Higher-Order Quasi-Monte Carlo Training of Deep Neural Networks

DOI10.1137/20M1369373zbMath1492.65043arXiv2009.02713OpenAlexW3216945074MaRDI QIDQ5015302

Marcello Longo, Christoph Schwab, T. Konstantin Rusch, Siddhartha Mishra

Publication date: 7 December 2021

Published in: SIAM Journal on Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2009.02713

zbMATH Keywords

machine learninghigh-dimensional approximationparametric PDEsdeep neural networkshigher-order quasi-Monte Carlo

Mathematics Subject Classification ID

Artificial neural networks and deep learning (68T07) Monte Carlo methods (65C05) PDEs with randomness, stochastic partial differential equations (35R60) Algorithms for approximation of functions (65D15) Numerical integration (65D30)

Related Items (4)

Convergence Analysis of a Quasi-Monte CarloBased Deep Learning Algorithm for Solving Partial Differential Equations ⋮ Wavenumber-Explicit Parametric Holomorphy of Helmholtz Solutions in the Context of Uncertainty Quantification ⋮ An overview on deep learning-based approximation methods for partial differential equations ⋮ Greedy training algorithms for neural networks and applications to PDEs

Uses Software

• redbKIT
• PyTorch
• Adam

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Sparse adaptive approximation of high dimensional parametric initial value problems
• Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs
• Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic SPDEs
• Deep learning observables in computational fluid dynamics
• Reconstructing a neural net from its output
• Iterative surrogate model optimization (ISMO): an active learning algorithm for PDE constrained optimization with deep neural networks
• Shape holomorphy of the Calderón projector for the Laplacian in \(\mathbb{R}^2\)
• Solving the Kolmogorov PDE by means of deep learning
• Exponential ReLU DNN expression of holomorphic maps in high dimension
• Higher order quasi-Monte Carlo integration for Bayesian PDE inversion
• Machine learning for fast and reliable solution of time-dependent differential equations
• A quasi-Monte Carlo data compression algorithm for machine learning
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Certified Reduced Basis Methods for Parametrized Partial Differential Equations
• Computational Higher Order Quasi-Monte Carlo Integration
• Multilevel higher-order quasi-Monte Carlo Bayesian estimation
• Low-discrepancy point sets obtained by digital constructions over finite fields
• Shape Holomorphy of the Stationary Navier--Stokes Equations
• Higher Order Quasi--Monte Carlo Integration for Holomorphic, Parametric Operator Equations
• Richardson Extrapolation of Polynomial Lattice Rules
• Domain Uncertainty Quantification in Computational Electromagnetics
• Solving high-dimensional partial differential equations using deep learning
• Enhancing Accuracy of Deep Learning Algorithms by Training with Low-Discrepancy Sequences
• A multi-level procedure for enhancing accuracy of machine learning algorithms
• Optimal Approximation with Sparsely Connected Deep Neural Networks
• Convergence rates of high dimensional Smolyak quadrature
• Deep ReLU networks and high-order finite element methods
• Deep neural network expression of posterior expectations in Bayesian PDE inversion
• Higher Order QMC Petrov--Galerkin Discretization for Affine Parametric Operator Equations with Random Field Inputs
• High-dimensional integration: The quasi-Monte Carlo way
• Advanced Lectures on Machine Learning
• Reduced Basis Methods for Partial Differential Equations
• On the distribution of points in a cube and the approximate evaluation of integrals

This page was built for publication: Higher-Order Quasi-Monte Carlo Training of Deep Neural Networks