Uniform error estimates for artificial neural network approximations for heat equations

Abstract

Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results in the scientific literature which examine the approximation capabilities of such deep learning based approximation algorithms for PDEs. These mathematical results from the scientific literature prove in part that algorithms based on ANNs are capable of overcoming the curse of dimensionality in the numerical approximation of high-dimensional PDEs. In these mathematical results from the scientific literature usually the error between the solution of the PDE and the approximating ANN is measured in the $L^p$-sense with respect to some $p\in [1,\infty )$ and some probability measure. In many applications it is, however, also important to control the error in a uniform $L^{\infty }$-sense. The key contribution of the main result of this article is to develop the techniques to obtain error estimates between solutions of PDEs and approximating ANNs in the uniform $L^{\infty }$-sense. In particular, we prove that the number of parameters of an ANN to uniformly approximate the classical solution of the heat equation in a region $[a,b{]}^d$ for a fixed time point $T\in (0,\infty )$ grows at most polynomially in the dimension $d\in \mathbb{N}$ and the reciprocal of the approximation precision $\epsilon >0$. This shows that ANNs can overcome the curse of dimensionality in the numerical approximation of the heat equation when the error is measured in the uniform $L^{\infty }$-norm.

References in corpus (22)

• Very Deep Convolutional Networks for Large-Scale Image Recognition
• DGM: A deep learning algorithm for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• A unified deep artificial neural network approach to partial differential equations in complex geometries
• Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
• Deep learning observables in computational fluid dynamics
• A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
• Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
• Convergence of the Deep BSDE Method for Coupled FBSDEs
• Deep splitting method for parabolic PDEs
• DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing
• A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
• Loss of regularity for Kolmogorov equations
• Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations
• Neural Networks Trained to Solve Differential Equations Learn General Representations
• A Discussion on Solving Partial Differential Equations using Neural Networks
• Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems
• Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions
• Deep Neural Network Framework Based on Backward Stochastic Differential Equations for Pricing and Hedging American Options in High Dimensions
• Space-time error estimates for deep neural network approximations for differential equations
• Deep Curve-dependent PDEs for affine rough volatility

Cited by in corpus (12)

• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
• DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing
• An overview on deep learning-based approximation methods for partial differential equations
• Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems
• Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
• Numerical methods for backward stochastic differential equations: A survey
• Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
• Deep neural network approximations for Monte Carlo algorithms
• Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
• Unbiased deep solvers for linear parametric PDEs
• High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations
• Neural variance reduction for stochastic differential equations