CaLMFlow: Volterra Flow Matching using Causal Language Models

We introduce CaLMFlow (Causal Language Models for Flow Matching), a novel framework that casts flow matching as a Volterra integral equation (VIE), leveraging the power of large language models (LLMs) for continuous data generation. CaLMFlow enables the direct application of LLMs to learn complex flows by formulating flow matching as a sequence modeling task, bridging discrete language modeling and continuous generative modeling. Our method implements tokenization across space and time, thereby solving a VIE over these domains. This approach enables efficient handling of high-dimensional data and outperforms ODE solver-dependent methods like conditional flow matching (CFM). We demonstrate CaLMFlow's effectiveness on synthetic and real-world data, including single-cell perturbation response prediction, showcasing its ability to incorporate textual context and generalize to unseen conditions. Our results highlight LLM-driven flow matching as a promising paradigm in generative modeling, offering improved scalability, flexibility, and context-awareness. Recent advances in deep learning have revolutionized generative modeling for complex, highdimensional data. In particular, methods based on ordinary differential equations (ODEs), such as continuous normalizing flows (CNFs) (Chen et al., 2018) and flow matching (Lipman et al., 2022), have emerged as efficient tools for modeling continuous data distributions. However, many ODE systems suffer from stiffness making them numerically unstable and computationally expensive to solve accurately (Kushnir & Rokhlin, 2012; Zappala et al., 2024). Recent work in operator learning (Xiong et al., 2021; Cao, 2021; Zappala et al., 2024) has also connected solving integral equations with transformers, the foundational architecture of large language models (LLMs), inspiring the use of LLMs to model dynamical systems through the lens of IEs.