Deep Energy-Based Discrete-Time Physical Model for Reproducing Energetic Behavior

Modeling and simulating physical phenomena, especially those governed by partial differential equations (PDEs), pose significant challenges in computational physics and scientific machine learning. While neural network approaches have made strides in learning continuous-time dynamics, they have struggled with discrete-time scenarios and often fail to adhere to fundamental laws of physics, such as the conservation of energy and mass. This study addresses this gap by introducing a novel deep energy-based discrete-time model. In the real world, energy-based modeling theories like Hamiltonian mechanics and the Landau theory are pivotal, as they support various laws of physics. By integrating differential geometric structures into neural networks as coefficient matrices, our model successfully simulates the conservation and dissipation laws of energy and mass. Furthermore, we propose an automatic discrete differentiation algorithm, which enables neural networks to utilize the discrete gradient method, ensuring adherence to these laws in discrete-time settings. This capability also facilitates the identification of such laws directly from data by learning matrices that represent geometric structures. These advantages are verified using simulation results of physical phenomena, namely the 1- and 2-D Korteweg-de Vries (KdV) equation and the Cahn-Hilliard equation.