Transport-based methods have emerged as a leading paradigm for building generative models from large, clean datasets. However, in many scientific and engineering domains, clean data are often unavailable: instead, we only observe measurements corrupted through a noisy, ill-conditioned channel. A generative model for the original data thus requires solving an inverse problem at the level of distributions. In this work, we introduce a novel approach to this task based on Stochastic Interpolants: we iteratively update a transport map between corrupted and clean data samples using only access to the corrupted dataset as well as black box access to the corruption channel. Under appropriate conditions, this iterative procedure converges towards a self-consistent transport map that effectively inverts the corruption channel, thus enabling a generative model for the clean data. The resulting method (i) is computationally efficient compared to variational alternatives, (ii) highly flexible, handling arbitrary nonlinear forward models with only black-box access, and (iii) enjoys theoretical guarantees. We demonstrate superior performance on inverse problems in natural image processing and scientific reconstruction, and establish convergence guarantees of the scheme under appropriate assumptions.

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.

Rare events such as conformational changes in biomolecules, phase transitions, and chemical reactions are central to the behavior of many physical systems, yet they are extremely difficult to study computationally because unbiased simulations seldom produce them. Transition Path Theory (TPT) provides a rigorous statistical framework for analyzing such events: it characterizes the ensemble of reactive trajectories between two designated metastable states (reactant and product), and its central object–the committor function, which gives the probability that the system will next reach the product rather than the reactant–encodes all essential kinetic and thermodynamic information. We introduce a framework that casts committor estimation as a stochastic optimal control (SOC) problem. In this formulation the committor defines a feedback control–proportional to the gradient of its logarithm–that actively steers trajectories toward the reactive region, thereby enabling efficient sampling of reactive paths. To solve the resulting hitting-time control problem we develop two complementary objectives: a direct backpropagation loss and a principled off-policy Value Matching loss, for which we establish first-order optimality guarantees. We further address metastability, which can trap controlled trajectories in intermediate basins, by introducing an alternative sampling process that preserves the reactive current while lowering effective energy barriers. On benchmark systems, the framework yields markedly more accurate committor estimates, reaction rates, and equilibrium constants than existing methods.

We develop a kernel method for generative modeling within the stochastic interpolant framework, replacing neural network training with linear systems. The drift of the generative SDE is b̂_t(x) = (x)^_t, where η_t ∈ℝ^P solves a P× P system computable from data, with P independent of the data dimension d. Since estimates are inexact, the diffusion coefficient D_t affects sample quality; the optimal D_t^* from Girsanov diverges at t=0, but this poses no difficulty and we develop an integrator that handles it seamlessly. The framework accommodates diverse feature maps: scattering transforms, pretrained generative models, etc, enabling generation and model combination without neural network training. We demonstrate the approach on financial time series, turbulence, and image generation.

Understanding the geometry of learned distributions is fundamental to improving and interpreting diffusion models, yet systematic tools for exploring their landscape remain limited. Standard latent-space interpolations fail to respect the structure of the learned distribution, often traversing low-density regions. We introduce a framework based on the string method that computes continuous paths between samples by evolving curves under the learned score function. Operating on pretrained models without retraining, our approach interpolates between three regimes: pure generative transport, which yields continuous sample paths; gradient-dominated dynamics, which recover minimum energy paths (MEPs); and finite-temperature string dynamics, which compute principal curves -- self-consistent paths that balance energy and entropy. We demonstrate that the choice of regime matters in practice. For image diffusion models, MEPs contain high-likelihood but unrealistic ''cartoon'' images, confirming prior observations that likelihood maxima appear unrealistic; principal curves instead yield realistic morphing sequences despite lower likelihood. For protein structure prediction, our method computes transition pathways between metastable conformers directly from models trained on static structures, yielding paths with physically plausible intermediates. Together, these results establish the string method as a principled tool for probing the modal structure of diffusion models -- identifying modes, characterizing barriers, and mapping connectivity in complex learned distributions.

We present Free energy Estimators with Adaptive Transport (FEAT), a novel framework for free energy estimation---a critical challenge across scientific domains. FEAT leverages learned transports implemented via stochastic interpolants and provides consistent, minimum-variance estimators based on escorted Jarzynski equality and controlled Crooks theorem, alongside variational upper and lower bounds on free energy differences. Unifying equilibrium and non-equilibrium methods under a single theoretical framework, FEAT establishes a principled foundation for neural free energy calculations. Experimental validation on toy examples, molecular simulations, and quantum field theory demonstrates promising improvements over existing learning-based methods. Our PyTorch implementation is available at https://github.com/jiajunhe98/FEAT.

Building on the framework proposed in Boffi et al. (2024), we present a systematic approach for learning flow maps associated with flow and diffusion models. Flow map-based models, commonly known as consistency models, encompass recent efforts to improve the efficiency of generative models based on solutions to differential equations. By exploiting a relationship between the velocity field underlying a continuous-time flow and the instantaneous rate of change of the flow map, we show how to convert existing distillation schemes into direct training algorithms via self-distillation, eliminating the need for pre-trained models. We empirically evaluate several instantiations of our framework, finding that high-dimensional tasks like image synthesis benefit from objective functions that avoid temporal and spatial derivatives of the flow map, while lower-dimensional tasks can benefit from objectives incorporating higher-order derivatives to capture sharp features.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

We study the design of interpolation schedules in the stochastic interpolants framework for flow and diffusion-based generative models. We show that while all scalar interpolation schedules achieve identical statistical efficiency under Kullback-Leibler divergence in path space after optimal diffusion coefficient tuning, their numerical efficiency can differ substantially. This observation motivates focusing on numerical properties of the resulting drift fields rather than statistical criteria for schedule design. We propose averaged squared Lipschitzness minimization as a principled criterion for numerical optimization, providing an alternative to kinetic energy minimization used in optimal transport approaches. A transfer formula is derived that enables conversion between different schedules at inference time without retraining neural networks. For Gaussian distributions, our optimized schedules achieve exponential improvements in Lipschitz constants over standard linear schedules, while for Gaussian mixtures, they reduce mode collapse in few-step sampling. We also validate our approach on high-dimensional invariant distributions from stochastic Allen-Cahn equations and Navier-Stokes equations, demonstrating robust performance improvements across resolutions.

Flow-based generative models can face significant challenges when modeling scientific data with multiscale Fourier spectra, often producing large errors in fine-scale features. We address this problem within the framework of stochastic interpolants, via principled design of noise distributions and interpolation schedules. The key insight is that the noise should not be smoother than the target data distribution – measured by Fourier spectrum decay rates – to ensure bounded drift fields near the initial time. For Gaussian and near-Gaussian distributions whose fine-scale structure is known, we show that spectrum-matched noise improves numerical efficiency compared to standard white-noise approaches. For complex non-Gaussian distributions, we develop scale-adaptive interpolation schedules that address the numerical ill-conditioning arising from rougher-than-data noise. Numerical experiments on synthetic Gaussian random fields and solutions to the stochastic Allen-Cahn and Navier-Stokes equations validate our approach and demonstrate its ability to generate high-fidelity samples at lower computational cost than traditional approaches.

We present a novel simulation-free framework for training continuous-time diffusion processes over very general objective functions. Existing methods typically involve either prescribing the optimal diffusion process -- which only works for heavily restricted problem formulations -- or require expensive simulation to numerically obtain the time-dependent densities and sample from the diffusion process. In contrast, we propose a coupled parameterization which jointly models a time-dependent density function, or probability path, and the dynamics of a diffusion process that generates this probability path. To accomplish this, our approach directly bakes in the Fokker-Planck equation and density function requirements as hard constraints, by extending and greatly simplifying the construction of Neural Conservation Laws. This enables simulation-free training for a large variety of problem formulations, from data-driven objectives as in generative modeling and dynamical optimal transport, to optimality-based objectives as in stochastic optimal control, with straightforward extensions to mean-field objectives due to the ease of accessing exact density functions. We validate our method in a diverse range of application domains from modeling spatio-temporal events to learning optimal dynamics from population data.

A common recipe to improve diffusion models at test-time so that samples score highly against a user-specified reward is to introduce the gradient of the reward into the dynamics of the diffusion itself. This procedure is often ill posed, as user-specified rewards are usually only well defined on the data distribution at the end of generation. While common workarounds to this problem are to use a denoiser to estimate what a sample would have been at the end of generation, we propose a simple solution to this problem by working directly with a flow map. By exploiting a relationship between the flow map and velocity field governing the instantaneous transport, we construct an algorithm, Flow Map Trajectory Tilting (FMTT), which provably performs better ascent on the reward than standard test-time methods involving the gradient of the reward. The approach can be used to either perform exact sampling via importance weighting or principled search that identifies local maximizers of the reward-tilted distribution. We demonstrate the efficacy of our approach against other look-ahead techniques, and show how the flow map enables engagement with complicated reward functions that make possible new forms of image editing, e.g. by interfacing with vision language models.

Active systems comprise a class of nonequilibrium dynamics in which individual components autonomously dissipate energy. Efforts towards understanding the role played by activity have centered on computation of the entropy production rate (EPR), which quantifies the breakdown of time reversal symmetry. A fundamental difficulty in this program is that high dimensionality of the phase space renders traditional computational techniques infeasible for estimating the EPR. Here, we overcome this challenge with a novel deep learning approach that estimates probability currents directly from stochastic system trajectories. We derive a new physical connection between the probability current and two local definitions of the EPR for inertial systems, which we apply to characterize the departure from equilibrium in a canonical model of flocking. Our results highlight that entropy is produced and consumed on the spatial interface of a flock as the interplay between alignment and fluctuation dynamically creates and annihilates order. By enabling the direct visualization of when and where a given system is out of equilibrium, we anticipate that our methodology will advance the understanding of a broad class of complex nonequilibrium dynamics.

Recent works have shown that diffusion models can undergo phase transitions, the resolution of which is needed for accurately generating samples. This has motivated the use of different noise schedules, the two most common choices being referred to as variance preserving (VP) and variance exploding (VE). Here we revisit these schedules within the framework of stochastic interpolants. Using the Gaussian Mixture (GM) and Curie-Weiss (CW) data distributions as test case models, we first investigate the effect of the variance of the initial noise distribution and show that VP recovers the low-level feature (the distribution of each mode) but misses the high-level feature (the asymmetry between modes), whereas VE performs oppositely. We also show that this dichotomy, which happens when denoising by a constant amount in each step, can be avoided by using noise schedules specific to VP and VE that allow for the recovery of both high- and low-level features. Finally we show that these schedules yield generative models for the GM and CW model whose probability flow ODE can be discretized using Θ_d(1) steps in dimension d instead of the Θ_d(√(d)) steps required by constant denoising.

In living matter, concentration gradients of nutrients carve the motility of microorganisms in a heterogeneous environment. Here, we use swimming droplets as a model system to study how swimmer-trail interactions guide locomotion. Combining experiments and theory, we show that our non-Markovian droplet model quantitatively captures droplet motility. The two fit parameters provide the first estimate of the effective temperature arising from hydrodynamic flows and the coupling strength of the propulsion force. This framework is general and explains memory effects, droplet hovering, and enhanced collective motion.

We introduce the Non-Equilibrium Transport Sampler (NETS), an algorithm for sampling from unnormalized probability distributions. NETS builds on non-equilibrium sampling strategies that transport a simple base distribution into the target distribution in finite time, as pioneered in Neal's annealed importance sampling (AIS). In the continuous-time setting, this transport is accomplished by evolving walkers using Langevin dynamics with a time-dependent potential, while simultaneously evolving importance weights to debias their solutions following Jarzynski's equality. The key innovation of NETS is to add to the dynamics a learned drift term that offsets the need for these corrective weights by minimizing their variance through an objective that can be estimated without backpropagation and provably bounds the Kullback-Leibler divergence between the estimated and target distributions. NETS provides unbiased samples and features a tunable diffusion coefficient that can be adjusted after training to maximize the effective sample size. In experiments on standard benchmarks, high-dimensional Gaussian mixtures, and statistical lattice field theory models, NETS shows compelling performances.

Diffusion- and flow-based policies deliver state-of-the-art performance on long-horizon robotic manipulation and imitation-learning tasks. However, these controllers employ a fixed inference budget at every control step, regardless of task complexity, leading to computational inefficiency for simple subtasks while potentially underperforming on challenging ones. To address these issues, we introduce Difficulty-Aware Stochastic Interpolant Policy (DA-SIP), a framework that enables robotic controllers to adaptively adjust their integration horizon in real-time based on task difficulty. Our approach employs a difficulty classifier that analyzes RGB-D observations to dynamically select the step budget, the optimal solver variant, and ODE/SDE integration at each control cycle. DA-SIP builds upon the stochastic interpolant formulation to provide a unified framework that unlocks diverse training and inference configurations for diffusion- and flow-based policies. Through comprehensive benchmarks across diverse manipulation tasks, DA-SIP achieves 2.6-4.4× reduction in total computation time while maintaining task-success rates comparable to fixed maximum-computation baselines. By implementing adaptive computation within this framework, DA-SIP transforms generative robot controllers into efficient, task-aware systems that intelligently allocate inference resources where they provide the greatest benefit.

We propose a framework for learning maps between probability distributions that broadly generalizes the time dynamics of flow and diffusion models. To enable this, we generalize stochastic interpolants by replacing the scalar time variable with vectors, matrices, or linear operators, allowing us to bridge probability distributions across multiple dimensional spaces. This approach enables the construction of versatile generative models capable of fulfilling multiple tasks without task-specific training. Our operator-based interpolants not only provide a unifying theoretical perspective for existing generative models but also extend their capabilities. Through numerical experiments, we demonstrate the zero-shot efficacy of our method on conditional generation and inpainting, fine-tuning and posterior sampling, and multiscale modeling, suggesting its potential as a generic task-agnostic alternative to specialized models.

Training nonlinear parametrizations such as deep neural networks to numerically approximate solutions of partial differential equations is often based on minimizing a loss that includes the residual, which is analytically available in limited settings only. At the same time, empirically estimating the training loss is challenging because residuals and related quantities can have high variance, especially for transport-dominated and high-dimensional problems that exhibit local features such as waves and coherent structures. Thus, estimators based on data samples from un-informed, uniform distributions are inefficient. This work introduces Neural Galerkin schemes that estimate the training loss with data from adaptive distributions, which are empirically represented via ensembles of particles. The ensembles are actively adapted by evolving the particles with dynamics coupled to the nonlinear parametrizations of the solution fields so that the ensembles remain informative for estimating the training loss. Numerical experiments indicate that few dynamic particles are sufficient for obtaining accurate empirical estimates of the training loss, even for problems with local features and with high-dimensional spatial domains.

We present Scalable Interpolant Transformers (SiT), a family of generative models built on the backbone of Diffusion Transformers (DiT). The interpolant framework, which allows for connecting two distributions in a more flexible way than standard diffusion models, makes possible a modular study of various design choices impacting generative models built on dynamical transport: learning in discrete or continuous time, the objective function, the interpolant that connects the distributions, and deterministic or stochastic sampling. By carefully introducing the above ingredients, SiT surpasses DiT uniformly across model sizes on the conditional ImageNet 256x256 and 512x512 benchmark using the exact same model structure, number of parameters, and GFLOPs. By exploring various diffusion coefficients, which can be tuned separately from learning, SiT achieves an FID-50K score of 2.06 and 2.62, respectively.