Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems
This work addresses the computational bottleneck in probabilistic numerics for PDEs, enabling scalable uncertainty quantification in scientific computing.
The authors tackled the high computational cost of probabilistic PDE solvers by developing a stochastic dual descent algorithm and an adaptive collocation point selection strategy, achieving linear per-iteration complexity and scalability to large numbers of points.
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.