We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the data-driven model by design. We provide a rigorous convergence statement of the method. The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus. Moreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory. This is illustrated on the example of the discrete wave equation and Schr\"odinger equation. The article constitutes an extension of our previous article arXiv:2404.19626 for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes.

翻译：我们提出了一种基于高斯过程回归的方法，用于从场论观测解中识别离散变分原理。该方法基于数据驱动的离散拉格朗日密度识别，是一种几何机器学习技术——其设计使得真实场论的变分结构在数据驱动模型中得以体现。我们给出了该方法的严格收敛性证明，该证明通过规避变分法反问题中离散拉格朗日密度多义性带来的挑战而实现。此外，本方法可用于量化运动方程及离散场论任意线性可观测量中的模型不确定性，并以离散波动方程和薛定谔方程为例进行说明。本文是我们先前论文arXiv:2404.19626的拓展，将变分动力学（离散）拉格朗日量的数据驱动识别从常微分方程框架推广至离散偏微分方程框架。