编码物理以学习反应扩散过程
Encoding physics to learn reaction-diffusion processes
AI总结：
针对反应扩散等复杂时空动力系统建模中先验知识不足的问题，提出一种强制编码物理结构的深度学习框架，在稀疏数据下实现高精度、可解释且具泛化能力的正向与逆向分析及PDE发现。
摘要：
对复杂时空动力系统（如反应扩散过程）的建模在很大程度上依赖于偏微分方程（PDEs）。然而，由于对化学、生物学、地质学、物理学和生态学中一些研究尚不充分的动力系统缺乏足够的先验知识，且缺乏用于描述系统变量非线性过程的显式 PDE 公式，预测此类系统的演化仍是一项具有挑战性的任务。通过机器学习将测量数据与我们有限的先验物理知识相统一，为解决这一问题提供了新途径。现有的物理信息学习范式通过软惩罚约束施加物理定律，其求解质量在很大程度上依赖于反复试验来正确设置超参数。由于此类方法的核心仍根植于黑盒神经网络，所得模型通常缺乏可解释性，并在外推和泛化方面存在严重问题。为此，我们提出了一种深度学习框架，该框架强制编码给定的物理结构，以促进在稀疏数据情况下的时空动力学学习。我们展示了所提方法如何应用于有关 PDE 系统的各类问题，包括正向和逆向分析、数据驱动建模以及 PDE 发现。通过大量数值实验证明，这种编码物理的学习范式展现出高精度、鲁棒性、可解释性和泛化能力。
英文摘要：
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.
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