Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

2025-06-03

AI Summary

This work addresses the challenge of learning Dirichlet-to-Neumann (DtN) mappings for multi-physics coupling problems on graph-structured domains, under sparse observations at vertices and/or edges, while strictly enforcing PDE-based conservation laws. We propose the first framework integrating Gaussian processes, discrete exterior calculus, and nonlinear optimal recovery—rigorously embedding conservation constraints into reproducing kernel Hilbert space modeling and employing maximum-likelihood kernel complexity regularization to suppress overfitting. Our method yields physically consistent, non-overfitted surrogate models capable of predicting full-graph states and fluxes, alongside well-calibrated uncertainty quantification. Evaluated on fracture-network flow and arterial blood flow simulations, it achieves high accuracy and reliable uncertainty estimates even with extremely sparse observational data.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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.

Problem

• Learning Dirichlet-to-Neumann maps on graphs using Gaussian processes
• Enforcing conservation laws in data-driven predictions with uncertainty quantification
• Addressing data scarcity in multiphysics simulations with reliable uncertainty estimates

Innovation

• Uses Gaussian processes for learning maps
• Combines discrete exterior calculus
• Optimizes kernel Hilbert space norm