Modeling complex spatiotemporal dynamical systems, such as the reaction-diffusion processes, have largely relied on partial differential equations (PDEs). However, due to insufficient prior knowledge on some under-explored dynamical systems, such as those in chemistry, biology, geology, physics and ecology, and the lack of explicit PDE formulation used for describing the nonlinear process of the system variables, to predict the evolution of such a system remains a challenging task. Unifying measurement data and our limited prior physics knowledge via machine learning provides us with a new path to solving this problem. Existing physics-informed learning paradigms impose physics laws through soft penalty constraints, whose solution quality largely depends on a trial-and-error proper setting of hyperparameters. Since the core of such methods is still rooted in black-box neural networks, the resulting model generally lacks interpretability and suffers from critical issues of extrapolation and generalization. To this end, we propose a deep learning framework that forcibly encodes given physics structure to facilitate the learning of the spatiotemporal dynamics in sparse data regimes. We show how the proposed approach can be applied to a variety of problems regarding the PDE system, including forward and inverse analysis, data-driven modeling, and discovery of PDEs. The resultant learning paradigm that encodes physics shows high accuracy, robustness, interpretability and generalizability demonstrated via extensive numerical experiments.

翻译：对复杂时空动态系统（如反应-扩散过程）的建模在很大程度上依赖偏微分方程。然而，由于对某些探索不足的动态系统（如化学、生物学、地质学、物理学和生态学中的系统）缺乏充分的先验知识，且缺乏用于描述系统变量非线性过程的显式PDE公式，预测此类系统的演变仍是一项具有挑战性的任务。通过机器学习融合测量数据与有限的先验物理知识，为解决该问题提供了新途径。现有的物理信息学习范式通过软惩罚约束施加物理定律，其解质量很大程度上依赖于通过试错法适当设置超参数。由于这类方法的核心仍植根于黑箱神经网络，所得模型通常缺乏可解释性，并面临外推与泛化的关键问题。为此，我们提出一种深度学习框架，通过强制编码给定的物理结构，促进稀疏数据场景下对时空动态的学习。我们展示了该方法如何应用于PDE系统的多种问题，包括正反问题分析、数据驱动建模以及PDE的发现。大量数值实验表明，这种编码物理的学习范式具有高精度、强鲁棒性、可解释性与泛化能力。