Equivariant Multiscale Models

• Equivariant models of multiscale phenomena are frameworks that enforce symmetry constraints across different spatial and temporal scales to capture intricate interactions.
• They employ mathematical principles and data-driven techniques, including group convolutions, sparse regression, and hierarchical pipelines for robust closure and accurate predictions.
• Recent developments show state-of-the-art performance in applications like turbulence modeling, medical imaging, and molecular generation, ensuring interpretability and sample efficiency.

Equivariant models of multiscale phenomena are mathematical, data-driven, or algorithmic frameworks designed to model systems with processes occurring at multiple spatial or temporal scales, while enforcing exact equivariance with respect to problem-specific symmetries (e.g., translations, rotations, scaling, or more general group actions). These models represent a principled approach to capturing fine- and coarse-scale interactions in fields such as turbulence, molecular dynamics, medical imaging, and computer vision, with guarantees that their predictive laws or learned representations transform consistently with the symmetries present in the data or underlying physical laws. Recent developments provide frameworks that systematically enforce symmetry constraints throughout the modeling pipeline, leading to interpretable, robust, and sample-efficient models that generalize across scale and orientation.

1. Mathematical Foundations of Equivariance in Multiscale Systems

Equivariance is defined relative to a symmetry group $G$ acting on the data space $X$ and function space $Y$, such that a mapping $F: X \to Y$ satisfies $F(g\cdot x) = g'\cdot F(x)$ for all $g\in G$, $x \in X$, where $g'$ is the appropriate induced action on $Y$. In multiscale phenomena, relevant groups include:

• Translation group $T = (\mathbb{R}^d, +)$
• Rotation group $SO(d)$ or $SE(3)$ for 3D
• Scaling group $S = (\mathbb{R}_+, \cdot)$
• Semidirect products and general Lie groups; e.g., $S \ltimes T$ encodes scaling and translations simultaneously (Zhu et al., 2019, Wimmer et al., 2023, Sangalli et al., 2021)

Symmetry constraints ensure the model outputs are consistent under these group actions. In turbulence modeling, enforcing translation, rotation, and Galilean invariance is essential for representing the closure terms in filtered equations (Choi et al., 13 Nov 2025). In deep learning, group-equivariant convolutions are necessary and sufficient conditions for equivariant representations (Zhu et al., 2019, Wimmer et al., 2023). PDE-based approaches realize symmetry through G-invariant metrics and operators on manifolds (Diop et al., 2024).

2. Data-driven Equivariant Closure and Effective Field Theories

A central challenge in multiscale modeling is closure: representing the effect of unresolved (small) scales on resolved (large) scales. The equivariant, data-driven closure framework proceeds as follows (Choi et al., 13 Nov 2025):

1. Scale Decomposition: Start with a "fundamental" PDE (e.g., incompressible Navier–Stokes). Introduce spatial coarse-graining via a filter operator $\mathcal{F}_{\Delta}$.
2. Decomposition: Split each field $u = u_L + u_S$ into resolved ($u_L$) and unresolved ($u_S$) components.
3. Explicit LES System: Filtering the PDE yields large-scale equations with unclosed subgrid-scale (SGS) stresses $\tau_{ij}$.
4. Library Construction: Build equivariant tensor libraries (basis functions) preserving translation, rotation, Galilean invariance, and dimensional consistency.
5. Sparse Regression: Fit explicit, sparse linear combinations of basis terms using simulation data in a weak (integral) sense; ensure each candidate respects the symmetry group.
6. Closure System: Obtain algebraic-differential, interpretable evolution equations for large and small-scale fields that close the system.

For 2D turbulence, the inferred system involves:

• Augmented Navier–Stokes for resolved scales,
• A closure for $\tau_{ij}$ decomposed into Leonard, cross, and Reynolds terms,
• Dynamic evolution equations for the Reynolds stress tensor $R_{ij}$ capturing memory effects and backscatter, i.e., small-to-large scale energy fluxes—previously inaccessible to traditional closure models.

This approach yields state-of-the-art accuracy for SGS stress and energy flux across multiple flow regimes and Reynolds numbers (LCR closure achieving correlation $C_\tau>99.3\%$, $C_\Pi>96\%$) and fully captures localized backscatter (Choi et al., 13 Nov 2025).

3. Equivariant Neural Architectures for Multiscale Phenomena

Equivariance in deep neural networks for multiscale modeling is enforced via group convolutions, scale-space lifting, and morphological PDEs:

• Group Convolutional Neural Networks: Construct convolutions over $(s, t)$ in $S\ltimes T$ or $HT$ for 3D data. Each feature channel is indexed by both position and scale; operations are tied across scales and positions to preserve equivariance. ScDCFNet achieves exact scaling-translation equivariance and offers reduced parameter complexity via low-frequency decomposition of filters in spatial and scale bases (Zhu et al., 2019).
• Scale-space Lifting and Semigroup Cross-correlation: Inputs are lifted into a joint space of position and scale (e.g., via Gaussian or morphological scale-spaces). Cross-correlation operators defined over the scaling-translation semigroup yield equivariant intermediate features. This approach empirically enables CNNs to generalize across a 16$\times$ scale range and delivers superior performance for both classification and segmentation, especially with morphological scale-space lifting; the latter is particularly effective in preserving sharp geometric features (Sangalli et al., 2021).
• Scale-Equivariant 3D CNNs: For 3D data, scale-steerable filter bases (e.g., Hermite–Gaussian) allow efficient formal scaling and analytic expansion, sidestepping interpolation artifacts. Plugging such convolutions into U-Net architectures results in models with guaranteed equivariance across discrete scales, superior data efficiency, and generalization in medical imaging tasks (Wimmer et al., 2023).
• Higher-Order Gauge-Equivariant Convolutions: On manifolds, higher-order (Volterra) gauge-equivariant convolutions using steerable kernels of degree 1 and 2 model both micro- and macroscopic interactions while ensuring full group equivariance (e.g., SO(3) on $S^2$). Coupled with convolutional kernel networks (CKNs) for the large scale, this compound architecture is highly parameter-efficient and achieves high classification accuracy in neuroimaging and spherical vision benchmarks (Cortes et al., 2023).
• PDE-based Morphological Equivariant Layers: Nonlinear layers formulated as Hamilton–Jacobi PDEs for erosion and dilation, solved in closed form on G-invariant Riemannian manifolds, yield explicit equivariant nonlinearities. Stacking such layers in generator architectures for GANs yields state-of-the-art sample efficiency, particularly for thin geometric structures (Diop et al., 2024).

4. Hierarchical, Coarse-to-Fine, and Multiscale Learning Pipelines

Multiscale equivariant models often employ hierarchical or coarse-to-fine learning strategies:

• Diffusion Models with Multiscale Equivariance: Equivariant Blurring Diffusion (EBD) implements a two-stage pipeline for molecular conformer generation: (i) coarse-grained structure generation at the fragment level, (ii) refinement at the atomic level via SE(3)-equivariant diffusion that blurs and then de-blurs toward the coarse scaffold. SE(3)-equivariance is enforced using relative displacements and invariant/equivariant feature sharing between atoms and fragments. EBD matches or outperforms state-of-the-art diffusion models (e.g., GeoDiff) with an order of magnitude fewer sampling steps while retaining better coverage and quantum property matching (Park et al., 2024).
• Multiscale Invertible Architectures in Inverse Problems: LIRE+ for cone-beam CT reconstruction leverages a coarse-to-fine (50% to 100% resolution) primal-dual iterative scheme, with patch-wise, reversible residual networks to reduce memory and computational cost. Rotation equivariance with respect to discrete subgroups ensures robustness to patient re-orientation, avoiding severe performance drops encountered by non-equivariant baselines (Moriakov et al., 2024).

These architectures integrate multiscale transitions, adaptively refine features, and preserve symmetry at each level, enabling efficient, robust inference and generation across domains.

5. Quantitative Results and Empirical Performance

Equivariant multiscale models consistently achieve superior performance in domains where relevant symmetries are exploited:

Application	Model	Key Metric(s)	Performance	Symmetries Enforced
2D Turbulence	Equivariant Closure	$C_\tau$, $C_\Pi$, $q_\Pi$	$C_\tau>$99.3%, $C_\Pi>96\%$, $q_\Pi=102$–$105\%$	Translation, Rotation, Galilean
Medical 3D Imaging	SE-U-Net	Dice Coefficient	$0.882$ vs. $0.846$ (baseline), 3–4 pp improvement on test scales	Scale, Translation
Molecular Generative	EBD	COV-R, MAT-R	COV-R: 92.6% vs 89.4% (GeoDiff); MAT-R: 0.822Å vs 0.857Å	SE(3)
Tomographic Recon.	LIRE+	PSNR, Robustness	PSNR: 35.38 dB (+0.24 vs LIRE); $<$5dB loss under rotations for baselines, 0 for LIRE+	Rotational ($P_4$ subgroup)
Vision	ScDCFNet, Morph.-lift	Top-1 Accuracy, IoU	Outperforms baseline CNN by 0.3–1.0%, higher IoU on scale shifts	Scaling, Translation
Neuroimaging	GEVNet+CKN	Classification accuracy	AD vs DLB: 92.86%; AD vs PD: 98.36%; DLB vs PD: 98.27%	Gauge (SO(2)), Global (SO(3))
GANs/images	GM-GAN	FID, KL	FID: 0.93 (GM-GAN) vs. 15.55 (DCGAN); KL: 0.95 vs. 1.07	Lie group, Morphological

Such results indicate that precise symmetry enforcement not only confers theoretical guarantees but also enhances robustness, sample efficiency, and generalization in practice. In all cases, explicit equivariant algorithms outperform conventional CNNs or non-equivariant generative models for data exhibiting multiscale and geometric variability.

6. Extensions, Generality, and Open Directions

The frameworks summarized are general in the following respects:

• Minimal Physical Assumptions: Most approaches require only the fundamental PDE (for physics), relevant symmetry group, hierarchy of tensor bases or filter parameterizations, and high-quality data. No problem-specific physics or empirically tuned closures are introduced unless symmetry demands.
• Flexible to Domain and Symmetry: The core approach adapts to fluid turbulence, magnetohydrodynamics, plasma physics, molecular generation, tomographic inversion, and is equally compatible with generalization to compressible or non-Newtonian fluids, non-Euclidean geometries, arbitrary Riemannian manifolds, or higher-dimensional group actions (Choi et al., 13 Nov 2025, Diop et al., 2024, Cortes et al., 2023).
• Open Problems:
  • Systematic learning of functional dependencies of closure coefficients on invariants (such as dynamic dependence in turbulence models)
  • Optimal design of filters and aliasing control in hierarchical and scale-space models
  • Robustness to noise and uncertainty quantification, critical for real-world deployment
  • Extending to irregular or point-cloud representations in 3D, joint treatment of rotation-scale equivariance, and continuous (rather than discrete) scale representations
  • Efficient implementation of higher-order, nonlinear, or PDE-based equivariant layers without excessive computational overhead

A plausible implication is that as empirical and theoretical advances continue, explicit symmetry enforcement will provide the foundation for interpretable, sample-efficient, and robust multiscale models in a broad range of scientific and engineering applications.