[Paper Review] Rotational and Reflectional Equivariant Convolutional Neural Network for data-limited applications: Multiphase Flow demonstration

B. Siddani, S. Balachandar+1 more|arXiv (Cornell University)|Aug 7, 2021

Model Reduction and Neural Networks49 references24 citations

TL;DR

This paper proposes an SE(3)-equivariant CNN that enforces 3D rotational and reflectional symmetry in fluid flow prediction for particle-laden flows, significantly improving generalization under data-scarce conditions. By embedding SO(3) symmetry directly into the network architecture, the model achieves higher accuracy than standard CNNs or data-augmented baselines, especially with limited training data, while reducing parameter count and implicitly augmenting data through symmetry constraints.

ABSTRACT

This article deals with approximating steady-state particle-resolved fluid flow around a fixed particle of interest under the influence of randomly distributed stationary particles in a dispersed multiphase setup using Convolutional Neural Network (CNN). The considered problem involves rotational symmetry about the mean velocity (streamwise) direction. Thus, this work enforces this symmetry using $\mathbf{ extbf{SE(3)-equivariant}}$, special Euclidean group of dimension 3, CNN architecture, which is translation and three-dimensional rotation equivariant. This study mainly explores the generalization capabilities and benefits of SE(3)-equivariant network. Accurate synthetic flow fields for Reynolds number and particle volume fraction combinations spanning over a range of [86.22, 172.96] and [0.11, 0.45] respectively are produced with careful application of symmetry-aware data-driven approach.

Motivation & Objective

• To address the challenge of poor generalization in data-driven fluid flow modeling under limited training data.
• To enforce rotational and reflectional symmetries inherent in particle-resolved multiphase flows through architectural design rather than data augmentation.
• To demonstrate that symmetry-aware neural networks yield more accurate and statistically consistent predictions of velocity and pressure fields.
• To show that SE(3)-equivariant networks require fewer trainable parameters and generalize better than standard CNNs when data is scarce.
• To validate the method on steady-state particle-resolved direct numerical simulations (PR-DNS) across varying Reynolds numbers and particle volume fractions.

Proposed method

• The study employs an SE(3)-equivariant CNN architecture that is invariant to translations and equivariant to 3D rotations and reflections (SO(3) group).
• The network is trained to predict full velocity and pressure fields around a reference particle using local particle positions and flow parameters as input.
• Spatial derivatives (e.g., velocity gradients) are computed using second-order central difference stencils, implemented as convolutional layers with fixed kernels.
• Loss functions include normalized mean squared error (NMSE) between predicted and DNS-derived flow fields and their derivatives.
• Symmetry enforcement is achieved through irreducible representations of SO(3) in the network’s intermediate layers, ensuring consistent transformation behavior under rotation and reflection.
• The model is compared against a standard 3D CNN and a data-augmented CNN (with rotated/reflected training samples) across multiple test cases.

Experimental results

Research questions

• RQ1 Can an SE(3)-equivariant CNN outperform standard and data-augmented CNNs in predicting fluid flow fields under data-limited conditions?
• RQ2 To what extent does architectural enforcement of rotational and reflectional symmetries improve statistical consistency in flow predictions?
• RQ3 How does symmetry enforcement affect the generalization capability of data-driven models when training data is scarce?
• RQ4 Does the SE(3)-CNN maintain symmetry properties in predicted flow fields even when the input is slightly rotated from the canonical axis system?
• RQ5 Can the SE(3)-CNN achieve high accuracy with fewer trainable parameters compared to standard CNNs?

Key findings

• The SE(3)-CNN achieved the lowest normalized mean squared error (NMSE) for all flow field components, including velocity gradients, across all test cases.
• For case-1, the NMSE for ∂p′/∂x was 0.2272 with SE(3)-CNN, compared to 0.2549 and 0.2593 for data-augmented and simple CNNs, respectively.
• The NMSE values for transverse derivatives ∂p′/∂y and ∂p′/∂z were nearly identical (0.3277 and 0.3276) under SE(3)-CNN, indicating strong statistical symmetry preservation.
• In low-data regimes (fewer than 20 training samples), the SE(3)-CNN maintained symmetry-consistent predictions, while data-augmented and simple CNNs showed significantly higher symmetry errors.
• The SE(3)-CNN demonstrated superior generalization, achieving high accuracy even with limited data, due to implicit data augmentation via symmetry constraints.
• The model’s intermediate layers naturally represent irreducible SO(3) representations, enabling interpretable and physically consistent flow predictions.