Equivariant U-Shaped Neural Operator (E-UNO)

• The paper demonstrates that E-UNO significantly reduces prediction errors in Cahn–Hilliard simulations, achieving up to an order-of-magnitude improvement over FNO.
• The method employs a unique combination of global spectral convolution and hierarchical U-shaped architecture to capture both global coupling and local high-frequency structures.
• Equivariance regularization with respect to the dihedral group stabilizes training, enforces symmetry, and ensures physically consistent phase evolution.

Equivariant U-Shaped Neural Operator (E-UNO) is a neural-operator architecture introduced for the Cahn–Hilliard phase-field model, where the objective is to learn the evolution of the phase-field variable from short histories of past dynamics. The model combines global spectral convolution, a multi-resolution U-shaped architecture, and an explicit equivariance regularization aligned with the symmetries of the underlying dynamics. In the reported study, E-UNO is presented as an efficient surrogate for complex phase-field systems, with improved accuracy over standard Fourier neural operator (FNO) and U-shaped neural operator (UNO) baselines, particularly on fine-scale and high-frequency structures (Xue et al., 1 Sep 2025).

1. Governing equation and operator-learning setting

The target physical system is the Cahn–Hilliard model for phase separation in binary mixtures. In the formulation used for E-UNO, the free energy is

$$F[\Phi] = \int_\Omega \left( \frac{\lambda}{\epsilon} W(\Phi) + \frac{\lambda \epsilon}{2} |\nabla \Phi|^2 \right) dx, \qquad W(\Phi)=\frac{1}{4}(\Phi^2-1)^2,$$

on a domain $\Omega \subset \mathbb{R}^d$ with $d=2,3$ and time interval $T=(0,T_f)$. The strong form is

$$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$

$$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$

with homogeneous Neumann boundary conditions

$$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$

The model satisfies mass conservation and energy dissipation,

$$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$

For constant mobility $\gamma \equiv 1$, the formulation recovers classical spinodal-decomposition dynamics (Xue et al., 1 Sep 2025).

A spectral weak formulation on a periodic $2$D grid $\Omega \subset \mathbb{R}^d$0 is written by expanding $\Omega \subset \mathbb{R}^d$1 and $\Omega \subset \mathbb{R}^d$2 in Fourier modes $\Omega \subset \mathbb{R}^d$3 and enforcing

$\Omega \subset \mathbb{R}^d$4

Within the operator-learning viewpoint, the short-window temporal evolution is treated as an operator

$\Omega \subset \mathbb{R}^d$5

mapping

$\Omega \subset \mathbb{R}^d$6

between the Bochner spaces

$\Omega \subset \mathbb{R}^d$7

In practice, space is discretized on $\Omega \subset \mathbb{R}^d$8 and time-stepped with $\Omega \subset \mathbb{R}^d$9, while a neural operator $d=2,3$0 is trained by minimizing a data loss in either an $d=2,3$1 or $d=2,3$2 sense.

2. Architectural composition of E-UNO

E-UNO is defined by three coupled design elements: global spectral convolution, a U-shaped multi-resolution backbone, and equivariance regularization. The architecture is intended to capture both nonlocal coupling and multiscale structure, which is central for Cahn–Hilliard dynamics (Xue et al., 1 Sep 2025).

The Fourier layers implement global spectral convolution. At layer $d=2,3$3, a feature field $d=2,3$4 is mapped to $d=2,3$5 by

$d=2,3$6

where

$d=2,3$7

and $d=2,3$8 is learned for modes $d=2,3$9 and zero otherwise. The stated effect is a mesh-independent, global, translation-invariant convolution in spectral space.

The U-shaped encoder–decoder contains $T=(0,T_f)$0 spectral blocks with feature widths

$T=(0,T_f)$1

and spatial scalings

$T=(0,T_f)$2

The architecture uses average-pool by $T=(0,T_f)$3 at layers $T=(0,T_f)$4–$T=(0,T_f)$5, a bottleneck at layer $T=(0,T_f)$6 with $T=(0,T_f)$7, bilinear-interpolate by $T=(0,T_f)$8 at layers $T=(0,T_f)$9–$$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$0, and skip connections that add encoder features to decoder features at matching resolutions. The input $$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$1 is lifted via $$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$2, processed through the $$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$3 spectral U-blocks, and projected back to the phase-field space via $$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$4.

A concise architectural summary is given below.

Component	Specification	Role
Spectral blocks	$$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$5	Core U-shaped backbone
Channel widths	$$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$6	Multi-resolution feature hierarchy
Spatial scalings	$$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$7	Down/up-sampling schedule

This arrangement differs from plain FNO by combining Fourier layers with explicit hierarchical resolution changes and skip connections. The paper attributes the resulting gain to the ability of U-shape multi-resolution blocks to capture both global coupling and local high-frequency structures.

3. Equivariance regularization and symmetry handling

A defining element of E-UNO is its equivariance regularization with respect to the dihedral group $$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$8, comprising rotations and reflections of the square. The regularization term is

$$\frac{\partial \Phi}{\partial t} = \nabla \cdot (\gamma(\Phi)\nabla \mu),$$9

Operationally, the procedure is specified as follows: apply each $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$0 to the input snapshot sequence; feed the transformed sequence through the same UNO network; apply $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$1 to the network’s output; and penalize the difference to the original untransformed output (Xue et al., 1 Sep 2025).

The stated purpose of minimizing $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$2 is to bias $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$3 to commute with the dihedral group actions, thereby reducing redundancy and improving generalization. The paper further states that encoding $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$4-equivariance reduces data redundancy, stabilizes training, and yields more uniform predictions across rotated and reflected patterns.

A common point of interpretation is the status of equivariance in E-UNO. In the formulation reported here, equivariance is encouraged by an explicit loss term rather than imposed by a strictly equivariant layer construction. This suggests that the model seeks approximate commutation with group actions through optimization, while still retaining the standard UNO computational pathway.

4. Training protocol and implementation details

The reported dataset consists of 300 independent COMSOL Multiphysics simulations on $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$5 with a $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$6 grid, $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$7, and $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$8. Each simulation is run from uniform $$\mu = \frac{\delta F}{\delta \Phi} = \lambda \left( \frac{\Phi(\Phi^2-1)}{\epsilon} - \epsilon \Delta \Phi \right),$$9 plus small noise for spinodal decomposition. From each run, 30 sub-trajectories are extracted; sampling uses $$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$0; and the predictive task uses $$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$1 past frames to predict $$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$2 future frames. The train/validation/test split is $$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$3 (Xue et al., 1 Sep 2025).

Training uses the total loss

$$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$4

with $$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$5. The data term is either

$$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$6

or the gradient-enhanced form

$$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$7

Optimization uses Adam with cosine-annealing learning-rate schedule over 200 epochs, with learning rate decreasing from $$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$8 to $$n\cdot \nabla \Phi = 0,\qquad n\cdot \nabla \mu = 0 \quad \text{on } \partial \Omega.$$9. The batch size is 20 trajectories, or sub-windows, per batch. No extra data augmentation is used beyond the $$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$0 transformations employed in $$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$1.

These implementation choices place the method squarely in a short-horizon autoregressive or windowed-forecasting regime. A plausible implication is that the architecture is optimized for local-in-time operator approximation rather than direct end-to-end long-rollout prediction.

5. Empirical performance

The reported test-set comparison evaluates FNO, UNO, and E-UNO using three-step predictive error. Median relative $$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$2 error $$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$3 is given as approximately $$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$4–$$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$5 for FNO, $$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$6–$$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$7 for UNO, and $$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$8–$$\frac{d}{dt} F[\Phi] = - \int_\Omega \gamma |\nabla \mu|^2 dx \le 0.$$9 for E-UNO. The paper also states that E-UNO reduces the max early-stage $\gamma \equiv 1$0 error by $\gamma \equiv 1$1 relative to UNO and by an order of magnitude relative to FNO (Xue et al., 1 Sep 2025).

The reported inference speed is $\gamma \equiv 1$2 per sample on an NVIDIA A100 for E-UNO and UNO, compared with $\gamma \equiv 1$3 for COMSOL on CPU, corresponding to an $\gamma \equiv 1$4 speed-up. FNO and E-FNO are reported at approximately $\gamma \equiv 1$5.

A compact summary of the quantitative comparison is as follows.

Model	Median relative $\gamma \equiv 1$6 error $\gamma \equiv 1$7	Inference speed
FNO	$\gamma \equiv 1$8–$\gamma \equiv 1$9	$2$0
UNO	$2$1–$2$2	$2$3
E-UNO	$2$4–$2$5	$2$6

The ablation studies isolate the effects of equivariance, gradient enhancement, and hierarchical multi-resolution structure. FNO with equivariance, denoted E-FNO, shows a $2$7–$2$8 drop in median error in dynamic regimes, specifically time windows $2$9–$\Omega \subset \mathbb{R}^d$00. UNO with equivariance, denoted E-UNO, yields an additional $\Omega \subset \mathbb{R}^d$01–$\Omega \subset \mathbb{R}^d$02 error reduction versus UNO. Adding the gradient term $\Omega \subset \mathbb{R}^d$03 in $\Omega \subset \mathbb{R}^d$04 reduces early-stage peak error from $\Omega \subset \mathbb{R}^d$05 to $\Omega \subset \mathbb{R}^d$06 and tightens error variance. UNO architectures outperform FNO by approximately $\Omega \subset \mathbb{R}^d$07–$\Omega \subset \mathbb{R}^d$08 in $\Omega \subset \mathbb{R}^d$09 error, which the paper interprets as confirmation of the benefit of hierarchical features.

6. Physical consistency, super-resolution, and broader scope

The qualitative analysis emphasizes that E-UNO reproduces complex finger-like spinodal patterns. In the reported three-step rollout from $\Omega \subset \mathbb{R}^d$10, the model achieves maximum absolute error below $\Omega \subset \mathbb{R}^d$11. Free-energy trajectories are also used as a physical-consistency diagnostic: the E-UNO and ground-truth $\Omega \subset \mathbb{R}^d$12 curves are described as almost indistinguishable, with ensemble spread below $\Omega \subset \mathbb{R}^d$13 at all times (Xue et al., 1 Sep 2025).

The study also reports super-resolution behavior. An E-UNO model trained on a $\Omega \subset \mathbb{R}^d$14 grid is directly applied to a $\Omega \subset \mathbb{R}^d$15 grid and is reported to capture fine interfacial structures with max error below $\Omega \subset \mathbb{R}^d$16. This result is consistent with the paper’s description of spectral convolution as mesh-independent and global, although the super-resolution claim itself is empirical rather than a formal guarantee.

The broader significance is framed in terms of symmetry and scale hierarchy. The paper states that U-shape multi-resolution blocks capture both global coupling through Fourier layers and local high-frequency structures through skip-connected down/up-sampling, and that the synergy between spectral nonlocality and hierarchical locality leads to superior accuracy on multiscale phase-field dynamics. It further proposes that the same framework can be applied to N-phase Cahn–Hilliard, Cahn–Hilliard–Navier–Stokes, Allen–Cahn, and phase-field–crystal models by replacing the temporal history length or augmenting the loss with model-specific invariants such as momentum conservation or curvature flow. Other symmetry groups, including $\Omega \subset \mathbb{R}^d$17 and translation in inhomogeneous domains, are likewise identified as enforceable through a similar group-equivariance loss. Super-resolution inference and online coupling to finite-element solvers are described as immediate next steps for multiscale PDE surrogates.

Within this framing, E-UNO occupies a specific position among neural operators: it is not only a surrogate for short-horizon Cahn–Hilliard evolution, but also a design pattern combining Fourier nonlocality, U-shaped scale hierarchy, and explicit symmetry regularization. The reported results suggest that this combination is particularly effective when the target dynamics are simultaneously multiscale, translation-structured, and approximately symmetric under discrete geometric transformations.