Core Viewpoint - The article introduces a novel energy guidance theoretical framework for flow matching models, addressing the gap in energy guidance algorithms within this context and proposing various practical algorithms suitable for different tasks [2][3][27]. Summary by Sections Research Background - Energy guidance is a crucial technique in the application of generative models, ideally altering the distribution of generated samples to align with a specific energy function while maintaining adherence to the training set distribution [7][9]. - Existing energy guidance algorithms primarily focus on diffusion models, which differ fundamentally from flow matching models, necessitating a general energy guidance theoretical framework for flow matching [9]. Method Overview - The authors derive a general flow matching energy guidance vector field from the foundational definitions of flow matching models, leading to the formulation of three categories of practical, training-free energy guidance algorithms [11][12]. - The guidance vector field is designed to direct the original vector field towards regions of lower energy function values [12]. Experimental Results - Experiments were conducted on synthetic data, offline reinforcement learning, and image linear inverse problems, demonstrating the effectiveness of the proposed algorithms [20][22]. - In synthetic datasets, the Monte Carlo sampling-based guidance algorithm achieved results closest to the ground truth distribution, validating the correctness of the flow matching guidance framework [21]. - In offline reinforcement learning tasks, the Monte Carlo sampling guidance exhibited the best performance due to the need for stable guidance samples across different time steps [23]. - For image inverse problems, the Gaussian approximation guidance and GDM showed optimal performance, while the Monte Carlo sampling struggled due to high dimensionality [25]. Conclusion - The work fills a significant gap in energy guidance algorithms for flow matching models, providing a new theoretical framework and several practical algorithms, along with theoretical analysis and experimental comparisons to guide real-world applications [27].