Machine learning based data-driven discovery of nonlinear phase-field dynamics

Abstract

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.

References in corpus (20)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• DeepXDE: A deep learning library for solving differential equations
• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• DGM: A deep learning algorithm for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• Modeling Elasticity in Crystal Growth
• Modeling Elastic and Plastic Deformations in Non-Equilibrium Processing Using Phase Field Crystals
• Learning data driven discretizations for partial differential equations
• Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks
• Continuum field description of crack propagation
• Training convolutional neural networks to estimate turbulent sub-grid scale reaction rates
• Phase field modeling of crack propagation
• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
• Coarse-scale PDEs from fine-scale observations via machine learning
• Consistent Hydrodynamics for Phase Field Crystals
• Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations
• A new dimension to Turing patterns
• Particles to Partial Differential Equations Parsimoniously

Cited by in corpus (2)

• A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data
• Black and Gray Box Learning of Amplitude Equations: Application to Phase Field Systems