# Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

arXiv: https://arxiv.org/abs/2506.02337 · HTML: https://mlsanity.com/paper/2506.02337/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 43 references that are in the map.
- First posted on arXiv 3 Jun 2025.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Learning dynamical systems from data: a simple cross-validation perspective](https://mlsanity.com/paper/2007.05074/) (2020)
- [Learning Reduced-Order Models for Cardiovascular Simulations with Graph Neural Networks](https://mlsanity.com/paper/2303.07310/) (2023)
- [Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods](https://mlsanity.com/paper/2103.10935/) (2021)
- [GINNs: Graph-Informed Neural Networks for Multiscale Physics](https://mlsanity.com/paper/2006.14807/) (2020)
- [Error Bounds For Gaussian Process Regression Under Bounded Support Noise With Applications To Safety Certification](https://mlsanity.com/paper/2408.09033/) (2024)
- [Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation](https://mlsanity.com/paper/2405.13149/) (2024)
- [Dynamic flux surrogate-based partitioned methods for interface problems](https://mlsanity.com/paper/2402.03560/) (2024)

## Built on by (most cited first, 1 of 1)

- [Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification](https://mlsanity.com/paper/2606.11650/) (2026)

## Abstract (arXiv)

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.