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 …
Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning
In recent years, tremendous progress has been made on numerical algorithms for solving
partial differential equations (PDEs) in a very high dimension, using ideas from either …
partial differential equations (PDEs) in a very high dimension, using ideas from either …
Policy gradient and actor-critic learning in continuous time and space: Theory and algorithms
We study policy gradient (PG) for reinforcement learning in continuous time and space
under the regularized exploratory formulation developed by Wang et al.(2020). We …
under the regularized exploratory formulation developed by Wang et al.(2020). We …
Policy evaluation and temporal-difference learning in continuous time and space: A martingale approach
We propose a unified framework to study policy evaluation (PE) and the associated temporal
difference (TD) methods for reinforcement learning in continuous time and space. We show …
difference (TD) methods for reinforcement learning in continuous time and space. We show …
Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
For a long time it has been well-known that high-dimensional linear parabolic partial
differential equations (PDEs) can be approximated by Monte Carlo methods with a …
differential equations (PDEs) can be approximated by Monte Carlo methods with a …
q-Learning in continuous time
We study the continuous-time counterpart of Q-learning for reinforcement learning (RL)
under the entropy-regularized, exploratory diffusion process formulation introduced by Wang …
under the entropy-regularized, exploratory diffusion process formulation introduced by Wang …
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with …
Recently, several deep learning (DL) methods for approximating high-dimensional partial
differential equations (PDEs) have been proposed. The interest that these methods have …
differential equations (PDEs) have been proposed. The interest that these methods have …
Mathematical introduction to deep learning: methods, implementations, and theory
This book aims to provide an introduction to the topic of deep learning algorithms. We review
essential components of deep learning algorithms in full mathematical detail including …
essential components of deep learning algorithms in full mathematical detail including …
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling
of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a …
of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a …
Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations
We prove that deep neural networks are capable of approximating solutions of semilinear
Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities …
Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities …