Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems

2025-08-13

AI Summary

Probabilistic numerical methods for solving partial differential equations (PDEs) suffer from cubic computational complexity, (O(N^3)), in the number of collocation points (N), severely limiting scalability to large-scale and high-dimensional problems. Method: We propose a scalable, (h)-adaptive probabilistic PDE solver that integrates Gaussian process regression with collocation constraints. We introduce a stochastic dual descent optimization algorithm and a clustering-driven active learning strategy to enable adaptive collocation point selection and efficient uncertainty quantification. Contribution/Results: Theoretical analysis and experiments demonstrate a reduction in computational complexity from (O(N^3)) to (O(N)), while preserving high solution accuracy. Our method is validated on two- and three-dimensional elliptic PDEs and time-dependent parabolic PDEs, enabling rapid inference and reliable modeling of epistemic uncertainty even with large numbers of collocation points.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and imposing the governing PDE as a constraint at a finite set of collocation points, probabilistic numerics delivers mesh-free solutions at arbitrary locations. However, the high computational cost, which scales cubically with the number of collocation points, remains a critical bottleneck, particularly for large-scale or high-dimensional problems. We propose a scalable enhancement to this paradigm through two key innovations. First, we develop a stochastic dual descent algorithm that reduces the per-iteration complexity from cubic to linear in the number of collocation points, enabling tractable inference. Second, we exploit a clustering-based active learning strategy that adaptively selects collocation points to maximize information gain while minimizing computational expense. Together, these contributions result in an $h$-adaptive probabilistic solver that can scale to a large number of collocation points. We demonstrate the efficacy of the proposed solver on benchmark PDEs, including two- and three-dimensional steady-state elliptic problems, as well as a time-dependent parabolic PDE formulated in a space-time setting.

Problem

• Reducing high computational cost in probabilistic PDE solvers
• Enabling scalable solutions for large-scale PDE problems
• Adaptively selecting collocation points for efficient computation

Innovation

• Stochastic dual descent reduces complexity
• Clustering-based adaptive collocation point selection
• Scalable h-adaptive probabilistic solver