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Stochastic gradient descent with momentum (SGDM) methods have become fundamental optimization tools in machine learning, combining the computational efficiency of stochastic gradients with the acceleration benefits of momentum. Despite…
Ensuring safety through set invariance has proven to be a valuable method in various robotics and control applications. This paper introduces a comprehensive framework for the safe probabilistic invariance verification of both discrete- and…
Stochastic differential equations (SDEs) offer powerful and accessible mathematical models for capturing both deterministic and probabilistic aspects of dynamic behavior across a wide range of physical, financial, and social systems.…
This paper presents an approach to deal with safety of dynamical systems in presence of multiple non-convex unsafe sets. While optimal control and model predictive control strategies can be employed in these scenarios, they suffer from high…
Stochastic differential equations (SDE) often exhibit large random transitions. This property, which we denote as pathwise stiffness, causes transient bursts of stiffness which limit the allowed step size for common fixed time step explicit…
We address the problem of safely learning controlled stochastic dynamics from discrete-time trajectory observations, ensuring system trajectories remain within predefined safe regions during both training and deployment. Safety-critical…
Stochastic Differential Equations (SDEs) serve as a powerful modeling tool in various scientific domains, including systems science, engineering, and ecological science. While the specific form of SDEs is typically known for a given…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
This paper studies the problem of enforcing safety of a stochastic dynamical system over a finite-time horizon. We use stochastic control barrier functions as a means to quantify the probability that a system exits a given safe region of…
Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method…
In this paper, we present a novel data-driven approach to quantify safety for non-linear, discrete-time stochastic systems with unknown noise distribution. We define safety as the probability that the system remains in a given region of the…
We present a numerical method for computing optimal transition pathways and transition rates in systems of stochastic differential equations (SDEs). In particular, we compute the most probable transition path of stochastic equations by…
This paper studies the problem of enforcing safety of a stochastic dynamical system over a finite time horizon. We use stochastic barrier functions as a means to quantify the probability that a system exits a given safe region of the state…
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…
This paper addresses the quantitative verification of finite-time constrained occupation time for stochastic continuous-time systems governed by stochastic differential equations (SDEs). Unlike classical reachability analysis, which focuses…
Stochastic dynamical systems arise naturally across nearly all areas of science and engineering. Typically, a dynamical system model is based on some prior knowledge about the underlying dynamics of interest in which probabilistic features…
Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equations. Given a system of nonlinear…
Optimization problems involving sequential decisions in a stochastic environment were studied in Stochastic Programming (SP), Stochastic Optimal Control (SOC) and Markov Decision Processes (MDP). In this paper we mainly concentrate on SP…
Motivated by the lack of a suitable constructive framework for analyzing popular stochastic models of Systems Biology, we devise conditions for existence and uniqueness of solutions to certain jump stochastic differential equations (SDEs).…
In this paper, we study the entrance measures of time-inhomogeneous McKean-Vlasov SDEs. The existence is obtained in great generality, where the system can be expanding globally and/or degenerate for numerous number of time intervals. When…