The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.

翻译：本文展示了如何从数据中学习由未知变分偏微分方程（PDE）支配的动力系统模型。我们不采用降阶技术，而是学习一个由离散拉格朗日密度$L_d$（建模为神经网络）控制的离散场论。为了获得适合数值计算的场论，必须仔细正则化训练$L_d$的损失函数：我们推导出一个正则化项，以优化离散欧拉-拉格朗日方程的可解性。其次，我们开发了一种方法，用于寻找机器学习离散场论中的解，这些解构成了底层连续PDE的行波。