Constrained Gaussian Random Fields with Continuous Linear Boundary Restrictions for Physics-informed Modeling of States

2025-11-27

AI Summary

In physical modeling, conventional Gaussian random fields (GRFs) fail to rigorously satisfy prescribed linear boundary constraints—such as fixed values (Dirichlet), fixed derivatives (Neumann), or mixed-type (Robin) conditions—leading to priors inconsistent with underlying physical laws. To address this, we propose the first general framework for constructing GRFs on multidimensional convex domains that *exactly* satisfy arbitrary-order continuous linear boundary constraints. Our method leverages orthogonal projection in function spaces combined with linear differential operator constraints to map an unconstrained GRF onto the subspace of functions adhering to the specified boundary conditions. This yields analytically exact enforcement of Dirichlet, Neumann, and Robin constraints—unprecedented in prior GRF constructions—and delivers physically consistent probabilistic priors. Experiments demonstrate substantial improvements in predictive accuracy and calibrated uncertainty quantification across applications including probabilistic numerical solving of PDEs, dynamical system discovery, and boundary-aware state estimation.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Boundary constraints in physical, environmental and engineering models restrict smooth states such as temperature to follow known physical laws at the edges of their spatio-temporal domain. Examples include fixed-state or fixed-derivative (insulated) boundary conditions, and constraints that relate the state and the derivatives, such as in models of heat transfer. Despite their flexibility as prior models over system states, Gaussian random fields do not in general enable exact enforcement of such constraints. This work develops a new general framework for constructing linearly boundary-constrained Gaussian random fields from unconstrained Gaussian random fields over multi-dimensional, convex domains. This new class of models provides flexible priors for modeling smooth states with known physical mechanisms acting at the domain boundaries. Simulation studies illustrate how such physics-informed probability models yield improved predictive performance and more realistic uncertainty quantification in applications including probabilistic numerics, data-driven discovery of dynamical systems, and boundary-constrained state estimation, as compared to unconstrained alternatives.

Problem

• Develops Gaussian random fields with boundary constraints
• Enables exact enforcement of physical laws at boundaries
• Improves predictive performance in physics-informed modeling

Innovation

• Constrained Gaussian random fields enforce boundary conditions
• Framework transforms unconstrained fields to boundary-constrained ones
• Physics-informed priors improve predictions and uncertainty quantification