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.

翻译：狄利克雷-诺依曼映射通过确保人工界面处状态变量与通量的连续性，实现了跨计算子域的多物理场模拟耦合。本文提出一种利用高斯过程学习图上狄利克雷-诺依曼映射的新方法，特别适用于数据遵循底层偏微分方程守恒约束的问题。该方法结合离散外微积分与非线性最优恢复理论，以推断顶点值与边值之间的关系。该框架能够生成覆盖全图的数据驱动预测，并附带不确定性量化，即使在观测数据仅限于部分顶点和边的情况下依然适用。通过在再生核希尔伯特空间范数上进行优化，同时对核复杂度施加最大似然估计惩罚，我们的方法确保生成的代理模型严格强制执行守恒定律且避免过拟合。我们在两个典型应用场景中验证了该方法：地下裂缝网络与动脉血流模拟。结果表明，即使在数据极度匮乏的情况下，该方法仍能保持高精度和良好校准的不确定性估计，突显了其在数据有限且可靠性不确定性量化至关重要的科学应用中的潜力。