This study introduces a modified score matching method aimed at generating molecular structures with high energy accuracy. The denoising process of score matching or diffusion models mirrors molecular structure optimization, where scores act like physical force fields that guide particles toward equilibrium states. To achieve energetically accurate structures, it can be advantageous to have the score closely approximate the gradient of the actual potential energy surface. Unlike conventional methods that simply design the target score based on structural differences in Euclidean space, we propose a Riemannian score matching approach. This method represents molecular structures on a manifold defined by physics-informed internal coordinates to efficiently mimic the energy landscape, and performs noising and denoising within this space. Our method has been evaluated by refining several types of starting structures on the QM9 and GEOM datasets, demonstrating that the proposed Riemannian score matching method significantly improves the accuracy of the generated molecular structures, attaining chemical accuracy. The implications of this study extend to various applications in computational chemistry, offering a robust tool for accurate molecular structure prediction.

Transporting between arbitrary distributions is a fundamental goal in generative modeling. Recently proposed diffusion bridge models provide a potential solution, but they rely on a joint distribution that is difficult to obtain in practice. Furthermore, formulations based on continuous domains limit their applicability to discrete domains such as graphs. To overcome these limitations, we propose Discrete Diffusion Schrödinger Bridge Matching (DDSBM), a novel framework that utilizes continuous-time Markov chains to solve the SB problem in a highdimensional discrete state space. Our approach extends Iterative Markovian Fitting to discrete domains, and we have proved its convergence to the SB. Furthermore, we adapt our framework for the graph transformation and show that our design choice of underlying dynamics characterized by independent modifications of nodes and edges can be interpreted as the entropy-regularized version of optimal transport with a cost function described by the graph edit distance. To demonstrate the effectiveness of our framework, we have applied DDSBM to molecular optimization in the field of chemistry. Experimental results demonstrate that DDSBM effectively optimizes molecules' property-of-interest with minimal graph transformation, successfully retaining other features. Transporting an initial distribution to a target distribution is a foundational concept in modern generative modeling. Denoising diffusion models (DDMs) have been highly influential in this area, with a primary focus on generating data distributions from simple prior (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020; Song et al., 2020; Kim et al., 2024b). Despite their promising results, setting the initial distribution as a simple prior makes DDMs hard to work in tasks where the initial distribution becomes a data distribution, such as image-to-image translation. To tackle this, diffusion bridge models (DBMs) extend DDMs to transport data between arbitrary distributions (Liu & Wu, 2023; Liu et al., 2023; Zhou et al., 2023).

As quantum chemical properties have a dependence on their geometries, graph neural networks (GNNs) using 3D geometric information have achieved high prediction accuracy in many tasks. However, they often require 3D geometries obtained from high-level quantum mechanical calculations, which are practically infeasible, limiting their applicability to real-world problems. To tackle this, we propose a new training framework, GeoTMI, that employs denoising process to predict properties accurately using easy-to-obtain geometries (corrupted versions of correct geometries, such as those obtained from low-level calculations). Our starting point was the idea that the correct geometry is the best description of the target property. Hence, to incorporate information of the correct, GeoTMI aims to maximize mutual information between three variables: the correct and the corrupted geometries and the property. GeoTMI also explicitly updates the corrupted input to approach the correct geometry as it passes through the GNN layers, contributing to more effective denoising. We investigated the performance of the proposed method using 3D GNNs for three prediction tasks: molecular properties, a chemical reaction property, and relaxed energy in a heterogeneous catalytic system. Our results showed consistent improvements in accuracy across various tasks, demonstrating the effectiveness and robustness of GeoTMI.

The exploration of transition state (TS) geometries is crucial for elucidating chemical reaction mechanisms and modeling their kinetics. Recently, machine learning (ML) models have shown remarkable performance for prediction of TS geometries. However, they require 3D conformations of reactants and products often with their appropriate orientations as input, which demands substantial efforts and computational cost. Here, we propose a generative approach based on the stochastic diffusion method, namely TSDiff, for prediction of TS geometries just from 2D molecular graphs. TSDiff outperformed the existing ML models with 3D geometries in terms of both accuracy and efficiency. Moreover, it enables to sample various TS conformations, because it learned the distribution of TS geometries for diverse reactions in training. Thus, TSDiff was able to find more favorable reaction pathways with lower barrier heights than those in the reference database. These results demonstrate that TSDiff shows promising potential for an efficient and reliable TS exploration.