We present a method for the data-driven learning of physical phenomena whose evolution in time depends on history terms. It is well known that a Mori-Zwanzig-type projection produces a description of the physical phenomena that depends on history, and also incorporates noise. If the data stream is sampled from the projected Mori-Zwanzig manifold, the description of the phenomenon will always depend on one or more unresolved variables, a priori unknown, and will also incorporate noise.

The present work introduces a novel technique able to unveil the presence of such internal variables—although without giving it a precise physical meaning—and to minimize the inherent noise. The method is based upon a refinement of the scale at which the phenomenon is described by means of kernel-PCA techniques. By learning the metriplectic form of the evolution of the physics, the resulting approximation satisfies basic thermodynamic principles such as energy conservation and positive entropy production.

Examples are provided that show the potential of the method in both discrete and continuum mechanics.

我们提出了一种数据驱动学习物理现象的方法，该方法的时间演变取决于历史术语。众所周知，Mori-Zwanzig型投影可以描述依赖于历史的物理现象，并且还包含噪声。如果从预计的Mori-Zwanzig流形中采样数据流，则现象的描述将始终取决于一个或多个未解决的变量（先验未知），并且还会包含噪声。

本工作介绍了一种新颖的技术，该技术能够揭示这种内部变量的存在（尽管没有给出精确的物理含义），并且可以最小化固有噪声。该方法基于通过核-PCA技术描述现象的尺度的细化。通过学习物理演化的中电形式，得出的近似值满足基本的热力学原理，例如能量守恒和正熵产生。

提供的示例显示了该方法在离散和连续力学中的潜力。