Test-Time Guidance for Flow-Based Generative Models via Parallel Tempering on Source Distributions

Generative models that transport a simple source distribution to a complex data distribution-such as diffusion and flow-based models-are central to high-fidelity data generation. Test-time guidance can further steer pretrained models toward user-specified high-reward regions without costly retraining. However, existing guidance methods face critical limitations: they struggle with non-differentiable rewards, fail to navigate complex landscapes, and often lack theoretical guarantees on generation performance. We propose Source Parallel Tempering (SPT), a gradient-free test-time guidance framework that operates entirely in source space, leveraging its simpler geometry to avoid the complexities of the data manifold. SPT couples a local exploration kernel with parallel tempering, enabling efficient barrier crossing and robust discovery of high-reward modes. Theoretically, we provide a new error bound linking training-time approximation error to test-time guidance performance. Empirically, SPT significantly improves over state-of-the-art methods on benchmark tasks in conditional image synthesis and dynamical system trajectory sampling. Code is available at https://github. com/Utah-Math-Data-Science/SPT.