Many inverse problems require reconstructing physical fields from limited and noisy data while incorporating known governing equations. A growing body of work within probabilistic numerics formalizes such tasks via Bayesian inference in function spaces by assigning a physically meaningful prior to the latent field. In this work, we demonstrate that Brownian bridge Gaussian processes can be viewed as a softly-enforced physics-constrained prior for the Poisson equation. We first show equivalence between the variational problem associated with the Poisson equation and a kernel ridge regression objective. Then, through the connection between Gaussian process regression and kernel methods, we identify a Gaussian process for which the posterior mean function and the minimizer to the variational problem agree, thereby placing this PDE-based regularization within a fully Bayesian framework. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We then develop a finite-dimensional representation in function space and prove convergence of the projected prior and resulting posterior in Wasserstein distance. Finally, we connect the method to the important problem of identifying model-form error in applications, providing a diagnostic for model misspecification.

翻译：许多反问题需要从有限且含噪声的数据中重建物理场，同时结合已知的控制方程。概率数值计算领域的研究日益增多，通过在函数空间中为隐场赋予具有物理意义的先验，将此类任务形式化为贝叶斯推断。本文证明，布朗桥高斯过程可视为泊松方程的一种软约束物理先验。我们首先展示了与泊松方程相关的变分问题与核岭回归目标之间的等价性。接着，通过高斯过程回归与核方法的联系，我们确定了一个高斯过程，其后验均值函数与该变分问题的极小化解一致，从而将这种基于偏微分方程的正则化置于完整的贝叶斯框架中。这一关联使我们能够探讨不同的理论问题，例如反问题的收敛性和行为。随后，我们在函数空间中构建有限维表示，并证明了投影先验及所得后验在Wasserstein距离下的收敛性。最后，我们将该方法与应用中重要的模型形式误差识别问题联系起来，为模型设定错误提供诊断工具。