Periodic Topological Deep Learning for Polymer Design and Discovery

Polymers underpin applications across energy, healthcare, and materials science, yet their vast chemical space makes systematic discovery challenging. Most machine learning approaches represent polymers as molecular graphs of a single repeating unit, thereby missing both the periodicity of polymer chains and many-body interactions beyond pairwise bonds. We introduce Periodic-TDL, a deep learning framework built on periodic Vietoris-Rips complexes that capture many-body interactions across multiple spatial scales, followed by a hierarchical simplicial message-passing (HSMP) encoder that propagates information from long-range interactions to covalent bonds, yielding representations enriched by higher-order topological features. Periodic-TDL outperforms all state-of-the-art models across polymer property prediction tasks spanning electronic, optical, physical, and thermal targets. Furthermore, we quantitatively validate how ester-to-amide substitution and $α$-methylation enhance thermal stability. Using a computationally synthesized dataset of 48,208 structures-generated via systematic substitution of acrylate and acrylamide polymers-we observed a mean $T_g$ increase of $\sim 55^\circ$C for ester-to-amide substitutions and $\sim 14^\circ$C for backbone $α$-methylation across matched polymer pairs. To verify these predicted trends, we use our Periodic-TDL model to analyze six novel polymer pairs from independent experimental measurements, including three newly synthesized polymers previously unreported in the literature. The experimental data successfully confirmed the model's predictions. Ultimately, these findings demonstrate that Periodic-TDL captures the underlying physical effects of specific functional group modifications, rather than merely optimizing predictive performance on benchmark datasets.

A roadmap for curvature-based geometric data analysis and learning

Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and interpretable concept that captures intrinsic geometric structure and underpins numerous tasks, from community detection to geometric deep learning. A wide range of discrete curvature models have been proposed for various data representations, including graphs, simplicial complexes, cubical complexes, and point clouds sampled from manifolds. These models not only provide efficient characterizations of data geometry but also constitute essential components in geometric learning frameworks. In this paper, we present the first comprehensive review of existing discrete curvature models, covering their mathematical foundations, computational formulations, and practical applications in data analysis and learning. In particular, we discuss discrete curvature from both Riemannian and metric geometry perspectives and propose a systematic pipeline for curvature-driven data analysis. We further examine the corresponding computational algorithms across different data representations, offering detailed comparisons and insights. Finally, we review state-of-the-art applications of curvature in both supervised and unsupervised learning. This survey provides a conceptual and practical roadmap for researchers to gain a better understanding of discrete curvature as a fundamental tool for geometric understanding and learning.