The rational design of catalyst structures tailored to target performance is an ambitious and profoundly impactful goal. Key challenges include achieving refined representations of the three-dimensional structure of active sites and imbuing models with robust physical interpretability. Herein, we developed a topology-based variational autoencoder framework (PGH-VAEs) to enable the interpretable inverse design of catalytic active sites. Leveraging high-entropy alloys as a case, we demonstrate that persistent GLMY homology, an advanced topological algebraic analysis tool, enables the quantification of three-dimensional structural sensitivity and establishes correlations with adsorption properties. The multi-channel PGH-VAEs illustrate how coordination and ligand effects shape the latent space and influence the adsorption energies. Building on the inverse design results from PGH-VAEs, the strategies to optimize the composition and facet structures to maximize the proportion of optimal active sites are proposed. This interpretable inverse design framework can be extended to diverse systems, paving the way for AI-driven catalyst design.

Change points caused by extreme events in global economic markets have been widely studied in the literature. However, existing techniques to identify change points rely on subjective judgments and lack robust methodologies. The objective of this paper is to generalize a novel approach that leverages topological data analysis (TDA) to extract topological features from time series data using persistent homology. In this approach, we use Taken’s embedding and sliding window techniques to transform the initial time series data into a high-dimensional topological space. Then, in this topological space, persistent homology is used to extract topological features which can give important information related to change points. As a case study, we analyzed 26 stocks over the last 12 years by using this method and found that there were two financial market volatility indicators derived from our method, denoted as $L_1$ and $L_2$. They serve as effective indicators of long-term and short-term financial market fluctuations, respectively. Moreover, significant differences are observed across markets in different regions and sectors by using these indicators. By setting a significance threshold of 98 % for the two indicators, we found that the detected change points correspond exactly to four major financial extreme events in the past twelve years: the intensification of the European debt crisis in 2011, Brexit in 2016, the outbreak of the COVID-19 pandemic in 2020, and the energy crisis triggered by the Russia–Ukraine war in 2022. Furthermore, benchmark comparisons with established univariate and multivariate CPD methods confirm that the TDA-based indicators consistently achieve superior F1 scores across different tolerance windows, particularly in capturing widely recognized consensus events.

The stability of topological persistence is one of the fundamental issues in topological data analysis. Numerous methods have been proposed to address the stability of persistent modules or persistence diagrams. Recently, the concept of persistent Laplacians has emerged as a novel approach to topological persistence, attracting significant attention and finding applications in various fields. In this paper, we investigate the stability of persistent Laplacians. We introduce the notion of ``Laplacian trees'', which captures the collection of persistent Laplacians that persist from a given parameter. To formalize our study, we construct the category of Laplacian trees and establish an algebraic stability theorem for persistent Laplacian trees. Notably, our stability theorem is applied to the real-valued functions on simplicial complexes and digraphs.