Reduced-order prediction model for the Cahn-Hilliard equation based on deep learning

This study presents an end-to-end deep learning framework for nonlinear reduced-order modeling and prediction, combining Variational Autoencoders (VAE) for feature extraction and Long Short-Term Memory (LSTM) networks for temporal prediction. The framework simplifies the modeling process by integrating multiple steps into a unified architecture, improving both design and training efficiency. The VAE compresses input data into a low-dimensional latent space while using a progressive channel reduction strategy to retain key features and minimize redundancy. The LSTM network captures temporal dependencies, ensuring accurate predictions based on historical data. The framework is validated through applications to the Cahn-Hilliard (CH) equation, demonstrating superior performance over traditional dimensionality reduction and prediction models. A comprehensive hyperparameter analysis identifies optimal configurations, and the model's extrapolation capabilities and computational efficiency are thoroughly assessed. Results highlight the framework's potential as an effective tool for modeling and predicting complex dynamic systems governed by partial differential equations.