Improving Flow Matching by Aligning Flow Divergence

Conditional flow matching (CFM) stands out as an efficient, simulation-free approach for training flow-based generative models, achieving remarkable performance for data generation. However, CFM is insufficient to ensure accuracy in learning probability paths. In this paper, we introduce a new partial differential equation characterization for the error between the learned and exact probability paths, along with its solution. We show that the total variation gap between the two probability paths is bounded above by a combination of the CFM loss and an associated divergence loss. This theoretical insight leads to the design of a new objective function that simultaneously matches the flow and its divergence. Our new approach improves the performance of the flow-based generative model by a noticeable margin without sacrificing generation efficiency. We showcase the advantages of this enhanced training approach over CFM on several important benchmark tasks, including generative modeling for dynamical systems, DNA sequences, and videos. Code is available at \href{https://github.com/Utah-Math-Data-Science/Flow_Div_Matching}{Utah-Math-Data-Science}.

Towards Multiscale Graph-based Protein Learning with Geometric Secondary Structural Motifs

Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale representations and modeling long-range dependencies efficiently. In this work, we propose an efficient multiscale graph-based learning framework tailored to proteins. Our proposed framework contains two crucial components: (1) It constructs a hierarchical graph representation comprising a collection of fine-grained subgraphs, each corresponding to a secondary structure motif (e.g., $α$-helices, $β$-strands, loops), and a single coarse-grained graph that connects these motifs based on their spatial arrangement and relative orientation. (2) It employs two GNNs for feature learning: the first operates within individual secondary motifs to capture local interactions, and the second models higher-level structural relationships across motifs. Our modular framework allows a flexible choice of GNN in each stage. Theoretically, we show that our hierarchical framework preserves the desired maximal expressiveness, ensuring no loss of critical structural information. Empirically, we demonstrate that integrating baseline GNNs into our multiscale framework remarkably improves prediction accuracy and reduces computational cost across various benchmarks.

Learning Decentralized Swarms Using Rotation Equivariant Graph Neural Networks

The orchestration of agents to optimize a collective objective without centralized control is challenging yet crucial for applications such as controlling autonomous fleets, and surveillance and reconnaissance using sensor networks. Decentralized controller design has been inspired by self-organization found in nature, with a prominent source of inspiration being flocking; however, decentralized controllers struggle to maintain flock cohesion. The graph neural network (GNN) architecture has emerged as an indispensable machine learning tool for developing decentralized controllers capable of maintaining flock cohesion, but they fail to exploit the symmetries present in flocking dynamics, hindering their generalizability. We enforce rotation equivariance and translation invariance symmetries in decentralized flocking GNN controllers and achieve comparable flocking control with 70% less training data and 75% fewer trainable weights than existing GNN controllers without these symmetries enforced. We also show that our symmetry-aware controller generalizes better than existing GNN controllers. Code and animations are available at http://github.com/Utah-Math-Data-Science/Equivariant-Decentralized-Controllers.