Technology Category
Reasoning under Uncertainty: Probabilistic ProgrammingMachine Learning: Probabilistic Circuits and Graphical ModelsCognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields
🤖 AI Summary
To address scalability bottlenecks in large-scale probabilistic PDE solving caused by dense covariance matrices, this paper proposes a scalable physics-informed Bayesian solver. Methodologically, it introduces the first coupling of Gaussian Markov random fields (GMRFs) with stochastic PDE (SPDE) priors, yielding an SPDE-GMRF joint prior that inherits Markovian sparsity; combined with a variational inference framework leveraging sparse linear algebra and embedded physical constraints, it explicitly models discretization error, parameter uncertainty, and observational noise. Compared to conventional dense Gaussian process approaches, the method achieves order-of-magnitude reductions in memory and runtime overhead for nonlinear PDE inverse problems, while significantly accelerating convergence. The core contribution is a unified modeling paradigm that overcomes expressivity limitations of standard covariance functions—simultaneously ensuring physical interpretability, computational scalability, and rigorous uncertainty quantification.
Technology Category
Application Category
📝 Abstract
Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers -- Bayesian statistical models mostly based on Gaussian process (GP) priors which seamlessly combine empirical measurements and mechanistic knowledge. As such, they quantify uncertainties arising from e.g. noisy or missing data, unknown PDE parameters or discretization error by design. Prior work has established connections to classical PDE solvers and provided solid theoretical guarantees. However, scaling such methods to large-scale problems remains a fundamental challenge primarily due to dense covariance matrices. Our approach addresses the scalability issues by leveraging the Markov property of many commonly used GP priors. It has been shown that such priors are solutions to stochastic PDEs (SPDEs) which when discretized allow for highly efficient GP regression through sparse linear algebra. In this work, we show how to leverage this prior class to make probabilistic PDE solvers practical, even for large-scale nonlinear PDEs, through greatly accelerated inference mechanisms. Additionally, our approach also allows for flexible and physically meaningful priors beyond what can be modeled with covariance functions. Experiments confirm substantial speedups and accelerated convergence of our physics-informed priors in nonlinear settings.
Problem
Research questions and friction points this paper is trying to address.
Address scalability issues in probabilistic PDE solvers
Enable efficient GP regression using sparse linear algebra
Provide flexible, physics-informed priors for large-scale nonlinear PDEs
Innovation
Methods, ideas, or system contributions that make the work stand out.
Leverages Gaussian Markov Random Fields for scalability
Uses sparse linear algebra for efficient GP regression
Enables flexible, physics-informed priors for large-scale PDEs
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