In this talk, we consider the Schrödinger Bridge problem as a fixed-point problem and propose an approach based on minimizing the empirical discrepancy computed from $n$ random points. The method reformulates the fixed-point equation over a suitable class of transformed potentials. Under rather general conditions, we establish a convergence rate of order $1/n$. We further discuss applications of this framework to generative modeling of complex distributions.
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We present an adjoint method for optimization of the spatially homogeneous Boltzmann equation for rarefied gas dynamics. The adjoint method is derived using a “discretize then optimize” approach. Discretization (in time and velocity) is via the Direct Simulation Monte Carlo (DSMC) method, and adjoint equations are derived from an augmented Lagrangian. After a forward (in time) solution of DSMC, the adjoint variables are found by a backwards solver. They are equal to velocity derivatives of an objective function, and are used for optimization of the Boltzmann equation. For general collision models, DSMC requires the use of a rejection sampling step, which involves discontinuities that lead to a new term, involving a “score function”. This is joint work with Yunan Yang (Cornell) and Denis Silantyev (U Colorado, Colorado Springs).
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The principle behind magnetic fusion is to confine high-temperature plasma inside a device in such a way that the nuclei of deuterium and tritium joining together can release energy. The high temperatures generated require the plasma to be isolated from the wall of the device to avoid damage, and the purpose of external magnetic fields is to achieve this goal.
In this talk, we address this challenge from a numerical perspective by introducing an instantaneous control approach to steer plasma into a prescribed spatial region. Our modeling framework is based on the Vlasov equation in a bounded domain, incorporating a self-induced electric field and a strong external magnetic field.
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Generative models parametrize flexible families of distributions capable of fitting complex datasets such as images or text. These models can generate independent samples from intricate high-dimensional distributions at negligible cost. In contrast, sampling exactly from a given target distribution—such as the Boltzmann distribution of a physical system—is often a major challenge due to high dimensionality, multimodality, ill-conditioning, or a combination of these factors. This raises the question: how can generative models be leveraged to assist in the sampling task? A key difficulty in this setting is the lack of an extensive dataset to learn from upfront. In this talk, I will focus in particular on sampling from multimodal distributions and present recent attempts inspired by diffusion models to sample using a denoising process. The talk is mainly based on works with Louis Grenioux, Maxence Noble and Alain Oliviero Durmus.
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I will introduce orthogonal coupling dynamics (OCD), a dynamical formulation closely linked to the optimal transport, that creates coupling between two particle clouds. The dynamics evolves the joint distribution by projecting gradient descent onto a marginal-preserving subspace, ensuring that each marginal remains fixed while the transport cost between samples decreases monotonically. This projection, expressed through conditional expectations, defines a concise mechanism by which independent samples are dynamically paired. At the conceptual level, OCD reveals how optimal transport can be estimated through regression-like projections and interactions analogous to kinetic systems. The model exhibits favorable computational scaling in estimating optimal transport maps. I will discuss the structure, examples, and computational implications of this dynamics and conclude by outlining several open problems.
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The existence of random dynamical systems for McKean–Vlasov SDEs is established. This is approached by considering the joint dynamics of the corresponding nonlinear Fokker-Planck equation governing the law of the system and the underlying stochastic differential equation (SDE) as a dynamical system on the product space $\RR^d \times \mathcal{P}(\RR^d)$. The proof relies on two main ingredients: At the level of the SDE, a pathwise rough path-based solution theory for SDEs with time-dependent coefficients is implemented, while at the level of the PDE a well-posedness theory is developed, for measurable solutions and allowing for degenerate diffusion coefficients. The results apply in particular to the so-called ensemble Kalman sampler (EKS), proving the existence of an associated RDS under some assumptions on the posterior, as well as to the Lagrangian formulation of the Landau equation with Maxwell molecules. As a by-product of the main results, the uniqueness of solutions non-linear Fokker–Planck equations associated to the EKS is shown. This is going wont with Benjamin Gess and Shanshan Hu.
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We study systems of interacting stochastic particles approximating McKean-Vlasov SDEs and establish both strong and weak convergence rates for the mean-field limit. Using classical stochastic analysis tools, including Ito’s formula, the Kolmogorov-Chentsov theorem, and the Kolmogorov backward equation, we prove existence and uniqueness of strong solutions, derive L^p strong convergence rates, and show that the weak error for a P-particle approximation is of order P^{-1}.
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Why are score-based models so successful? The conventional view is that they recover the data distribution in the low-noise limit. In this talk, we propose an alternative hypothesis: their effectiveness arises from an implicit bias toward manifold learning. We show that in the small-$\sigma$ regime, score estimation necessarily captures geometric information about the data manifold before encoding distributional information. Moreover, the accuracy required for geometry is asymptotically weaker: concentration on the manifold is guaranteed under a score error of $o(\sigma^{-2})$, whereas recovering the exact data distribution demands the much stricter condition $o(1)$. This rate separation leads to three consequences: (i) diffusion models detect data support under $o(\sigma^{-2})$ accuracy; (ii) uniform distributions on the manifold are also $O(\sigma^{-2})$ easier to learn; and (iii) in Bayesian inverse problems, maximum entropy priors are $O(\sigma^{-2})$ more robust than generic ones. We further validate these results with preliminary experiments on large-scale models, including Stable Diffusion.
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The Vlasov–Maxwell–Landau equation is widely regarded as a first-principles model for plasma dynamics. In this talk, we introduce a novel particle method for this equation that simultaneously accounts for particle transport, electromagnetic field effects, and Coulomb collisions. The method is derived from a regularization of the variational formulation of the Landau collision operator, leading to a deterministic particle discretization that conserves mass, momentum, and energy, while dissipating entropy. We will also present recent progress on energy-conserving time discretizations for the resulting particle system.
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I will present free energy adaptive biasing techniques, and discuss convergence results for these methods, in particular a recent work in collaboration with Xuyang Lin and Pierre Monmarché. Free-energy-based adaptive biasing methods, such as Metadynamics, the Adaptive Biasing Force (ABF) and their variants, are enhanced sampling algorithms widely used in molecular simulations. Although their efficiency has been empirically acknowledged for decades, providing theoretical insights via a quantitative convergence analysis is a difficult problem, in particular for the kinetic Langevin diffusion, which is non-reversible and hypocoercive. We obtain the first exponential convergence result for such a process, in an idealized setting where the dynamics can be associated with a mean-field non-linear flow on the space of probability measures. A key of the analysis is the interpretation of the (idealized) algorithm as the gradient descent of a suitable functional over the space of probability distributions.
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Sampling or otherwise summarizing complex probability distributions is a central task in applied mathematics, statistics, and machine learning. Many modern algorithms for this task introduce dynamics in the space of probability measures, designing these dynamics to achieve good practical performance.
We will discuss two aspects of this broad design endeavor. First is the problem of optimal scheduling of dynamic transport, i.e., with what speed should one proceed along a prescribed path of probability measures? Though many popular methods seek straight line trajectories, i.e., trajectories with zero acceleration in a Lagrangian frame, we show how a specific class of “curved” trajectories can significantly improve approximation and learning. Specifically, we consider the unit-time interpolation of a given transport map with the identity, and introduce a “schedule” function that rescales time. We show that a schedule minimizing the spatial Lipschitz constant of velocity field, uniformly over time, can be computed in closed form, and that the resulting Lipschitz constant is exponentially smaller than that induced by an identity schedule (corresponding to, for instance, the Wasserstein geodesic). We then discuss extensions of this idea which seek not only schedules but paths that improve spatial regularity of the velocity.
Second, we discuss the problem of weighted quantization, i.e., summarizing a complex distribution with a small set of weighted Dirac measures. We study this problem from the perspective of minimizing maximum mean discrepancy via gradient flow in the Wasserstein–Fisher–Rao (WFR) geometry. This gradient flow yields an ODE system from which we further derive a fixed-point algorithm called mean shift interacting particles (MSIP). We show that MSIP extends the classical mean shift algorithm, used for identifying modes in kernel density estimates, and that it outperforms state-of-the-art methods for quantization. One lesson from these efforts is that gradient flows offer a unifying perspective on how to design sampling dynamics, but that it can be useful to depart from certain gradient flows in a principled way.
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Consensus-Based Optimization (CBO) has recently emerged as a new paradigm in collective particle optimization, providing a simple yet powerful mechanism where agents interact through consensus formation around promising regions of the search space. Unlike classical metaheuristics, CBO admits a rigorous mean-field description, enabling convergence analysis in high-dimensional and nonconvex settings.
In this talk, we present two recent advances motivated by the challenge of efficiently sampling and minimizing complex, high-dimensional landscapes. First, we extend CBO by introducing a superlinear drift inspired by quantum models, which accelerates convergence beyond the classical linear regime and yields provably faster concentration around global minimizers, while retaining the robustness of consensus dynamics. Second, we propose a swarm-based dynamic with random velocity jumps, formalized through a kinetic BGK framework. This formulation allows for general noise structures, including Cauchy-type heavy-tailed perturbations that enhance long-range exploration. We establish propagation of chaos, establish convergence to global minimizers, and show how the model recovers CBO in a diffusive scaling limit.
These are joint works with J. Franceschi (Ferrara), M. Zanella (Pavia), H. Im (Edinburgh), G. Borghi (Edinburgh).
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The Langevin dynamics, both in their overdamped and underdamped/kinetic form, is routinely used for sampling from probability distributions. We consider preconditioned Langevin dynamics, where the preconditioner is the mobility matrix in the overdamped case and the friction matrix in the kinetic one. We study the problem of calculating the optimal preconditioner, with respect to maximizing the spectral gap. Both log-concave and non-log-concave problems are considered. The talk is based on the articles Optimizing the diffusion coefficient of overdamped Langevin dynamics T. Lelièvre, G.A. Pavliotis, G. Robin, R. Santet, G. Stoltz Math. Computation (2026)
And Optimal friction matrix for underdamped Langevin sampling M Chak, N Kantas, T Lelièvre, GA Pavliotis ESAIM: Mathematical Modelling and Numerical Analysis 57 (6), 3335-3371 (2023)
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Estimating parameters from samples of an optimal probability distribution is essential in applications ranging from socio-economic modeling to biological system analysis. In these settings, the probability distribution arises as the solution to an optimization problem that captures either static interactions among agents or the dynamic evolution of a system over time. We introduce a general methodology based on a new class of loss functions, called sharpened Fenchel-Young losses, which measure the sub-optimality gap of the optimization problem over the space of probability measures.
We provide explicit stability guarantees for two relevant settings in the context of optimal transport: The first is inverse unbalanced optimal transport (iUOT) with entropic regularization, where the parameters to estimate are cost functions that govern transport computations; this method has applications such as link prediction in machine learning. The second is inverse gradient flow (iJKO), where the objective is to recover a potential function that drives the evolution of a probability distribution via the Jordan-Kinderlehrer-Otto (JKO) time-discretization scheme; this is particularly relevant for understanding cell population dynamics in single-cell genomics. We also establish source conditions to ensure stability of our method under mirror stratifiable regularizers (such as l1 or nuclear norm) that promote structure.
Finally, we present optimization algorithms specifically tailored to efficiently solve iUOT and iJKO problems. We validate our approach through numerical experiments on Gaussian distributions, where closed-form solutions are available, to demonstrate the practical performance of our methods.
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The talk outlines a novel extension of the classical Pontryagin minimum (maximum) principle to stochastic optimal control problems. Contrary to the well-known stochastic Pontryagin minimum principle involving forward-backward stochastic differential equations, the proposed formulation is deterministic and of mean-field type. The Hamiltonian structure of the proposed Pontryagin minimum principle is achieved via the introduction of an appropriate gauge variable. The gauge freedom can be used to decouple the forward and reverse time equations; hence simplifying the solution of the underlying boundary value problem. We also consider infinite horizon discounted cost optimal control problems. In this case, the mean-field formulation allows converting the computation of the desired optimal control law into solving a pair of forward mean-field ordinary differential equations. Application to diffusion generative modelling and Bayesian inference will be discussed.
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The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a~deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. In one dimension, this form of the kernel has been proposed by Duncan, Nuesken, Szpruch. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the target distribution. The main novelty in our approach is to interpret the Stein-Fisher information as a duality pairing between $H^{-1}$ and $H^{1}$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions. This is a joint work with J. A. Carrillo and J. Warnett.
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Motivated by the study of stochastic gradient descent for over-parameterized models, we consider Langevin dynamics with noise projected onto directions which are orthogonal to an isometric group action. Our main result identifies a novel form of implicit regularization: when the initial and target density are both invariant under the group action, projected Langevin is equivalent in law to Langevin with identity diffusion but with an additional drift term proportional to the negative log volume of the group orbit. The proof is based on constructing a certain stochastic process on the group itself, and identifies the extra drift term as the negative mean curvature of the group orbit. Based on ongoing joint work with Govind Menon and Adrien Vacher.
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Sampling complex distributions in high dimension—especially under rare events, mean-field interactions, or partial observation—benefits from principled structure.  I will present sampling and rare-event estimation, combining multilevel/hierarchical designs with importance sampling (IS) derived from stochastic optimal control (SOC).
For stochastic reaction networks, we introduce SOC-based IS driven by Markovian projection, which reduces high-dimensional path functionals to low-dimensional surrogates while preserving key marginals; learning-based controllers further automate variance reduction.
Then, for McKean–Vlasov Itô SDEs, we develop multilevel and double-loop IS estimators that exploit the mean-field structure to control both bias and variance, yielding efficiency gains and robust weak/strong convergence behavior for interacting particle systems.
Finally, for Itô SDE models, we design hierarchical IS estimators for occupation-time functionals and integrate rare-event tracking within Ensemble Kalman filtering.
Together, these flows—mean-field coupling, Markovian projection maps, and control-based particle transports—yield accurate and scalable samplers for complex distributions.
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The Poincaré inequality is a key property for the convergence analysis of many practical algorithms, including MCMC samplers, dimension reduction methods etc. In this talk, we introduce a Riemannian version of the Poincaré inequality where a positive definite weighting matrix field (i.e., a Riemannian metric) is introduced to improve the Poincaré constant, and therefore the convergence speed of the resulting preconditioned Langevin dynamics. By leveraging the notion of *moment measure*, we prove the existence of an optimal metric. We also present an implementable optimization algorithm to numerically obtain the optimal metric. The method’s effectiveness is illustrated through simple but non-trivial examples which reveals rather complex solutions. Lastly, we show how to design efficient sampling algorithm which enables rapid jump across various modes and tails of the measure to be sampled from.
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Recent advancements in generative modelling, including flow-based, diffusion-based, and autoregressive networks, have achieved remarkable success in data generation. However, understanding their performance and limitations, particularly in high-dimensional settings, remains an open challenge. This talk explores the intersection of generative models and statistical physics, leveraging insights from spin-glass theory and denoising frameworks.
We first examine the efficiency of generative models compared to classical methods like Monte Carlo and Langevin dynamics in sampling from complex distributions, focusing on phase transitions that impact sampling performance. Next, we analyze denoising autoencoders in high dimensions, providing closed-form results that reveal their advantage over simpler architectures. Finally, we analyze the training of flow-based generative models on limited samples, presenting sharp theoretical characterization of their learning dynamics.
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This talk features two recent works on gradient flows for MCMC sampling in machine learning. The first part of this talk focuses on the global exponential decay of the entropy functionals along the unbalanced-transport gradient flows, a.k.a. Hellinger-Kantorovich (HK) or Wasserstein-Fisher-Rao. As an example, we will construct a new formal dissipation geometry, which we term the interaction-force transport, that guarantees convergence of minimizing the maximum-mean discrepancy. This then results in a new algorithm for sampling and generative modeling. In the second part, I will introduce a gradient flow formulation to enhance the robustness of learning and optimization algorithms under data distribution shifts. Previously, researchers have interpreted these problems as interacting gradient flows. In comparison, our new formulation unifies the analysis and motivates the design of new and flexible algorithms. In particular, I will show using entropy-regularized optimal transport for learning robustly under distribution shifts.
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Diffusion models are widely regarded as state-of-the-art generative samplers, but their iterative denoising dynamics define a stochastic process whose geometry remains only partially understood. In this work, we investigate how these dynamics encode information about the underlying data distribution and how such structure can be exploited for inference. By analyzing denoising trajectories of noisy observations, we observe class-dependent convergence behavior and geometric patterns that distinguish different modes of the distribution. These trajectory-based signatures provide a basis for zero-shot inference tasks such as classification, anomaly detection, and complexity assessment, carried out without task-specific retraining. Our study suggests that diffusion models, beyond their generative capabilities, realize implicit inference dynamics whose geometry offers new perspectives on statistical learning and the analysis of high-dimensional distributions.

Data assimilation aims to sequentially integrate observational data into computational models to improve state estimation in complex, high-dimensional systems. A widely used approach is the ensemble Kalman filter (EnKF), a sequential Monte Carlo method applied in fields such as geosciences and hydrology. However, EnKF performance is often limited by linearization assumptions, difficulties in handling nonlinear observation operators, and reduced accuracy with small ensemble sizes. To address these challenges, we propose a Neural Filter, a data-driven assimilation framework that learns the mapping from prior to posterior distributions directly within the observation space. This approach eliminates the need for linearization, naturally accommodates nonlinear observation models, and enhances assimilation accuracy. Moreover, by formulating the update in the observation space, the Neural Filter scales effectively to large systems, offering a flexible and powerful alternative to traditional ensemble-based methods.

Particle-based stochastic approximations of the Boltzmann equation are popular tools for simulations of non-equilibrium gas flows, for which the Navier-Stokes-Fourier equations fail to provide accurate description. However, these numerical methods are computationally demanding, especially in the near-continuum regime, where the collisions become overwhelming. On the other hand, the Fokker-Planck kinetic models offer an efficient alternative, as the binary collisions are described by a diffusive process. Despite the intuitive advantage, rigorous and efficient Fokker-Planck approximations of the Boltzmann equation remain an open problem. On one hand, the moment projection of the Fokker–Planck operator should be consistent with that of the Boltzmann operator. On the other hand, the Fokker–Planck model should be constructed in such a way that the H-theorem is satisfied. The central aim of this study is fulfilling these two categorically different constraints, i.e. moment matching and entropy dissipation, within a flexible and tractable Fokker–Planck framework. To this end, we introduce a Fisher information-based entropic constraint and demonstrate that, with a suitable polynomial expansion of the drift term, it is possible to simultaneously achieve weak moment matching while honouring the H-theorem. We support our theoretical result by numerical experiments on the shock problem, validating our Fisher Entropic Fokker–Planck framework.

We introduce a multifidelity variant of Stein Variational Gradient Descent (SVGD), a widely used particle-based variational inference method. SVGD iteratively transports particles to approximate a target distribution by following the steepest descent of the Kullback-Leibler (KL) divergence. Although highly effective, its structure often makes it computationally expensive. As the complexity and size of the studied systems grow, it becomes increasingly important to use computational resources as efficiently as possible. To enhance efficiency, we propose a multifidelity approach that leverages varying levels of fidelity characterized by different number of particles, step sizes, and likelihood approximations. Numerical experiments, including distribution approximations and Bayesian inverse problems, demonstrate the method’s effectiveness and computational advantages

When gases are far from thermal equilibrium, conventional continuum models fail to accurately describe their behavior. Instead, a kinetic model that describes gases statistically is necessary so that molecular interactions can be taken into account. The Boltzmann equation is the most widely used kinetic model and is typically solved by particle Monte-Carlo methods. These methods offer high physical accuracy but are computationally expensive, especially for near-equilibrium flows, as collisions must be calculated explicitly.
An alternative kinetic model is given by the Fokker-Planck equation, which approximates the Boltzmann equation by modeling the effect of binary collisions as a drift-diffusion process. This approach allows for a more efficient particle method via the underlying Langevin equation, eliminating the need for explicit collision calculations.
To further improve efficiency, we replace the random numbers used in the stochastic diffusion term of the velocity update with quasi-random numbers. Unlike standard pseudo-random numbers, which are scattered irregularly, quasi random number are constructed to fill the space more evenly. This reduces the statistical noise inherent in particle simulations and allows accurate results to be obtained with fewer particles. 

We propose a Boltzmann-type collisional dynamics equipped with an efficient Monte Carlo algorithm to approximate solutions of the optimal transport problem through randomized pairwise swapping of sample indices. The linear computational complexity and memory requirements of the method with respect to number of samples makes it particularly suitable for high-dimensional settings. Through several experiments, we demonstrate the effectiveness of our approach in training the Denoising Diffusion Probabilistic Model (DDPM) as generative model to sample high-dimensional datasets such as Food101, CelebA, and MNIST.

One of the main challenges in Bayesian inference on the parameters of a diffusion model from discrete observations of the process is the unavailability of an analytical expression for the transition density function between consecutive observation times. Extending previous studies that solve Fokker-Planck (FP) type PDEs with Normalizing flows, we propose an operator learning framework to learn the transition density function between two observation times (solution of the Fokker-Planck equation associated to the SDE) for any values of the initial datum and the parameters of the SDE describing the drift and the diffusion terms. By implementing Neural Galerkin schemes, such an advection dominated PDE is solved with high accuracy, allowing the use of the surrogate transition density function in Bayesian inference problems. MCMC routines are finally used to sample from the target posterior. After the offline training phase, online evaluation is much cheaper than an approach that solves the FP equation for each parameter proposed by the MCMC sampler.

Model reduction techniques have emerged as a powerful paradigm across different fronts of scientific computing. Despite their success, the provided tools and methodologies remain limited if high-dimensional dynamical systems subject to initial uncertainty and/or stochastic noise are encountered; in particular if rare events are of interest. We address this
open challenge by borrowing ideas from Mori-Zwanzig formalism and Chorin’s optimal prediction method. The novelty of our work lies on employing time-dependent optimal projection of the dynamic on a desired set of resolved variables. We show several theoretical and numerical properties of our model reduction approach. In particular, we show that the devised surrogate trajectories are consistent with the probability flow of the full-order system. Furthermore, we identify the measure underlying the projection through polynomial chaos expansion technique. This allows us to efficiently compute the projection even for trajectories that are initiated on low probability events. Moreover, we investigate the introduced model-reduction error of the surrogate trajectories on a standard setup, characterizing the convergence behaviour of the scheme. Several numerical results highlight the computational advantages of the proposed scheme in comparison to Monte-Carlo and optimal prediction method. Through this framework, we demonstrate that by tracking the measure along with the consistent projection of the dynamic we are able to access accurate estimates of different statistics including observables conditional on a given initial configuration.

The Fokker-Planck equation plays a pivotal role in describing the time evolution of probability density functions for systems subject to stochastic dynamics. In this work, we propose a conditional normalizing flow-based method for efficiently approximating the solution operator of the Fokker-Planck equation across a range of initial conditions. Unlike traditional numerical solvers, our method eliminates the need to repeatedly resolve the equation when a new initial condition is specified. Numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method.