Nicholas M. Boffi

Rishal Aggarwal, David Ryan Koes, Nicholas M. Boffi et al.

Path sampling methods generate ensembles of reactive trajectories connecting metastable states, but extracting mechanistic insight from these data remains nontrivial. We introduce Flux Matching, a framework that learns two complementary objects directly from reactive trajectory data: a current velocity $u(z)$, whose streamlines trace the dominant reaction pathways, and a scalar potential $h(z)$, obtained from a weighted Helmholtz-Hodge decomposition of the reactive current, that serves as a data-driven reaction coordinate. Both minimize quadratic functionals over the reactive path ensemble, analogous to the flow matching loss in generative modeling, and require no knowledge of the underlying dynamics or stationary distribution. Unlike committor-based methods, $u$ and $h$ remain well-defined under projection onto non-Markovian collective variables, and their level sets in turn provide adaptive interfaces for improved sampling with enhanced sampling methods. Flux Matching is validated through the generation of current velocity trajectories and rate constant calculations on molecular systems.

Jerry Y. Huang, Justin Lin, Sheel Shah et al.

In generative modeling, we often wish to produce samples that maximize a user-specified reward such as aesthetic quality or alignment with human preferences, a problem known as guidance. Despite their widespread use, existing guidance methods either require expensive multi-particle, many-step schemes or rely on poorly understood approximations. We reformulate guidance as a deterministic optimal control problem, yielding a hierarchy of algorithms that subsumes existing approaches at the coarsest level. We show that the flow map, an object of significant recent interest for its role in fast inference, arises naturally in the optimal solution. Based on this observation, we propose Flow Map Reward Guidance (FMRG): a training-free, single-trajectory framework that uses the flow map to both integrate and guide the flow. At text-to-image scale, FMRG matches or surpasses baselines across inverse problems, style transfer, human preferences, and VLM rewards with as few as 3 NFEs, giving at least an order-of-magnitude speedup in comparison to prior state of the art.

RuiKang OuYang, Hanlin Yu, Xinyue Ai et al.

Recent progress in flow-based generative modeling has led to models that output high-quality samples while using only a small number of function evaluations. However, at present, there is a lack of similar advances in estimating the model likelihood. In particular, most existing methods either rely on restrictive architectures that enable exact calculations, or use stochastic approximations such as Hutchinson's trace estimator that introduce substantial variance. In this work, we introduce SCAlable LikeLihood distillation of flOw maPs (SCALLOP). SCALLOP builds on the recently proposed F2D2, a likelihood flow map model that can generate samples and their densities in a small number of function evaluations. While F2D2 uses Hutchinson's estimator during training, we introduce an alternative and more scalable likelihood distillation objective that is Hutchinson-free and admits a vectorized formulation. Empirically, we demonstrate the effectiveness of SCALLOP as a Boltzmann generator in molecular science, and further validate its benefit on image datasets. SCALLOP significantly reduces both training variance and training time while consistently improving performance compared to F2D2, and is competitive with the state-of-the-art while achieving up to 10x inference speedup over the fastest baseline.

Manan Agarwal, Sheel Shah, Chanhyuk Lee et al.

Non-autoregressive generation offers a powerful paradigm for iterative refinement, allowing models to recursively critique, erase and regenerate arbitrary subsets of tokens. However, existing non-autoregressive models fail to realize this potential. Masked Diffusion Models (MDMs) suffer from factorization error, causing sample quality to collapse when generating multiple tokens simultaneously. Flow Map Language Models (FMLMs) circumvent this bottleneck via joint sequence transport for excellent few-step generation, but sacrifice the inference-time flexibility of MDMs. We introduce FMLM+, a framework that bridges this gap by equipping FMLM with masking-style noise schedules. While generating the full sequence in a single step, FMLM+ simultaneously scores the global consistency of each token a posteriori. We leverage this to introduce Posterior Refinement, a novel inference-time refinement strategy that enables the model to adaptively self-correct its outputs, matching the performance of discrete baselines with 32x fewer NFEs. Across diverse benchmarks, we demonstrate that FMLM+ with Posterior Refinement improves the speed--quality tradeoff over both MDM and FMLM families, providing a scalable foundation for high-fidelity language modeling.

Sanjit Dandapanthula, Nicholas M. Boffi

Reward guidance algorithms steer a learned generative process toward the reward-tilted measure at inference time. While empirically powerful, these methods are prone to reward hacking: the guided model over-optimizes the reward at the cost of fidelity to the learned distribution. Prior work has attributed this to the complexity of neural reward functions or implicit biases in diffusion training, but its fundamental origins remain poorly understood. We show that reward hacking arises from an approximation made in most practical implementations of reward-guided diffusion -- finite-particle plug-in estimation of the Doob h-function -- even in the simplest non-trivial settings of Gaussian and Gaussian mixture targets with quadratic rewards. In closed form, we isolate two distinct failure modes of the plug-in estimator: it leads to reward hacking within each mode and it cannot select high-reward modes. We propose a closed-form reward damping schedule that corrects the within-mode bias with no additional compute, and clarify the role of best-of-n sampling in compensating for the mode selection failure. Experiments on Gaussian mixture targets, a 2D checkerboard, and FLUX.1 text-to-image generation confirm that our theoretical insights carry over to practical settings.