Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

TL;DR

This work tackles learning Dirichlet-to-Neumann maps on graphs to enable efficient coupling of subdomain simulations under conservation laws. The authors propose a framework that blends Gaussian processes with discrete exterior calculus (DEC) to enforce a global divergence-free constraint, yielding an RKHS-minimal surrogate for edge flux given boundary data. By embedding the learning problem in a nonlinear optimal recovery setting (CGC) and solving a constrained LCQP via KKT with a Schur complement, they obtain predictive means and calibrated uncertainty across the entire graph. The method delivers accurate boundary predictions and physically meaningful, globally conservative interior predictions on challenging domains (subsurface fracture networks and arterial graphs) even under severe data scarcity, with formal error bounds and uncertainty quantification. This approach offers a principled, data-driven alternative to repeated subdomain solves in multiscale simulators, with verifiable trust through probabilistic guarantees.

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.

Figures (7)

• Figure 1: Toy circuit problem. Left: a toy circuit with three edges in series. Right: inference results for leftmost boundary edge. Training points are marked in red, with the test predictions and target values in purple and blue, respectively. The pink shading indicates the estimated error bound from \ref{['lem:error']} for $\delta=0.05,$ corresponding to $95\%$ confidence. Note that the error bound grows away from the cluster of training points.
• Figure 2: Histogram of $\log_2(\lvert\text{error}\rvert/\lvert\text{bound}\rvert)$ for different test-set sizes (color-coded). Values below zero indicate that the estimated bound safely over-predicts the true error; only a few cases in the sample of size 500 lie above zero, as expected from the $95\%$ confidence level.
• Figure 3: Example subsurface fracture dataset. A) Graph topology with boundary edges and vertices in red; B) Flow velocity on edges; C) Hydraulic head on vertices. Flow proceeds predominantly along darker-red edges from yellow vertices (with high hydraulic head) to black vertices (with low hydraulic head).
• Figure 4: Example results for the subsurface fracture network problem. A) Predicted hydraulic head; B) Ground truth hydraulic head; C) Error (true-predicted) on edges and vertices; D) Predicted flow velocity; E) Ground truth flow velocity; F) Inferred D2N relationship for boundary edge 25, with posterior mean in red, $95\%$ confidence band in pink, training data in red, test predictions in purple, and ground truth test values in blue.
• Figure 5: Example arterial dataset. A) Topology of the dataset, highlighting boundary vertices and edges in red; B) Topology of the dataset, highlighting interior vertices and edges of interest in blue; C) Flow rate on graph edges; D) Pressure on graph vertices. Blood flows from the right-most boundary through the rest of the arterial branches.

Theorems & Definitions (11)

• Theorem 1
• Definition 1: Incidence matrix
• Definition 2: Graph gradient operator
• Definition 3: Graph divergence operator
• Theorem 2
• proof
• Remark
• Theorem 2
• proof
• Lemma 1