Approximation Error Analysis of Some Deep Backward Schemes for Nonlinear PDEs

DOI10.1137/20M1355355zbMath1490.65231arXiv2006.01496WikidataQ114074176 ScholiaQ114074176MaRDI QIDQ5021399

Huyên Pham, Xavier Warin, Maximilien Germain

Publication date: 13 January 2022

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

Full work available at URL: https://arxiv.org/abs/2006.01496

zbMATH Keywords

error estimatesneural networksnumerical approximationnonlinear PDEsbackward SDEsmultistep schemesdeep splitting

Mathematics Subject Classification ID

Probabilistic models, generic numerical methods in probability and statistics (65C20) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Computational methods for stochastic equations (aspects of stochastic analysis) (60H35) Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems (65M99)

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Gradient boosting-based numerical methods for high-dimensional backward stochastic differential equations ⋮ DeepSets and their derivative networks for solving symmetric PDEs ⋮ Convergence of deep fictitious play for stochastic differential games ⋮ A Neural Network Approach to High-Dimensional Optimal Switching Problems with Jumps in Energy Markets ⋮ Neural networks for first order HJB equations and application to front propagation with obstacle terms ⋮ Numerical methods for backward stochastic differential equations: a survey ⋮ Solving Kolmogorov PDEs without the curse of dimensionality via deep learning and asymptotic expansion with Malliavin calculus ⋮ Deep learning algorithms for solving high-dimensional nonlinear backward stochastic differential equations ⋮ An overview on deep learning-based approximation methods for partial differential equations

Uses Software

• Adam
• DGM

Cites Work

• Unnamed Item
• Adapted solution of a backward stochastic differential equation
• A probabilistic numerical method for fully nonlinear parabolic PDEs
• Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
• A numerical scheme for BSDEs
• Multilayer feedforward networks are universal approximators
• Branching diffusion representation of semilinear PDEs and Monte Carlo approximation
• A distribution-free theory of nonparametric regression
• DGM: a deep learning algorithm for solving partial differential equations
• Convergence of the deep BSDE method for coupled FBSDEs
• Error bounds for approximations with deep ReLU networks
• On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
• Machine learning for semi linear PDEs
• A forward scheme for backward SDEs
• Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations
• A regression-based Monte Carlo method to solve backward stochastic differential equations
• Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions
• Deep Splitting Method for Parabolic PDEs
• Deep backward schemes for high-dimensional nonlinear PDEs
• Solving high-dimensional partial differential equations using deep learning
• Deep Neural Networks Algorithms for Stochastic Control Problems on Finite Horizon: Convergence Analysis
• Backward Stochastic Differential Equations
• Breaking the Curse of Dimensionality with Convex Neural Networks

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