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Tensor Decomposition Methods for High-dimensional Hamilton--Jacobi--Bellman Equations

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TLDR
A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented in this article.
Abstract
A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton--Jacobi--Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The...

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Actor-Critic Method for High Dimensional Static Hamilton-Jacobi-Bellman Partial Differential Equations based on Neural Networks.

TL;DR: In this paper, an actor-critic framework inspired by reinforcement learning is proposed for high-dimensional elliptic partial differential equations (PDEs) with high dimensional value functions, where the authors employ a policy gradient approach to improve the control and derive a variance reduced least square temporal difference method (VR-LSTD) using stochastic calculus.
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Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-Dependent Riccati Equations

TL;DR: In this article, a stabilizing feedback law is trained from a dataset generated from State-dependent Riccati Equation solvers, enriched by the use of gradient information in the loss function.
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Error estimates for a tree structure algorithm solving finite horizon control problems.

TL;DR: A dynamic programming algorithm based on a tree structure built by the time discrete dynamics avoiding in this way the use of a fixed space grid which is the bottleneck for high-dimensional problems, this also drops the projection on the grid in the approximation of the value function.
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Data-Driven Recovery of Optimal Feedback Laws through Optimality Conditions and Sparse Polynomial Regression

TL;DR: An extended set of low and high-dimensional numerical tests in nonlinear optimal control reveal that enriching the dataset with gradient information reduces the number of training samples, and that the sparse polynomial regression consistently yields a feedback law of lower complexity.
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QRnet: Optimal Regulator Design With LQR-Augmented Neural Networks

TL;DR: In this paper, the authors proposed a new computational method for designing optimal regulators for high-dimensional nonlinear systems, which leverages physics-informed machine learning to solve highdimensional Hamilton-Jacobi-Bellman equations arising in optimal feedback control.
References
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Journal ArticleDOI

Tensor Decompositions and Applications

TL;DR: This survey provides an overview of higher-order tensor decompositions, their applications, and available software.
Journal ArticleDOI

Tensor-Train Decomposition

TL;DR: The new form gives a clear and convenient way to implement all basic operations efficiently, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.
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Solving high-dimensional partial differential equations using deep learning

TL;DR: A deep learning-based approach that can handle general high-dimensional parabolic PDEs using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function.
Journal ArticleDOI

DGM: A deep learning algorithm for solving partial differential equations

TL;DR: A deep learning algorithm similar in spirit to Galerkin methods, using a deep neural network instead of linear combinations of basis functions is proposed, and is implemented for American options in up to 100 dimensions.
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A sparse matrix arithmetic based on H -matrices. Part I: introduction to H -matrices

TL;DR: This paper is the first of a series and is devoted to the first introduction of the $\Cal H$-matrix concept, which allows the exact inversion of tridiagonal matrices.
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