Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

TL;DR

This paper introduces a novel Gaussian process-based method for learning Dirichlet-to-Neumann maps on graphs, enabling accurate, uncertainty-aware predictions in physics-informed graph problems with limited data.

Contribution

It combines discrete exterior calculus and nonlinear optimal recovery to create a data-driven, conservation-enforcing surrogate for Dirichlet-to-Neumann maps on graphs.

Findings

High accuracy in subsurface fracture network modeling

Well-calibrated uncertainty estimates with limited data

Effective in arterial blood flow simulations

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.