Related papers: Differential equation approximations of stochastic…
We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension…
Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular…
State estimation aims at approximately reconstructing the solution $u$ to a parametrized partial differential equation from $m$ linear measurements, when the parameter vector $y$ is unknown. Fast numerical recovery methods have been…
ODE solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity, and, implicitly, in stochastic optimisation. In this work, we propose and study the stochastic…
The paper studies an improved estimate for the rate of convergence for nonlinear homogeneous discrete-time Markov chains. These processes are nonlinear in terms of the distribution law. Hence, the transition kernels are dependent on the…
Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying…
We study the stochastic optimization problem from a continuous-time perspective, with a focus on the Stochastic Gradient Descent with Momentum (SGDM) method. We show that the trajectory of SGDM, despite its \emph{stochastic} nature,…
This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain H\"older-type classes in which a random field is…
We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $\alpha$-stable processes. Our approach utilizes compound Poisson particle…
In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent…
A numerical analysis for the fully discrete approximation of an operator Lyapunov equation related to linear SPDEs (stochastic partial differential equations) driven by multiplicative noise is considered. The discretization of the Lyapunov…
We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov--Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded…
Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…
We produce uniform and decaying bounds in time for derivatives of the solution to the backwards Kolmogorov equation associated to a stochastic processes governed by a time dependent dynamics. These hold under assumptions over the…
The stochastic interpolant framework offers a powerful approach for constructing generative models based on ordinary differential equations (ODEs) or stochastic differential equations (SDEs) to transform arbitrary data distributions.…
Stochastic chemical reaction networks (SRNs) in cellular systems are commonly modeled as continuous-time Markov chains (CTMCs) describing the dynamics of molecular copy numbers. The exact evaluation of transient copy number statistics is,…
Stochastic interpolants offer a robust framework for continuously transforming samples between arbitrary data distributions, holding significant promise for generative modeling. Despite their potential, rigorous finite-time convergence…
With the increasing ubiquity of safety-critical autonomous systems operating in uncertain environments, there is a need for mathematical methods for formal verification of stochastic models. Towards formally verifying properties of…
Hybrid systems whose mode dynamics are governed by non-linear ordinary differential equations (ODEs) are often a natural model for biological processes. However such models are difficult to analyze. To address this, we develop a…
We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated It\^{o}'s formula and Markov operator. As applications, we study the error bounds of the following…