Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes

We propose a framework for probabilistic forecasting of dynamical systems based on generative modeling. Given observations of the system state over time, we formulate the forecasting problem as sampling from the conditional distribution of the future system state given its current state. To this end, we leverage the framework of stochastic interpolants, which facilitates the construction of a generative model between an arbitrary base distribution and the target. We design a fictitious, non-physical stochastic dynamics that takes as initial condition the current system state and produces as output a sample from the target conditional distribution in finite time and without bias. This process therefore maps a point mass centered at the current state onto a probabilistic ensemble of forecasts. We prove that the drift coefficient entering the stochastic differential equation (SDE) achieving this task is non-singular, and that it can be learned efficiently by square loss regression over the time-series data. We show that the drift and the diffusion coefficients of this SDE can be adjusted after training, and that a specific choice that minimizes the impact of the estimation error gives a F\"ollmer process. We highlight the utility of our approach on several complex, high-dimensional forecasting problems, including stochastically forced Navier-Stokes and video prediction on the KTH and CLEVRER datasets.

SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers

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: using discrete vs. continuous time learning, deciding the objective for the model to learn, choosing the interpolant connecting the distributions, and deploying a deterministic or stochastic sampler. By carefully introducing the above ingredients, SiT surpasses DiT uniformly across model sizes on the conditional ImageNet 256x256 benchmark using the exact same backbone, 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.

Stochastic interpolants with data-dependent couplings

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to \textit{couple} the base and the target densities. This enables us to incorporate information about class labels or continuous embeddings to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.

Multimarginal generative modeling with stochastic interpolants

Given a set of $K$ probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

Deep learning probability flows and entropy production rates in active matter

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. These systems generically involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the magnitude of the steady-state probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry and the strength of nonequilibrium transport of measure. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework that estimates the score of this density. We show that the score, together with the microscopic equations of motion, gives direct access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles, spatial regions, and degrees of freedom. To represent the score, we introduce a novel, spatially-local transformer-based network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of interacting active particles undergoing motility-induced phase separation (MIPS). We show that a single instance of our network trained on a system of 4096 particles at one packing fraction can generalize to other regions of the phase diagram, including systems with as many as 32768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

Agile Catching with Whole-Body MPC and Blackbox Policy Learning

We address a benchmark task in agile robotics: catching objects thrown at high-speed. This is a challenging task that involves tracking, intercepting, and cradling a thrown object with access only to visual observations of the object and the proprioceptive state of the robot, all within a fraction of a second. We present the relative merits of two fundamentally different solution strategies: (i) Model Predictive Control using accelerated constrained trajectory optimization, and (ii) Reinforcement Learning using zeroth-order optimization. We provide insights into various performance trade-offs including sample efficiency, sim-to-real transfer, robustness to distribution shifts, and whole-body multimodality via extensive on-hardware experiments. We conclude with proposals on fusing "classical" and "learning-based" techniques for agile robot control. Videos of our experiments may be found at https://sites.google.com/view/agile-catching

Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

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. 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. Remarkably, we show that minimization of these quadratic objectives leads to control of the likelihood for any of our generative models built upon stochastic dynamics. By contrast, we establish that generative models based upon a deterministic dynamics must, in addition, control the Fisher divergence between the target and the model. We also construct estimators for the likelihood and the cross-entropy of interpolant-based generative models, discuss connections with other stochastic bridges, and demonstrate that such models recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant.

Probability flow solution of the Fokker-Planck equation

The method of choice for integrating the time-dependent Fokker-Planck equation in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation. Here, we introduce an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Unlike the stochastic dynamics, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. The method has the advantage of giving direct access to quantities that are challenging to estimate only given samples from the solution, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its "score"), and so is a-priori unknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of particles according to the instantaneous probability current. Our approach is based on recent advances in score-based diffusion for generative modeling, with the important difference that the training procedure is self-contained and does not require samples from the target density to be available beforehand. To demonstrate the validity of the approach, we consider several examples from the physics of interacting particle systems; we find that the method scales well to high-dimensional systems, and accurately matches available analytical solutions and moments computed via Monte-Carlo.

Adversarially Robust Stability Certificates can be Sample-Efficient

Motivated by bridging the simulation to reality gap in the context of safety-critical systems, we consider learning adversarially robust stability certificates for unknown nonlinear dynamical systems. In line with approaches from robust control, we consider additive and Lipschitz bounded adversaries that perturb the system dynamics. We show that under suitable assumptions of incremental stability on the underlying system, the statistical cost of learning an adversarial stability certificate is equivalent, up to constant factors, to that of learning a nominal stability certificate. Our results hinge on novel bounds for the Rademacher complexity of the resulting adversarial loss class, which may be of independent interest. To the best of our knowledge, this is the first characterization of sample-complexity bounds when performing adversarial learning over data generated by a dynamical system. We further provide a practical algorithm for approximating the adversarial training algorithm, and validate our findings on a damped pendulum example.