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Stochastic optimization (SO) considers the problem of optimizing an objective function in the presence of noise. Most of the solution techniques in SO estimate gradients from the noise corrupted observations of the objective and adjust…
An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener Chaos decomposition. The result is applied to the nonlinear filtering problem for the time…
Although generative diffusion models (GDMs) are widely used in practice, their theoretical foundations remain limited, especially concerning the impact of different discretization schemes applied to the underlying stochastic differential…
We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method…
Accurate state estimation requires careful consideration of uncertainty surrounding the process and measurement models; these characteristics are usually not well-known and need an experienced designer to select the covariance matrices. An…
A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of…
We address the problem of Bayesian inference for parameters in ordinary differential equation (ODE) models based on observational data. Conventional approaches in this setting typically rely on numerical solvers such as the Euler or…
In this paper, we study an online sampling problem of the Wiener process. The goal is to minimize the mean squared error (MSE) of the remote estimator under a sampling frequency constraint when the transmission delay distribution is…
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…
This paper investigates an issue of distributed fusion estimation under network-induced complexity and stochastic parameter uncertainties. First, a novel signal selection method based on event-trigger is developed to handle network-induced…
In this paper, we study a class of slow-fast stochastic partial differential equations with multiplicative Wiener noise. Under some appropriate conditions, we prove the slow component converges to the solution of the corresponding averaged…
Stochastic regularization of neural networks (e.g. dropout) is a wide-spread technique in deep learning that allows for better generalization. Despite its success, continuous-time models, such as neural ordinary differential equation (ODE),…
We consider the problem of distributed estimation of an unknown deterministic scalar parameter (the target signal) in a wireless sensor network (WSN), where each sensor receives a single snapshot of the field. We assume that the observation…
The wave speed of a stochastic wave equation driven by Riesz noise on the unbounded multidimensional spatial domain is estimated based on discrete measurements. Central limit theorems for second-order variations of the observations in…
Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) are fundamental for modeling stochastic dynamics across the natural sciences and modern machine learning. Learning their solution operators with…
This paper considers the implicit Euler discretization of Levant's arbitrary order robust exact differentiator in presence of sampled measurements. Existing implicit discretizations of that differentiator are shown to exhibit either…
This paper presents a general approach to linear stochastic processes driven by various random noises. Mathematically, such processes are described by linear stochastic differential equations of arbitrary order (the simplest non-trivial…
Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations.…
The understanding of adaptive algorithms for SDEs is an open area where many issues related to both convergence and stability (long time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based…
Ordinary and stochastic differential equations (ODEs and SDEs) are widely used to model continuous-time processes across various scientific fields. While ODEs offer interpretability and simplicity, SDEs incorporate randomness, providing…