Forecasting complex dynamical phenomena in settings where only partial knowledge of their dynamics is available is a prevalent problem across various scientific fields. While purely data-driven approaches are arguably insufficient in this context, standard physical modeling based approaches tend to be over-simplistic, inducing non-negligible errors. In this work, we introduce the APHYNITY framework, a principled approach for augmenting incomplete physical dynamics described by differential equations with deep data-driven models. It consists in decomposing the dynamics into two components: a physical component accounting for the dynamics for which we have some prior knowledge, and a data-driven component accounting for errors of the physical model. The learning problem is carefully formulated such that the physical model explains as much of the data as possible, while the data-driven component only describes information that cannot be captured by the physical model, no more, no less. This not only provides the existence and uniqueness for this decomposition, but also ensures interpretability and benefits generalization. Experiments made on three important use cases, each representative of a different family of phenomena, i.e. reaction-diffusion equations, wave equations and the non-linear damped pendulum, show that APHYNITY can efficiently leverage approximate physical models to accurately forecast the evolution of the system and correctly identify relevant physical parameters. Code is available at https://github.com/yuan-yin/APHYNITY .
翻译:在只能部分了解其动态的环境下预测复杂的动态现象是各种科学领域的一个普遍问题。虽然纯粹由数据驱动的方法在这方面可能不够充分,但标准的物理建模方法往往过于简单,引起不可忽略的错误。在这项工作中,我们引入了APHYNITY框架,这是一个原则性方法,用深层数据驱动模型来增强差异方程式描述的不完整的物理动态,它包括将动态分解成两个组成部分:一个物理组成部分,说明我们以前掌握一些知识的动态,一个数据驱动的构件,计算物理模型的错误。学习问题经过仔细制定,使物理模型尽可能解释数据,而数据驱动的构件只描述物理模型无法收集的信息,不胜于不减。这不仅为这种分解提供了存在和独特性,而且还确保了可解释性和效益的概括化。在三种重要用途案例中进行的实验,每个代表不同现象的类别,即反向增缩式方程式、波方程式方程方程式和不精确的物理演化模型,可以准确显示在物理系统/磁力流上可精确地显示相关的亚化系统。