Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

• URL: http://arxiv.org/abs/2506.02337v1
• Date: Tue, 03 Jun 2025 00:25:49 GMT
• Title: Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs
• Authors: Adrienne M. Propp, Jonas A. Actor, Elise Walker, Houman Owhadi, Nathaniel Trask, Daniel M. Tartakovsky,
• Abstract summary: We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes.<n>We show that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity.
• Score: 1.385411134620987
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation constraint from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By optimizing over the reproducing kernel Hilbert space norm while applying a maximum likelihood estimation penalty on kernel complexity, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface fracture networks and arterial blood flow. Our results show that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical.

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