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Nonlinear (systems of) ordinary differential equations (ODEs) are common tools in the analysis of complex one-dimensional dynamic systems. In this paper we propose a smoothing approach regularized by a quasilinearized ODE-based penalty in…
We consider strongly-convex-strongly-concave saddle-point problems with general non-bilinear objective and different condition numbers with respect to the primal and the dual variables. First, we consider such problems with smooth composite…
We consider saddle point problems which objective functions are the average of $n$ strongly convex-concave individual components. Recently, researchers exploit variance reduction methods to solve such problems and achieve linear-convergence…
This paper studies the semi-analytic solution (SAS) of a power system's differential-algebraic equation. A SAS is a closed-form function of symbolic variables including time, the initial state and the parameters on system operating…
The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs…
We characterize the solution to the entropically regularized optimal transport problem by a well-posed ordinary differential equation (ODE). Our approach works for discrete marginals and general cost functions, and in addition to two…
Stochastic approximation (SA) is a method for finding the root of an operator perturbed by noise. There is a rich literature establishing the asymptotic normality of rescaled SA iterates under fairly mild conditions. However, these…
We introduce a new second order stochastic algorithm to estimate the entropically regularized optimal transport cost between two probability measures. The source measure can be arbitrary chosen, either absolutely continuous or discrete,…
This work presents a numerical analysis of computing transition states of semilinear elliptic partial differential equations (PDEs) via the index-1 saddle dynamics, or equivalently, the gentlest ascent dynamics. To establish clear…
Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop…
We introduce OS-net (Orbitally Stable neural NETworks), a new family of neural network architectures specifically designed for periodic dynamical data. OS-net is a special case of Neural Ordinary Differential Equations (NODEs) and takes…
This work introduces and rigorously analyzes a novel operator-splitting finite element scheme for approximating viscosity solutions of a broad class of constrained second-order partial differential equations. By decoupling the primary PDE…
We adapt recent tools developed for the analysis of Stochastic Gradient Descent (SGD) in non-convex optimization to obtain convergence and sample complexity guarantees for the vanilla policy gradient (PG). Our only assumptions are that the…
We apply the sample average approximation (SAA) method to risk-neutral optimization problems governed by nonlinear partial differential equations (PDEs) with random inputs. We analyze the consistency of the SAA optimal values and SAA…
For the ordinary differential equation (ODE) $\dot{x}(t) = f(t,x)$, $x(0) = x_0$, $t\geq 0$, $x\in R^d$, assume $f$ to be at least continuous in $t$ and locally Lipshitz in $x$, and if necessary, several times continuously differentiable in…
This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our…
A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning…
Sample average approximation (SAA) is a widely popular approach to data-driven decision-making under uncertainty. Under mild assumptions, SAA is both tractable and enjoys strong asymptotic performance guarantees. Similar guarantees,…
The continuous dynamics of natural systems has been effectively modelled using Neural Ordinary Differential Equations (Neural ODEs). However, for accurate and meaningful predictions, it is crucial that the models follow the underlying rules…
Neural ordinary differential equations (Neural ODEs) define continuous time dynamical systems with neural networks. The interest in their application for modelling has sparked recently, spanning hybrid system identification problems and…