Equivariant Neural Networks (ENNs) have empowered numerous applications in scientific fields. Despite their remarkable capacity for representing geometric structures, ENNs suffer from degraded expressivity when processing symmetric inputs: the output representations are invariant to transformations that extend beyond the input's symmetries. The mathematical essence of this phenomenon is that a symmetric input, after being processed by an equivariant map, experiences an increase in symmetry. While prior research has documented symmetry increase in specific cases, a rigorous understanding of its underlying causes and general reduction strategies remains lacking. In this paper, we provide a detailed and in-depth characterization of symmetry increase together with a principled framework for its reduction: (i) For any given feature space and input symmetry group, we prove that the increased symmetry admits an infimum determined by the structure of the feature space; (ii) Building on this foundation, we develop a computable algorithm to derive this infimum, and propose practical guidelines for feature design to prevent harmful symmetry increases. (iii) Under standard regularity assumptions, we demonstrate that for most equivariant maps, our guidelines effectively reduce symmetry increase. To complement our theoretical findings, we provide visualizations and experiments on both synthetic datasets and the real-world QM9 dataset. The results validate our theoretical predictions.
Ning Lin, Jiacheng Cen, Anyi Li et al.· 1 citation
Crystals are cornerstone materials for semiconductors and renewable energy, yet their discovery is hindered by the prohibitive cost of Density Functional Theory (DFT). While geometric graph neural networks have advanced Crystal Structure Prediction (CSP) and energy estimation, existing methods treat these tasks as disparate problems with task-specific architectures, failing to exploit cross-task synergies. In this paper, we propose UniPath, a novel framework based on Learnable-Time Flow Matching that unifies crystal structure and energy prediction within a single probability path. Our key insight is modeling unstable crystals as an intermediate state along the probability path from a prior to the stable distribution. We introduce two pivotal mechanisms: (1) a learnable time parameter that dynamically identifies the intermediate state, and (2) an auxiliary velocity field that parameterizes tailored flows to synchronize the intermediate and unstable distributions. We theoretically prove that under optimal velocity fields, these paths converge to a consistent trajectory, enabling a unified backbone to maximize shared representation learning capabilities. Extensive evaluations on CSP and energy prediction benchmarks demonstrate that UniPath significantly outperforms task-specific baselines. Notably, UniPath achieves over 50% higher Match Rate on MPTS-52 and over 30% lower energy MAE on MPT-MPTS-52. Furthermore, UniPath generalizes effectively to small molecules, surpassing conventional models in molecular conformation generation. The source code is publicly available at https://github.com/GLAD-RUC/UniPath.
Songyou Li, Mingze Li, Qianpu Liu et al.· Proceedings of the 32nd ACM...· 1 citation
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