Data-driven sciences are widely regarded as the fourth paradigm of sciences that will fundamentally change the society and our everyday lives. Indeed, artificial intelligence (AI) models have already revolutionized and transformed various data-intensive industries.  Machine learning and deep learning models have achieved unprecedented extraordinary performance for image, text, audio, video, and network data analysis. The great successes are mainly due to three reasons, i.e., accumulation of the gigantic amount of data, ever-increasing computational power, and design of the highly efficient algorithms. Further, the remarkable achievement of AlphaFold2 for protein folding problem has ushered in a new era for AI-based molecular data analysis for materials, chemistry, and biology.

With the excitement and opportunities come challenges. Currently, one of the central challenges for AI-based molecular data analysis is molecular representation, which is to identify or design appropriate molecular descriptors or fingerprints. Proper descriptors should preserve the most important and intrinsic molecular properties and information that directly determinate molecular functions. In this way, they can be better “understood” by machine learning models. In fact, the performance of many learning methods is heavily dependent on the choice of data representation and featurization, which is a long-standing issue for cheminformatics and bioinformatics.  Traditional molecular descriptors are properties obtained from structural geometry/topology, chemical conformation, chemical graph, as well as molecular formula, hydrophobicity, steric properties, and electronic properties. These descriptors are widely used in the quantitative structure-activity relationship (QSAR) and learning models.

Mathematical AI for Molecular Sciences (IA matemática para ciencias moleculares, Matematička Ai Za Molekularne Znanosti) is proposed for molecular representation, featurization, and learning. As illustrated above, various types of data, in particular, molecular data from materials, chemistry, and biology, can be represented using topological models, including graphs, simplicial complexes, hypergraphs, etc. From these representations, various mathematical invariants are obtained by using advance mathematical models from algebraic topology, discrete geometry, combinatorics, etc. These mathematical invariants are used as input features for learning models. Dramatically different from previous models, molecular data are modelled using higher-dimensional topologies, such as simplicial complexes and hypergraphs, and filtration-induced multiscale representations. Further, mathematical invariant-based features characterize the most intrinsic and fundamental properties and have a better transferability for learning models.

A brief introduction of the area can be found in 2021 winter school lectures at Dalian, AATRN talk, report in Chinese for the series of talks on "Math for AI & AI for math", and Prof.Guowei Wei's works (SIAM news, Harvard talk, D3R news). We sincerely welcome highly motivated students and postdocs to join our group!

The structure-function relationship is of essential importance to the analysis of biomolecular flexibility, dynamics, interactions, and functions. Topology studies the network and connection information within the data and provides an effective way of structure characterization. As illustrated in the figures, there are three basic topological representations, including graph, simplicial complex, and hypergraph, for molecular structures.  Features for learning models can be obtained from these representations. The essential idea is to use eigen-spectrum-based properties as molecular descriptors.

Our persistent spectral (PerSpect) theory covers three basic models, i.e., PerSpect graph, PerSpect simplicial complex, and PerSpect hypergraph. These models are filtration-based multidimensional spectral methods. Mathematically, spectral graph theory, spectral simplicial complex, and spectral hypergraph have been developed based on graph, simplicial complex, and hypergraph. These models use different types of connection matrixes, in particular, Hodge (combinatorial) Laplacian matrixes, to represent structure connection. The multidimensional representation is achieved through a filtration process. The persistence and variation of eigen spectrum information during the filtration process are characterized by persistent functions or attributes, which are further used as  molecular features or fingerprints.

Reference: Zhenyu Meng and Kelin Xia, "Persistent spectral based machine learning (PerSpect ML) for protein-ligand binding affinity prediction", Science advances (2021) 

Ricci curvature is one of the fundamental concepts in differential geometry and theoretical physic. Two discrete Ricci curvature forms, i.e., Ollivier Ricci curvature (ORC) and Forman Ricci curvature (FRC), have been developed to characterize different aspects of the classical Ricci curvature. ORC is defined as the Wasserstein distance between two associated probability measurements on metric spaces. It captures clustering and coherence properties of global and local structures in networks. In contrast, FRC is defined as a combinatorial property of upper adjacent, lower-adjacent and parallel simplexes on CW complexes. This combinatorial curvature can be directly derived from the combinatorial Bochner-Weitzenbock decomposition. It characterizes geodesics dispersal property and algebraic topological information within networks. Even though the two discrete forms can have totally different values, sometimes even signs, for network substructures, they are found to be highly correlated in various complex networks. Generally speaking, positive ORCs or FRCs are commonly found in densely-packed clusters or “communities”, while negative ORCs or FRCs usually represent bridges or links between clusters.
The persistent Ricci curvature is proposed to combine filtration-based multiscale representations with Ricci curvatures for molecular featurization. Ricci curvatures are systematically evaluated on all the graphs/simplicial complexes/hypergraphs in the filtration process. The statistical and combinatorial properties of Ricci curvatures during the filtration are used as molecular descriptors.