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Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semilinear partial differential equations (PDEs) with general…

In this work we propose a deep adaptive sampling (DAS) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to…

In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential…

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…

We derive quantitative error bounds for deep neural networks (DNNs) approximating option prices on a $d$-dimensional risky asset as functions of the underlying model parameters, payoff parameters and initial conditions. We cover a general…

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental…

The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of high-dimensional nonlinear PDEs. In particular, there are…

This work proposes a sampling-based (non-intrusive) approach within the context of low-rank separated representations to tackle the issue of curse-of-dimensionality associated with the solution of models, e.g., PDEs/ODEs, with…

In this paper, we study the error in first order Sobolev norm in the approximation of solutions to linear parabolic PDEs. We use a Monte Carlo Euler scheme obtained from combining the Feynman--Kac representation with a Euler discretization…

We propose the first learning scheme for functional differential equations (FDEs). FDEs play a fundamental role in physics, mathematics, and optimal control. However, the numerical analysis of FDEs has faced challenges due to its…

We present an algorithm (CoDeFi) which overcomes the curse of dimensionality (CoD) in scientific computations and, especially, in mathematical finance (Fi). Our method applies a broad class of partial differential equations such as…

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose…

We show that deep neural networks (DNNs) can efficiently learn any composition of functions with bounded $F_{1}$-norm, which allows DNNs to break the curse of dimensionality in ways that shallow networks cannot. More specifically, we derive…

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…

In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of…

Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be…

In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied…

Policy gradient methods equipped with deep neural networks have achieved great success in solving high-dimensional reinforcement learning (RL) problems. However, current analyses cannot explain why they are resistant to the curse of…

Implementing deep neural networks for learning the solution maps of parametric partial differential equations (PDEs) turns out to be more efficient than using many conventional numerical methods. However, limited theoretical analyses have…

Neural network based methods have emerged as a promising paradigm for scientific computing, yet they face critical bottlenecks in high frequency function approximation and partial differential equation (PDE) solving.