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This work is motivated by the need to study the impact of data uncertainties and material imperfections on the solution to optimal control problems constrained by partial differential equations. We consider a pathwise optimal control…
There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include…
In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. A dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional…
In this paper, we study the optimal control of a discrete-time stochastic differential equation (SDE) of mean-field type, where the coefficients can depend on both a function of the law and the state of the process. We establish a new…
The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic…
Generative diffusion models have achieved remarkable success in producing high-quality images. However, these models typically operate in continuous intensity spaces, diffusing independently across pixels and color channels. As a result,…
This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These…
We investigate the extension of the multilevel Monte Carlo path simulation method to jump-diffusion SDEs. We consider models with finite rate activity, using a jump-adapted discretisation in which the jump times are computed and added to…
We introduce an inferential framework for a wide class of semi-linear stochastic differential equations (SDEs). Recent work has shown that numerical splitting schemes can preserve critical properties of such types of SDEs, give rise to…
We present a novel generative modeling method called diffusion normalizing flow based on stochastic differential equations (SDEs). The algorithm consists of two neural SDEs: a forward SDE that gradually adds noise to the data to transform…
Stochastic gradient descent (SGD) is a widely adopted iterative method for optimizing differentiable objective functions. In this paper, we propose and discuss a novel approach to scale up SGD in applications involving non-convex functions…
Differential equations (DE) constrained optimization plays a critical role in numerous scientific and engineering fields, including energy systems, aerospace engineering, ecology, and finance, where optimal configurations or control…
We propose a novel problem formulation of continuous-time information propagation on heterogenous networks based on jump stochastic differential equations (SDE). The structure of the network and activation rates between nodes are naturally…
We introduce a novel paradigm for learning non-parametric drift and diffusion functions for stochastic differential equation (SDE). The proposed model learns to simulate path distributions that match observations with non-uniform time…
Diffusion-based generative processes, formulated as differential equation solving, frequently balance computational speed with sample quality. Our theoretical investigation of ODE- and SDE-based solvers reveals complementary weaknesses: ODE…
In this paper we analyze the behaviour of the stochastic gradient descent (SGD), a widely used method in supervised learning for optimizing neural network weights via a minimization of non-convex loss functions. Since the pioneering work of…
Reward fine-tuning of diffusion and flow models and sampling from tilted or Boltzmann distributions can both be formulated as stochastic optimal control (SOC) problems, where learning an optimal generative dynamics corresponds to optimizing…
Stochastic mathematical models are essential tools for understanding and predicting complex phenomena. The purpose of this work is to study the exit times of a stochastic dynamical system-specifically, the mean exit time and the…
Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured…
Solving statistical learning problems often involves nonconvex optimization. Despite the empirical success of nonconvex statistical optimization methods, their global dynamics, especially convergence to the desirable local minima, remain…