To predict smooth physical phenomena from observations, spline interpolation provides an interpretable framework by minimizing an energy functional associated with the Laplacian operator. This work proposes a methodology to construct a spline predictor on a compact Riemannian manifold, while quantifying the uncertainty inherent in the classical deterministic solution. Our approach leverages the equivalence between spline interpolation and universal kriging with a specific covariance kernel. By adopting a Gaussian random field framework, we generate stochastic simulations that reflect prediction uncertainty. However, on compact manifolds, the covariance kernel depends on the generally unknown spectrum of the Laplace-Beltrami operator. To address this, we introduce a finite element approximation based on a triangulation of the manifold. This leads to the use of intrinsic Gaussian Markov Random Fields (GMRF) and allows for the incorporation of anisotropies through local modifications of the Riemannian metric. The method is validated using a temperature study on a sphere, where the operator's spectrum is known, and is further extended to a test case on a cylindrical surface.

翻译：为了从观测数据中预测平滑物理现象，样条插值通过最小化与拉普拉斯算子相关的能量泛函，提供了一种可解释的框架。本文提出了一种在紧致黎曼流形上构建样条预测器的方法，同时量化了经典确定性解中固有的不确定性。我们的方法利用了样条插值与具有特定协方差核的通用克里金法之间的等价性。通过采用高斯随机场框架，我们生成了反映预测不确定性的随机模拟。然而，在紧致流形上，协方差核依赖于通常未知的拉普拉斯-贝尔特拉米算子谱。为解决这一问题，我们引入了基于流形三角剖分的有限元近似。这导致采用内在高斯马尔可夫随机场（GMRF），并允许通过局部修改黎曼度量来纳入各向异性。该方法通过一个球面上的温度研究进行了验证（其中算子的谱已知），并进一步推广到圆柱面上的测试案例。