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

Area of Science:

• Computational Physics
• Scientific Machine Learning
• Numerical Analysis

Background:

• Neural networks struggle with discrete-time dynamics and preserving physical laws like energy and mass conservation.
• Existing models often fail to adhere to fundamental physical principles in simulations.
• Energy-based modeling theories are crucial for understanding physical laws.

Purpose of the Study:

• To develop a novel deep energy-based discrete-time model for simulating physical phenomena.
• To ensure adherence to conservation laws of energy and mass in discrete-time settings.
• To enable the identification of physical laws directly from data.

Main Methods:

• Integrating differential geometric structures into neural networks as coefficient matrices.
• Developing an automatic discrete differentiation algorithm for discrete gradient methods.
• Applying the model to simulate 1- and 2-D Korteweg-de Vries (KdV) and Cahn-Hilliard equations.

Main Results:

• The proposed model successfully simulates conservation and dissipation laws of energy and mass.
• The automatic discrete differentiation algorithm ensures adherence to physical laws in discrete-time.
• The model demonstrated effectiveness in simulating complex physical phenomena like KdV and Cahn-Hilliard equations.

Conclusions:

• The novel deep energy-based discrete-time model offers a robust solution for simulating physical phenomena governed by PDEs.
• Integrating differential geometry and discrete differentiation enhances the accuracy and physical consistency of neural network simulations.
• This approach advances scientific machine learning by enabling data-driven discovery of physical laws.