We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schr\"odinger equation.

翻译：我们从时空格点上的场观测数据出发，展示了如何学习离散场论。为此，我们训练了离散拉格朗日密度的神经网络模型，使得离散欧拉-拉格朗日方程与给定训练数据一致。由此，我们获得了一种保持结构的机器学习架构。拉格朗日密度并非由场论解唯一确定。我们引入了一种技术，用于推导训练过程中的正则化项，以优化离散场论的数值正则性。最小化这些正则化项可保证在训练数据附近，离散场论在数值模拟中表现出鲁棒性和高效性。此外，我们展示了如何识别底层连续场论的结构简单解（如行波）。即使训练数据中不存在行波，这一方法仍然可行。相比之下，基于数据驱动的模型降阶方法在训练数据中缺乏结构简单解时，难以识别合适的隐空间以包含此类解。我们通过波动方程和薛定谔方程的实例验证了上述思想。