Equivariant U-Shaped Neural Operators for the Cahn-Hilliard Phase-Field Model
This provides an efficient surrogate model for phase-field systems in materials science and soft matter, though it is incremental as it builds on existing neural operator architectures.
The paper tackled the problem of predicting phase separation in binary mixtures governed by the Cahn-Hilliard equation by developing an equivariant U-shaped neural operator (E-UNO), which achieved accurate predictions across space and time and outperformed standard baselines on fine-scale structures.
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.