Related papers: Nonlinear controllability and function representat…
Discontinuities and delayed terms are encountered in the governing equations of a large class of problems ranging from physics and engineering to medicine and economics. These systems cannot be properly modelled and simulated with standard…
We propose new limiting dynamics for stochastic gradient descent in the small learning rate regime called stochastic modified flows. These SDEs are driven by a cylindrical Brownian motion and improve the so-called stochastic modified…
The paper proposes a systematic framework for building data-driven stochastic differential equation (SDE) models from sparse, noisy observations. Unlike traditional parametric approaches, which assume a known functional form for the drift,…
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…
We consider some certain nonlinear perturbations of the stochastic linear-quadratic optimization problems and study the connections between their solutions and the corresponding Markovian backward stochastic diferential equations (BSDEs).…
This paper studies a class of non$-$Markovian singular stochastic control problems, for which we provide a novel probabilistic representation. The solution of such control problem is proved to identify with the solution of a $Z-$constrained…
Optimal control problems naturally arise in many scientific applications where one wishes to steer a dynamical system from a certain initial state $\mathbf{x}_0$ to a desired target state $\mathbf{x}^*$ in finite time $T$. Recent advances…
Dynamical systems are essential to model various phenomena in physics, finance, economics, and are also of current interest in machine learning. A central modeling task is investigating parameter sensitivity, whether tuning atmospheric…
We discover restrained numerical instabilities in current training practices of deep networks with stochastic gradient descent (SGD), and its variants. We show numerical error (on the order of the smallest floating point bit and thus the…
We consider stochastic differential equations (SDEs) driven by Feller processes which are themselves solutions of multivariate Levy driven SDEs. The solutions of these 'iterated SDEs' are shown to be non-Markovian. However, the process…
In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problem with non standard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a…
Parameter estimation for non-stationary stochastic differential equations (SDE) with an arbitrary nonlinear drift, and nonlinear diffusion is accomplished in combination with a non-parametric clustering methodology. Such a model-based…
This paper is concerned with necessary and sufficient conditions for near-optimal singular stochastic controls for systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on…
Objective. Precise control of neural systems is essential to experimental investigations of how the brain controls behavior and holds the potential for therapeutic manipulations to correct aberrant network states. Model predictive control,…
Although there is a substantial body of literature on control and optimization problems for parabolic and hyperbolic systems, the specific problem of controlling and optimizing the coefficients of the associated operators within such…
We develop a novel approach towards causal inference. Rather than structural equations over a causal graph, we learn stochastic differential equations (SDEs) whose stationary densities model a system's behavior under interventions. These…
We deal with the problem of parameter estimation in stochastic differential equations (SDEs) in a partially observed framework. We aim to design a method working for both elliptic and hypoelliptic SDEs, the latters being characterized by…
We study an optimal control problem on infinite horizon for a controlled stochastic differential equation driven by Brownian motion, with a discounted reward functional. The equation may have memory or delay effects in the coefficients,…
Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with representative datasets. Recently, an augmented framework has been…
Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore researchers often seek simpler…