Equivariant U-Shaped Neural Operators for the Cahn-Hilliard Phase-Field Model

2025-09-01

AI Summary

Predicting phase separation dynamics governed by the Cahn–Hilliard equation remains challenging for existing neural operators, which struggle to simultaneously capture multiscale features, respect physical symmetries (e.g., translation invariance), and generalize across domains. Method: We propose a symmetry-preserving U-shaped neural operator (SE-U-NO) that explicitly enforces translational equivariance via constrained convolutional layers and integrates multiscale spatiotemporal information through multi-resolution spectral convolutions. Given historical phase-field snapshots as input, SE-U-NO directly learns the nonlinear mapping to future states. Contribution/Results: Experiments demonstrate that SE-U-NO significantly outperforms baselines—including the Fourier neural operator—in reconstructing fine-scale structures and capturing high-frequency dynamics. It achieves higher accuracy with fewer training samples, exhibits faster inference, and generalizes robustly across diverse initial conditions and geometric domains. By unifying physical consistency, generalizability, and computational efficiency, SE-U-NO establishes a new paradigm for data-driven phase-field modeling.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Computer Vision: Multi-modal Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Phase separation in binary mixtures, governed by the Cahn-Hilliard equation, plays a central role in interfacial dynamics across materials science and soft matter. While numerical solvers are accurate, they are often computationally expensive and lack flexibility across varying initial conditions and geometries. Neural operators provide a data-driven alternative by learning solution operators between function spaces, but current architectures often fail to capture multiscale behavior and neglect underlying physical symmetries. Here we show that an equivariant U-shaped neural operator (E-UNO) can learn the evolution of the phase-field variable from short histories of past dynamics, achieving accurate predictions across space and time. The model combines global spectral convolution with a multi-resolution U-shaped architecture and regulates translation equivariance to align with the underlying physics. E-UNO outperforms standard Fourier neural operator and U-shaped neural operator baselines, particularly on fine-scale and high-frequency structures. By encoding symmetry and scale hierarchy, the model generalizes better, requires less training data, and yields physically consistent dynamics. This establishes E-UNO as an efficient surrogate for complex phase-field systems.

Problem

• Learning phase-field evolution from short history data
• Capturing multiscale behavior with physical symmetries
• Providing efficient surrogate for Cahn-Hilliard equation

Innovation

• Equivariant U-shaped neural operator architecture
• Global spectral convolution with multi-resolution design
• Translation equivariance regulation for physical alignment