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 high-dimensional 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. Source code is available [here](https://github.com/junhkim1226/DDSBM).

Understanding transition pathways between two meta-stable states of a molecular system is crucial to advance drug discovery and material design. However, unbiased molecular dynamics (MD) simulations are computationally infeasible because of the high energy barriers that separate these states. Although recent machine learning techniques are proposed to sample rare events, they are often limited to simple systems and rely on collective variables (CVs) derived from costly domain expertise. In this paper, we introduce a novel approach that trains diffusion path samplers (DPS) to address the transition path sampling (TPS) problem without requiring CVs. We reformulate the problem as an amortized sampling from the transition path distribution by minimizing the log-variance divergence between the path distribution induced by DPS and the transition path distribution. Based on the log-variance divergence, we propose learnable control variates to reduce the variance of gradient estimators and the off-policy training objective with replay buffers and simulated annealing techniques to improve sample efficiency and diversity. We also propose a scale-based equivariant parameterization of the bias forces to ensure scalability for large systems. We extensively evaluate our approach, termed TPS-DPS, on a synthetic system, small peptide, and challenging fast-folding proteins, demonstrating that it produces more realistic and diverse transition pathways than existing baselines. We also provide links to [project page](https://kiyoung98.github.io/tps-dps/) and [code](https://github.com/kiyoung98/tps-dps).

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.