Boundary constraints in physical, environmental and engineering models restrict smooth states such as temperature to follow known physical laws at the edges of their spatio-temporal domain. Examples include fixed-state or fixed-derivative (insulated) boundary conditions, and constraints that relate the state and the derivatives, such as in models of heat transfer. Despite their flexibility as prior models over system states, Gaussian random fields do not in general enable exact enforcement of such constraints. This work develops a new general framework for constructing linearly boundary-constrained Gaussian random fields from unconstrained Gaussian random fields over multi-dimensional, convex domains. This new class of models provides flexible priors for modeling smooth states with known physical mechanisms acting at the domain boundaries. Simulation studies illustrate how such physics-informed probability models yield improved predictive performance and more realistic uncertainty quantification in applications including probabilistic numerics, data-driven discovery of dynamical systems, and boundary-constrained state estimation, as compared to unconstrained alternatives.

翻译：物理、环境和工程模型中的边界约束要求平滑状态（如温度）在其时空域边缘遵循已知的物理规律。例如固定状态或固定导数（绝热）边界条件，以及关联状态与导数的约束（如热传导模型）。尽管高斯随机场作为系统状态的先验模型具有灵活性，但通常无法精确实施此类约束。本研究提出一种新通用框架，用于在多维凸域上从无约束高斯随机场构建线性边界约束的高斯随机场。该新模型类别为在已知边界物理机制下建模平滑状态提供了灵活先验。仿真研究表明，相较于无约束模型，此类物理信息概率模型在概率数值计算、数据驱动的动力系统发现及边界约束状态估计等应用中，能提升预测性能并提供更符合实际的不确定性量化。