TL;DR
The paper introduces Source-Guided Flow Matching (SGFM), a novel framework that guides generative models by directly modifying the source distribution, enabling flexible sampling and preserving the vector field, with theoretical guarantees and practical demonstrations.
Contribution
SGFM is a new guidance framework that directly alters the source distribution while keeping the vector field fixed, with theoretical recovery guarantees and flexible sampling options.
Findings
SGFM exactly recovers the target distribution under ideal conditions.
The framework provides bounds on Wasserstein error with approximate samplers.
Experimental results show effectiveness across various tasks.
Abstract
Guidance of generative models is typically achieved by modifying the probability flow vector field through the addition of a guidance field. In this paper, we instead propose the Source-Guided Flow Matching (SGFM) framework, which modifies the source distribution directly while keeping the pre-trained vector field intact. This reduces the guidance problem to a well-defined problem of sampling from the source distribution. We theoretically show that SGFM recovers the desired target distribution exactly. Furthermore, we provide bounds on the Wasserstein error for the generated distribution when using an approximate sampler of the source distribution and an approximate vector field. The key benefit of our approach is that it allows the user to flexibly choose the sampling method depending on their specific problem. To illustrate this, we systematically compare different sampling methods…
Peer Reviews
Decision·ICLR 2026 Poster
1. The core methodology is grounded in a theoretical result, Theorem 1, which states that if a vector field with flow map transports a source distribution to a target distribution, then for a desired new target distribution, there exists a modified source distribution that is precisely transported to the target one by the *same* vector field. This theorem provides the foundation for SGFM. 2. The framework also provides theoretical guarantees to ensure the quality of generated samples. Besides,
I am not familiar with the field. Here are a few comments: 1. Practical experimental setup: while this work provides solid theoretical derivations guaranteeing the quality of generated samples, all experiments were conducted on relatively small datasets. For large-scale datasets, such as the class-to-image task on ImageNet with vanilla diffusion model, i.e. DiT and SiT, it remains to be seen whether this work can also achieve high-quality generated samples. I hope the authors can provide implem
I appreciate the thorough theoretical analysis in presenting the method. The writing is clear, and all the theory was backed by intuition. The idea of modifying the source distribution through the constraint function is interesting and has not been explored as far as I am aware.
I am not very familiar with the baselines that have been compared with. The main weakness of this method is the weak empirical performance on the Darcy flow data. The promise of this method, as I see it, is in enforcing constraints (differentiable) in the generative modeling. Therefore, physical consistency is an important metric in which the method falls short compared to PnP. This is not a major weakness; however, I would request the authors to add a discussion on this.
• The paper is well-written and clearly-organized. The center of the proposed method based on modifying the source distribution is focused with in-depth discussion. • The framework is flexible to combine with different types of sampling methods based on the characteristics of the target distribution. The pros and cons of different methods are discussed in Section 4. • The experiments are selected from low to high dimensional problems, which is comprehensive.
• The clarity of the notations could be improved. For example, it would be better to denote samples from $q’_0$ by $x’_0$ rather than $x_0$ in Algorithm 1 to reduce ambiguity. • Even though the framework is flexible to incorporate different types of sampling methods, the mode collapse issue when the target distribution is highly non-concave still remains. In addition, the remedy to mitigate the mode collapse issue in line 309 only adapts to the Gaussian source distribution, it remains unknown w
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