An overview on deep learning-based approximation methods for partial differential equations
C Beck, M Hutzenthaler, A Jentzen… - arXiv preprint arXiv …, 2020 - arxiv.org
It is one of the most challenging problems in applied mathematics to approximatively solve
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
Crowd dynamics: Modeling and control of multiagent systems
This review aims to present recent developments in modeling and control of multiagent
systems. A particular focus is set on crowd dynamics characterized by complex interactions …
systems. A particular focus is set on crowd dynamics characterized by complex interactions …
Overcoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architectures
We propose new and original mathematical connections between Hamilton–Jacobi (HJ)
partial differential equations (PDEs) with initial data and neural network architectures …
partial differential equations (PDEs) with initial data and neural network architectures …
[HTML][HTML] Cross interpolation for solving high-dimensional dynamical systems on low-rank Tucker and tensor train manifolds
B Ghahremani, H Babaee - Computer Methods in Applied Mechanics and …, 2024 - Elsevier
We present a novel tensor interpolation algorithm for the time integration of nonlinear tensor
differential equations (TDEs) on the tensor train and Tucker tensor low-rank manifolds …
differential equations (TDEs) on the tensor train and Tucker tensor low-rank manifolds …
Solving high-dimensional parabolic PDEs using the tensor train format
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science
and engineering. However, their numerical treatment poses formidable challenges since …
and engineering. However, their numerical treatment poses formidable challenges since …
Actor-critic method for high dimensional static Hamilton--Jacobi--Bellman partial differential equations based on neural networks
We propose a novel numerical method for high dimensional Hamilton--Jacobi--Bellman
(HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as …
(HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as …
On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton–Jacobi partial differential equations
We propose novel connections between several neural network architectures and viscosity
solutions of some Hamilton–Jacobi (HJ) partial differential equations (PDEs) whose …
solutions of some Hamilton–Jacobi (HJ) partial differential equations (PDEs) whose …
Learning optimal feedback operators and their sparse polynomial approximations
K Kunisch, D Vásquez-Varas, D Walter - Journal of Machine Learning …, 2023 - jmlr.org
A learning based method for obtaining feedback laws for nonlinear optimal control problems
is proposed. The learning problem is posed such that the open loop value function is its …
is proposed. The learning problem is posed such that the open loop value function is its …
QRnet: Optimal regulator design with LQR-augmented neural networks
T Nakamura-Zimmerer, Q Gong… - IEEE Control Systems …, 2020 - ieeexplore.ieee.org
In this letter we propose a new computational method for designing optimal regulators for
high-dimensional nonlinear systems. The proposed approach leverages physics-informed …
high-dimensional nonlinear systems. The proposed approach leverages physics-informed …
Generalized eigenvalue for even order tensors via Einstein product and its applications in multilinear control systems
Y Wang, Y Wei - Computational and Applied Mathematics, 2022 - Springer
This paper devotes to the generalized eigenvalues for even order tensors. We extend
classical spectral theory for matrix pairs to the multilinear case, including the generalized …
classical spectral theory for matrix pairs to the multilinear case, including the generalized …