Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks

TL;DR

This work addresses the limitations of surface-only mesh methods in capturing thickness-driven behavior by introducing Thickness-aware E(3)-Equivariant Mesh Networks (T-EMNN). It couples a data-driven, E(3)-invariant coordinate system with a thickness processor that learns when and how to propagate information between opposing surfaces, enabling accurate node-level 3D deformation predictions while preserving computational efficiency. The approach combines invariant geometry encoding, a surface processor, and a thickness processor within an encoder–processor–decoder framework, and it learns a dynamic thickness threshold to distinguish true thickness from width. Validation on real-world injection molding data and dynamic deformation scenarios demonstrates state-of-the-art performance and practical applicability, with robust generalization under coordinate transformations and improved handling of thickness interactions in complex geometries.

Abstract

Mesh-based 3D static analysis methods have recently emerged as efficient alternatives to traditional computational numerical solvers, significantly reducing computational costs and runtime for various physics-based analyses. However, these methods primarily focus on surface topology and geometry, often overlooking the inherent thickness of real-world 3D objects, which exhibits high correlations and similar behavior between opposing surfaces. This limitation arises from the disconnected nature of these surfaces and the absence of internal edge connections within the mesh. In this work, we propose a novel framework, the Thickness-aware E(3)-Equivariant 3D Mesh Neural Network (T-EMNN), that effectively integrates the thickness of 3D objects while maintaining the computational efficiency of surface meshes. Additionally, we introduce data-driven coordinates that encode spatial information while preserving E(3)-equivariance or invariance properties, ensuring consistent and robust analysis. Evaluations on a real-world industrial dataset demonstrate the superior performance of T-EMNN in accurately predicting node-level 3D deformations, effectively capturing thickness effects while maintaining computational efficiency.

Figures (16)

• Figure 1: The left figures show a mesh, with two different target nodes ($\bullet$), their thickness paired nodes ($\bullet$), thickness distance ($\mathbf{-}$), and nearby nodes within a radius ($\bullet$). The right figures compare Pearson correlation and L2 Norm between the target node's deformation and its thickness paired / nearby nodes within a radius.
• Figure 2: Overview of T-EMNN.
• Figure 3: Our proposed data-driven coordinate system.
• Figure 4: The concept of thickness (left) and width (right).
• Figure 5: Learning curve of the thickness threshold $\tau$ during training across three seeds (left), and the distribution of thickness values $t(v_i)$ with the cutoff threshold (red dotted line, $t(v_i) = \tau$) used for message passing in the thickness processor (right).