论文标题
基于机器学习的非线性相位场动力学发现
Machine learning based data-driven discovery of nonlinear phase-field dynamics
论文作者
论文摘要
关于大规模复杂系统的主要问题之一是涉及从详细的微观特性中产生的有效相互作用和驱动力。粗粒元模型旨在用降低的自由度数量的粗尺度方程来描述复杂的系统。机器学习的最新发展（ML）算法已直接从数据中直接赋予了管理方程的发现过程。但是，很难发现具有高阶导数的部分微分方程（PDE）。在本文中，我们介绍了基于多层感知器（MLP），卷积神经网络（CNN）以及CNN和长期短期记忆（CNN-LSTM）结构的组合，以发现具有非现场模型的非现场模型的非线性运动方程的新型数据驱动体系结构。众所周知的Allen-Cahn，Cahn--Hilliard和相位场晶体（PFC）模型被用作测试用例。使用了两种概念上不同类型的实现：（a）在没有任何物理假设的情况下（Black-Box模型），以物理直觉（例如衍生物的局部依赖性）和（b）指导。我们表明，在两种情况下，我们不仅可以有效地学习该领域的时间导数，而且还可以使用数据驱动的PDE来及时传播该领域，并与原始PDE达成良好的一致性。
One of the main questions regarding complex systems at large scales concerns the effective interactions and driving forces that emerge from the detailed microscopic properties. Coarse-grained models aim to describe complex systems in terms of coarse-scale equations with a reduced number of degrees of freedom. Recent developments in machine learning (ML) algorithms have significantly empowered the discovery process of the governing equations directly from data. However, it remains difficult to discover partial differential equations (PDEs) with high-order derivatives. In this paper, we present new data-driven architectures based on multi-layer perceptron (MLP), convolutional neural network (CNN), and a combination of CNN and long short-term memory (CNN-LSTM) structures for discovering the non-linear equations of motion for phase-field models with non-conserved and conserved order parameters. The well-known Allen--Cahn, Cahn--Hilliard, and the phase-field crystal (PFC) models were used as the test cases. Two conceptually different types of implementations were used: (a) guided by physical intuition (such as local dependence of the derivatives) and (b) in the absence of any physical assumptions (black-box model). We show that not only can we effectively learn the time derivatives of the field in both scenarios, but we can also use the data-driven PDEs to propagate the field in time and achieve results in good agreement with the original PDEs.
