{"ID":2882694,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.09623","arxiv_id":"2508.09623","title":"Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems","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.","short_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, prob...","url_abs":"https://arxiv.org/abs/2508.09623","url_pdf":"https://arxiv.org/pdf/2508.09623v2","authors":"[\"Akshay Thakur\",\"Sawan Kumar\",\"Matthew Zahr\",\"Souvik Chakraborty\"]","published":"2025-08-13T08:54:18Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[]","has_code":false}