates a robust framework for physics-informed GP models, introducing significant computational efficiencies and enhancing predictive performance. By leveraging variational state-space methods and innovative approximations, PHYSS-GP stands well as a substantial contribution to the domain of physics-informed machine learning and offers a forward trajectory for integrating mechanistic models with data-driven techniques.