Related papers: Taming the diffusion approximation through a contr…

Populations are made up of an integer number of individuals and are subject to stochastic birth-death processes whose rates may vary in time. Useful quantities, like the chance of ultimate fixation, satisfy an appropriate difference…

Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and…

Stochastic fluctuations are central to the understanding of extinction dynamics. In the context of population models they allow for the description of the transition from the vicinity of a non-trivial fixed point of the deterministic…

We study large deviations of the time-averaged size of stochastic populations described by a continuous-time Markov jump process. When the expected population size $N$ in the steady state is large, the large deviation function (LDF) of the…

Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential…

Data assimilation (DA) addresses the problem of sequentially estimating the state of a dynamical system from noisy and incomplete observations. In this work, we employ a diffusion model as a world model to simulate and predict the system's…

We consider a stochastic population model where the intrinsic or demographic noise causes cycling between states before the population eventually goes extinct. A master equation approach coupled with a WKB (Wentzel-Kramers-Brillouin)…

Diffusion theory is a central tool of modern population genetics, yielding simple expressions for fixation probabilities and other quantities that are not easily derived from the underlying Wright-Fisher model. Unfortunately, the textbook…

Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear…

Stochastic diffusion is the noisy and uncertain process through which dynamics like epidemics, or agents like animal species, disperse over a larger area. Understanding these processes is becoming increasingly important as we attempt to…

We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary differential equations…

This paper introduces a novel deep learning method, called DeepWKB, for estimating the invariant distribution of randomly perturbed systems via its Wentzel-Kramers-Brillouin (WKB) approximation $u_\epsilon(x) = Q(\epsilon)^{-1}…

Generative models have had a profound impact on vision and language, paving the way for a new era of multimodal generative applications. While these successes have inspired researchers to explore using generative models in science and…

We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly…

Machine learning models struggle with generalization when encountering out-of-distribution (OOD) samples with unexpected distribution shifts. For vision tasks, recent studies have shown that test-time adaptation employing diffusion models…

Domain adaptation (DA) is a technique that transfers predictive models trained on a labeled source domain to an unlabeled target domain, with the core difficulty of resolving distributional shift between domains. Currently, most popular DA…

We describe a continuous-time modelling framework for biological population dynamics that accounts for demographic noise. In the spirit of the methodology used by statistical physicists, transitions between the states of the system are…

We study a family of numerical schemes applied to a class of multiscale systems of stochastic differential equations. When the time scale separation parameter vanishes, a well-known homogenization or Wong--Zakai diffusion approximation…

Transfer learning has achieved promising results by leveraging knowledge from the source domain to annotate the target domain which has few or none labels. Existing methods often seek to minimize the distribution divergence between domains,…

We study the stability of non-conservative deterministic cross diffusion models and prove that they are approximated by stochastic population models when the populations become locally large. In this model, the individuals of two species…