Minimum-Excess-Work Guidance: Score-Based Sampling with Experimental Data or Sparse Restraints

Surrogate models, such as Boltzmann generators (BGs) and emulators (BEs), based on deep generative models are becoming an important tool in molecular simulation. Often, we may want to use additional external information such as sparse experimental data to refine these models. However, there is no unique way to achieve this goal. Here, we propose a method inspired by thermodynamic work from statistical mechanics to regularize the guidance of pretrained probability flow generative models (e.g., continuous normalizing flows or diffusion models) to match additional sparse information. The regularization ensures that the excess work of the guidance procedure is minimized. We developed two guiding strategies based on this method: Path Guidance, which facilitates sampling of rare transition states by concentrating probability mass on user-defined subsets, and Observable Guidance, which aligns generated distributions with experimental observables while preserving entropy. We demonstrate the framework's versatility on two coarse-grained Boltzmann emulators, showcasing its ability to sample transition configurations and to correct systematic biases using experimental data on a variety of model protein systems. Finally, we provide bounds on the distributional differences between the guided and unguided distributions. The method bridges thermodynamic principles with modern generative architectures, offering a principled, efficient, and physics-inspired alternative to standard fine-tuning in data-scarce domains. Our results highlight improved sample efficiency and bias reduction, underscoring their applicability to molecular simulations and beyond.