In this work, we explore the trade-offs of explicit structural priors, particularly group-equivariance. We address this through theoretical analysis and a comprehensive empirical study focusing on point clouds. To enable controlled and fair comparisons, we introduce Rapidash, a unified group convolutional architecture that allows for different variants of equivariant and non-equivariant models. Our results suggest that more constrained equivariant models outperform less constrained alternatives when aligned with the geometry of the task, and increasing representation capacity does not fully eliminate performance gaps. We see improved performance of models with equivariance and symmetry-breaking through tasks like segmentation, regression, and generation across diverse datasets. Explicit symmetry breaking via geometric reference frames consistently improves performance, while breaking equivariance through geometric input features can be helpful when aligned with task geometry. Our results provide task-specific performance trends that offer a more nuanced way for model selection.

Learning dynamics is essential for model-based control and Reinforcement Learning in engineering systems, such as robotics and power systems.

Recently, Graph Neural Controlled Differential Equations (Graph Neural CDEs) successfully adapted Neural CDEs from paths on Euclidean domains to paths on graph domains. Building on this foundation, we introduce Permutation Equivariant Neural Graph CDEs, which project Graph Neural CDEs onto permutation equivariant function spaces. This significantly reduces the model's parameter count without compromising representational power, resulting in more efficient training and improved generalisation. We empirically demonstrate the advantages of our approach through experiments on simulated dynamical systems and real-world tasks, showing improved performance in both interpolation and extrapolation scenarios.

Equivariant networks enhance model efficiency and generalization by embedding symmetry priors into their architectures. However, most existing methods, primarily based on group convolutions and steerable convolutions, face significant limitations when dealing with complex transformation groups, particularly the projective group, which plays a crucial role in vision. In this work, we tackle the challenge by constructing projective equivariant networks based on differential invariants. Using the moving frame method with a carefully selected cross section tailored for multi-dimensional functions, we derive a complete and concise set of second-order fundamental differential invariants of the projective group. We provide a rigorous analysis of the properties and transformation relationships of their underlying components, yielding a further simplified and unified set of fundamental differential invariants, which facilitates both theoretical analysis and practical applications. Building on this foundation, we develop PDINet, the first framework for deep projective equivariant networks, achieving full projective equivariance without discretizing or sampling the group. Empirical results on the projectively transformed STL-10 and Imagenette datasets show that PDINet achieves improvements of 11.39% and 5.66% in accuracy over the respective standard baselines under out-of-distribution settings, demonstrating its strong generalization to complex geometric transformations.

Recent foundational models for tabular data, such as TabPFN, excel at adapting to new tasks via in-context learning, but remain constrained to a fixed, pre-defined number of target dimensions--often necessitating costly ensembling strategies. We trace this constraint to a deeper architectural shortcoming: these models lack target equivariance, so that permuting target dimension orderings alters their predictions. This deficiency gives rise to an irreducible "equivariance gap," an error term that introduces instability in predictions. We eliminate this gap by designing a fully target-equivariant architecture--ensuring permutation invariance via equivariant encoders, decoders, and a bi-attention mechanism. Empirical evaluation on standard classification benchmarks shows that, on datasets with more classes than those seen during pre-training, our model matches or surpasses existing methods while incurring lower computational overhead.

Geometric graph neural networks (GNNs) that respect E(3) symmetries have achieved strong performance on small molecule modeling, but they face scalability and expressiveness challenges when applied to large biomolecules such as RNA and proteins. These systems require models that can simultaneously capture fine-grained atomic interactions, long-range dependencies across spatially distant components, and biologically relevant hierarchical structure--such as atoms forming residues, which in turn form higher-order domains. Existing geometric GNNs, which typically operate exclusively in either Euclidean or Spherical Harmonics space, are limited in their ability to capture both the fine-scale atomic details and the long-range, symmetry-aware dependencies required for modeling the multi-scale structure of large biomolecules. We introduce DualEquiNet, a Dual-Space Hierarchical Equivariant Network that constructs complementary representations in both Euclidean and Spherical Harmonics spaces to capture local geometry and global symmetry-aware features. DualEquiNet employs bidirectional cross-space message passing and a novel Cross-Space Interaction Pooling mechanism to hierarchically aggregate atomic features into biologically meaningful units, such as residues, enabling efficient and expressive multi-scale modeling for large biomolecular systems. DualEquiNet achieves state-of-the-art performance on multiple existing benchmarks for RNA property prediction and protein modeling, and outperforms prior methods on two newly introduced 3D structural benchmarks demonstrating its broad effectiveness across a range of large biomolecule modeling tasks.

In domains such as molecular and protein generation, physical systems exhibit inherent symmetries that are critical to model. Two main strategies have emerged for learning invariant distributions: designing equivariant network architectures and using data augmentation to approximate equivariance. While equivariant architectures preserve symmetry by design, they often involve greater complexity and pose optimization challenges. Data augmentation, on the other hand, offers flexibility but may fall short in fully capturing symmetries. Our framework enhances both approaches by reducing training variance and providing a provably lower-variance gradient estimator.

Joint-embedding self-supervised learning (SSL) commonly relies on transformations such as data augmentation and masking to learn visual representations, a task achieved by enforcing invariance or equivariance with respect to these transformations applied to two views of an image. This dominant two-view paradigm in SSL often limits the flexibility of learned representations for downstream adaptation by creating performance trade-offs between high-level invariance-demanding tasks such as image classification and more fine-grained equivariance-related tasks. In this work, we propose seq-JEPA, a world modeling framework that introduces architectural inductive biases into joint-embedding predictive architectures to resolve this trade-off. Without relying on dual equivariance predictors or loss terms, seq-JEPA simultaneously learns two architecturally separate representations for equivariance-and invariance-demanding tasks. To do so, our model processes short sequences of different views (observations) of inputs.

A key challenge is integrating these modalities of different shapes while maintaining SE(3) equivariance for 3D coordinates. To achieve this, existing approaches typically maintain separate latent spaces for invariant and equivariant modalities, reducing efficiency in both training and sampling. In this work, we propose Unified Variational Auto-Encoder for 3DMolecular Latent Diffusion Modeling (UAE-3D), a multi-modal VAE that compresses 3D molecules into latent sequences from a unified latent space, while maintaining near-zero reconstruction error. This unified latent space eliminates the complexities of handling multi-modality and equivariance when performing latent diffusion modeling. We demonstrate this by employing the Diffusion Transformer-a general-purpose diffusion model without any molecular inductive bias-for latent generation. Extensive experiments on GEOM-Drugs and QM9 datasets demonstrate that our method significantly establishes new benchmarks in both de novo and conditional 3D molecule generation, achieving leading efficiency and quality. On GEOM-Drugs, it reduces FCD by 72.6% over the previous best result, while achieving over 70% relative average improvements in geometric fidelity. Our code is released at https://github.com/lyc0930/UAE-3D/.

Lorentz-equivariant neural networks are becoming the leading architectures for high-energy physics. Current implementations rely on specialized layers, limiting architectural choices. We introduce Lorentz Local Canonicalization (LLoCa), a general framework that renders any backbone network exactly Lorentz-equivariant. Using equivariantly predicted local reference frames, we construct LLoCatransformers and graph networks. We adapt a recent approach for geometric message passing to the non-compact Lorentz group, allowing propagation of space-time tensorial features. Data augmentation emerges from LLoCa as a special choice of reference frame. Our models achieve competitive and state-of-the-art accuracy on relevant particle physics tasks, while being 4 faster and using 10 fewer FLOPs.