One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear (partial and ordinary) differential equations together with their boundary conditions. We construct multi-output Gaussian process priors with realizations dense in the solution set of such systems, in particular any solution (and only such solutions) can be represented to arbitrary precision by Gaussian process regression. The construction is fully algorithmic via Gr\"obner bases and it does not employ any approximation. It builds these priors combining two parametrizations via a pullback: the first parametrizes the solutions for the system of differential equations and the second parametrizes all functions adhering to the boundary conditions.

中文翻译：

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具有边界条件的线性约束高斯过程

贝叶斯机器学习的一个目标是将先验知识编码为先验分布，以有效地对数据建模。我们考虑来自线性（偏和常）微分方程系统的先验知识及其边界条件。我们构建了多输出高斯过程先验，在这些系统的解决方案集中具有密集的实现，特别是任何解决方案（并且只有这样的解决方案）都可以通过高斯过程回归表示为任意精度。该构造是通过 Gr\"obner 基完全算法化的，它不使用任何近似值。它通过回拉将两个参数化组合起来构建这些先验：第一个参数化微分方程组的解，第二个参数化所有遵循边界条件。