Authors: Ning Lin, Luxi Chen, Huaguan Chen, Jiacheng Cen, Chongxuan Li, Wenbing Huang, Hao Sun
Abstract
Generating objects with specific symmetries is essential in various real-world scenarios. However, adapting existing 2D continuous representations to enforce planar group symmetry remains a challenge, as the transformation of non-reflective group elements may disrupt continuity. To overcome this limitation, we propose a symmetrization framework for arbitrary planar groups. Our method transforms any 2D continuous representation into a symmetric one while preserving continuity. We provide the mathematical formulation of this representation, demonstrate its approximation capability for symmetric functions, and detail the construction methodology. We validate our approach through three visual design tasks (pattern design, paper-cutting design and stylized topology design) and one material design task. Experiments confirm that our representation enables effective symmetry control and demonstrate its broader applicability.
Single-view 3D generative models have achieved impressive visual quality, yet they are not designed to satisfy structural or functional requirements, and in practice, often fall short. Symmetry is one such requirement: violations, even subtle ones, on symmetry can render a model physically unusable. We present SymTRELLIS, a method that enforces arbitrary finite point group symmetries (rotational, reflectional, and polyhedral) during the flow-based 3D generation of TRELLIS.2, without retraining the underlying VAE or flow model. Our key idea is to approximate the latent-space action of spatial transformations as a learned linear operator on voxel latents, implemented as a lightweight spatial-transform latent mapper trained on generic, non-symmetric 3D data. At generation time, we enforce symmetry by averaging predicted flow velocities across all symmetry-equivalent transformations at each ODE step, a process we call velocity symmetrization. The symmetry specification can be estimated automatically from an initial TRELLIS.2 generation or supplied by the user, enabling deliberate fold manipulation beyond what the input image suggests. On a curated benchmark of 266 strictly symmetric objects spanning 2- to 20-fold rotations and polyhedral symmetry groups, SymTRELLIS substantially reduces all symmetry error metrics compared to TRELLIS.2, Hunyuan3D-2.1, and TripoSG, while maintaining reconstruction accuracy comparable to the base model.
While transformers excel in many settings, their application in the field of automated planning is limited. Prior work like PlanGPT, a state-of-the-art decoder-only transformer, struggles with extrapolation from easy to hard planning problems. This in turn stems from problem symmetries: planning tasks can be represented with arbitrary variable names that carry no meaning beyond being identifiers. This causes a combinatorial explosion of equivalent representations that pure transformers cannot efficiently learn from. We propose a novel contrastive learning objective to make transformers symmetry-aware and thereby compensate for their lack of inductive bias. Combining this with architectural improvements, we show that transformers can be efficiently trained for either plan-generation or heuristic-prediction. Our results across multiple planning domains demonstrate that our symmetry-aware training effectively and efficiently addresses the limitations of PlanGPT.
Tackling the task of materials generation, we aim to enhance the previously proposed All-atom Diffusion Transformer (ADiT) by introducing SymADiT, a symmetry-aware variant. To do so, we use a representation of materials based on Wyckoff positions. We follow ADiT and perform generative modelling in latent space, adapted to our symmetry-aware representation. By forcing the output of the generative model to adhere to the symmetry restrictions imposed by the generated crystal's space group and each atom's Wyckoff-position, the generated materials exhibit more realistic symmetry properties. We benchmark our method against both symmetry-aware and symmetry-agnostic models for materials generation and show competitive performance, generating stable, symmetric materials with a simple Transformer architecture.