Variational Method-Based Operator Neural Network for Dynamic Systems in Energy Materials Governed by Gradient Flows

Variational methods have been well established to derive the governing equations of complex, coupled, and nonlinear systems. One example is the gradient flow that entails finding and constructing an appropriate potential energy and inner product to incorporate the kinetics into a variational framework. Gradient flows can be applied to a large variety of physics, including diffusion, phase separation, microstructure evolution, etc., where the governing partial differential equations (PDEs) can be eventually obtained. The conventional numerical methods, such as the finite element method (FEM), have been proven to be effective in solving PDEs. However, it is still challenging for systems with high dimensionality and nonlinearity. Recently, we have witnessed great successes in machine learning (ML) applications in many scientific disciplines. The concept of scientific machine learning was proposed and widely used by many research groups to solve variational problems. One of such approaches is approximating the solutions with ML models and training them by minimizing the energy functional, instead of solving a large set of non-linear equations. An example is the energy-based neural networks. Recently, another approach started to gain increasing attention, which is referred to as operator learning (OL). OL models the mapping from one functional space to another. It has the potential to incorporate solutions with different initial conditions into one algorithm. In this study, we proposed a general variational method-based operator neural network framework for dynamics systems governed by gradient flows. To validate the proposed framework, we investigated several dynamics systems that commonly exist in energy materials, including the linear relaxation kinetics, Allen-Cahn dynamics, and phase-field dynamic fracture. We compared the prediction of the proposed neural network with the FEM solution and found satisfactory agreements. We expect that the trained neural networks can work as surrogate models for energy materials to provide rapid predictions.