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

arXiv: https://arxiv.org/abs/2509.01293 · HTML: https://mlsanity.com/paper/2509.01293/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 52 references that are in the map.
- First posted on arXiv 1 Sep 2025.

## Builds on (most cited first, 11 of 11)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Rethinking materials simulations: Blending direct numerical simulations with neural operators](https://mlsanity.com/paper/2312.05410/) (2023)
- [Data-driven modelling of nonlinear spatio-temporal fluid flows using a deep convolutional generative adversarial network](https://mlsanity.com/paper/2004.00707/) (2020)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)

## Built on by (most cited first, 4 of 4)

- [Uni-Flow: a unified autoregressive-diffusion model for complex multiscale flows](https://mlsanity.com/paper/2602.15592/) (2026)
- [MENO: MeanFlow-Enhanced Neural Operators for Dynamical Systems](https://mlsanity.com/paper/2604.06881/) (2026)
- [Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution](https://mlsanity.com/paper/2603.09693/) (2026)
- [Fast-Forward Lattice Boltzmann: Learning Kinetic Behaviour with Physics-Informed Neural Operators](https://mlsanity.com/paper/2509.22411/) (2025)

## Abstract (arXiv)

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.