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We propose a novel numerical method for high dimensional Hamilton--Jacobi--Bellman (HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as optimal control problems, are tackled by the actor-critic framework…
Realistic physical phenomena exhibit random fluctuations across many scales in the input and output processes. Models of these phenomena require stochastic PDEs. For three-dimensional coupled (vector-valued) stochastic PDEs (SPDEs), for…
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).…
Traditional approaches to stabilizing hyperbolic PDEs, such as PDE backstepping, often encounter challenges when dealing with high-dimensional or complex nonlinear problems. Their solutions require high computational and analytical costs.…
This paper considers a non-Markov control problem arising in a financial market where asset returns depend on hidden factors. The problem is non-Markov because nonlinear filtering is required to make inference on these factors, and hence…
In this paper we make a survey on the so called randomization method, a recent methodology to study stochastic optimization problems. It allows to represent the value function of an optimal control problem by a suitable backward stochastic…
The paper is devoted to a stochastic optimal control problem for a two scale, infinite dimensional, stochastic system. The state of the system consists of slow and fast component and its evolution is driven by both continuous Wiener noises…
This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…
Deep learning needs high-precision handling of forwarding signals, backpropagating errors, and updating weights. This is inherently required by the learning algorithm since the gradient descent learning rule relies on the chain product of…
Neural operators have demonstrated promise in modeling and controlling systems governed by Partial Differential Equations (PDEs). Beyond PDEs, Stochastic Partial Differential Equations (SPDEs) play a critical role in modeling systems…
Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest…
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…
This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T.…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
This paper addresses the problem of robust stabilization for linear hyperbolic Partial Differential Equations (PDEs) with Markov-jumping parameter uncertainty. We consider a 2 x 2 heterogeneous hyperbolic PDE and propose a control law using…
Solving high dimensional optimal control problems and corresponding Hamilton-Jacobi PDEs are important but challenging problems in control engineering. In this paper, we propose two abstract neural network architectures which are…
Stochastic control with both inherent random system noise and lack of knowledge on system parameters constitutes the core and fundamental topic in reinforcement learning (RL), especially under non-episodic situations where online learning…
Modeling and controlling complex spatiotemporal dynamical systems driven by partial differential equations (PDEs) often necessitate dimensionality reduction techniques to construct lower-order models for computational efficiency. This paper…
The approximation of solutions to second order Hamilton--Jacobi--Bellman (HJB) equations by deep neural networks is investigated. It is shown that for HJB equations that arise in the context of the optimal control of certain Markov…
The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the…