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1. Tutorial: Langevin Dynamics methods for aerosol particle trajectory simulations and collision rate constant modeling

   https://doi.org/https://doi.org/10.1016/j.jaerosci.2021.105746  

   Suresh, Vikram; Gopalakrishnan, Ranganathan (February 2021, Journal of aerosol science)

   null (Ed.)

   The Langevin Dynamics (LD) method (also known in the literature as Brownian Dynamics) is routinely used to simulate aerosol particle trajectories for transport rate constant calculations as well as to understand aerosol particle transport in internal and external fluid flows. This tutorial intends to explain the methodological details of setting up a LD simulation of a population of aerosol particles and to deduce rate constants from an ensemble of classical trajectories. We discuss the applicability and limitations of the translational Langevin equation to model the combined stochastic and deterministic motion of particles in fields of force or fluid flow. The drag force and stochastic “diffusion” force terms that appear in the Langevin equation are discussed elaborately, along with a summary of common forces relevant to aerosol systems (electrostatic, gravity, van der Waals, …); a commonly used first order and a fourth order Runge-Kutta time stepping schemes for linear stochastic ordinary differential equations are presented. A MATLAB® implementation of a LD code for simulating particle settling under gravity using the first order scheme is included for illustration. Scaling analysis of aerosol transport processes and the selection of timestep and domain size for trajectory simulations are demonstrated through two specific aerosol processes: particle diffusion charging and coagulation. Fortran® implementations of the first order and fourth order time-stepping schemes are included for simulating the 3D motion of a particle in a periodic domain. Potential applications and caveats to the usage of LD are included as a summary. 

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2. Stochastic Bridges of Linear Systems

   https://doi.org/10.1109/TAC.2015.2440567  

   Chen, Yongxin; Georgiou, Tryphon (September 2019, IEEE Transactions on Automatic Control)

   We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein–Uhlenbeck process conditioned at the two end-points—a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions. 

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3. DIFFUSION-MODEL-ASSISTED SUPERVISED LEARNING OF GENERATIVE MODELS FOR DENSITY ESTIMATION

   https://doi.org/10.1615/JMachLearnModelComput.2024051346  

   Liu, Yanfang; Yang, Minglei; Zhang, Zezhong; Bao, Feng; Cao, Yanzhao; Zhang, Guannan (January 2024, Journal of Machine Learning for Modeling and Computing)

   We present a supervised learning framework of training generative models for density estimation.Generative models, including generative adversarial networks (GANs), normalizing flows, and variational auto-encoders (VAEs), are usually considered as unsupervised learning models, because labeled data are usually unavailable for training. Despite the success of the generative models, there are several issues with the unsupervised training, e.g., requirement of reversible architectures, vanishing gradients, and training instability. To enable supervised learning in generative models, we utilize the score-based diffusion model to generate labeled data. Unlike existing diffusion models that train neural networks to learn the score function, we develop a training-free score estimation method. This approach uses mini-batch-based Monte Carlo estimators to directly approximate the score function at any spatial-temporal location in solving an ordinary differential equation (ODE), corresponding to the reverse-time stochastic differential equation (SDE). This approach can offer both high accuracy and substantial time savings in neural network training. Once the labeled data are generated, we can train a simple, fully connected neural network to learn the generative model in the supervised manner. Compared with existing normalizing flow models, our method does not require the use of reversible neural networks and avoids the computation of the Jacobian matrix. Compared with existing diffusion models, our method does not need to solve the reverse-time SDE to generate new samples. As a result, the sampling efficiency is significantly improved. We demonstrate the performance of our method by applying it to a set of 2D datasets as well as real data from the University of California Irvine (UCI) repository. 

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4. Dynamics of Inertial Particles in Flows with Stochasticity

   https://doi.org/10.1175/JPO-D-24-0117.1  

   Rogers, Mason; Rypina, Irina I (July 2025, Journal of Physical Oceanography)

   Abstract Because of their buoyancy, rigidity, and finite size, inertial particles do not obey the same dynamics as fluid parcels. The motion of small spherical particles in a fluid flow is described by the Maxey–Riley equations and depends nonlinearly on the velocity of the fluid in which the particles are immersed. Fluid velocities in the ocean often have a strong small-scale turbulent component which is difficult to observe or model, presenting a challenge to predicting the evolution of distributions of inertial particles in the ocean. To overcome this challenge, we assume that the turbulent velocity imposes a random force on particles and consider a stochastic analog of the Maxey–Riley equations. By performing a perturbation analysis of the stochastic Maxey–Riley equations, we obtain a simple and accurate partial differential equation for the spatial distribution of particles. The equation is of the advection–diffusion type and handles the uncertainty introduced by unresolved turbulent flow features. In several numerical test cases, distributions of particles obtained by solving the newly derived equation compare favorably with distributions obtained from Monte Carlo simulations of individual particle trajectories and with theoretical predictions. The advection–diffusion form of our newly derived equation is amenable to inclusion within many existing ocean circulation models. Significance StatementWe introduce a new model for describing spatial distributions of small rigid objects, such as plastic debris, in the ocean. The model takes into account the effects of finite particle size and particle buoyancy, which cause particle trajectories to differ from fluid parcel trajectories. Our model also represents small-scale turbulence stochastically. 

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5. Probability flow solution of the Fokker–Planck equation

   https://doi.org/10.1088/2632-2153/ace2aa  

   Boffi, Nicholas M.; Vanden-Eijnden, Eric (July 2023, Machine Learning: Science and Technology)

   Abstract The method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). Here, we study an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, 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 isa-prioriunknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We show theoretically that the proposed approach controls the Kullback–Leibler (KL) divergence from the learned solution to the target, while learning on external samples from the SDE does not control either direction of the KL divergence. Empirically, we consider several high-dimensional FPEs from the physics of interacting particle systems. We find that the method accurately matches analytical solutions when they are available as well as moments computed via Monte-Carlo when they are not. Moreover, the method offers compelling predictions for the global entropy production rate that out-perform those obtained from learning on stochastic trajectories, and can effectively capture non-equilibrium steady-state probability currents over long time intervals. 

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