DGM: A deep learning algorithm for solving partial differential equations (2017-08-24T00:00:00.000000Z)

TL;DR

A deep learning algorithm similar in spirit to Galerkin methods, using a deep neural network instead of linear combinations of basis functions is proposed, and is implemented for American options in up to 100 dimensions.

Authors

Justin A. Sirignano

1 papers

K. Spiliopoulos

1 papers

References56 items

Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

TL;DR

Authors

References56 items

Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

Optimal approximation of piecewise smooth functions using deep ReLU neural networks

Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

Deep relaxation: partial differential equations for optimizing deep neural networks

Existence results for a Cauchy–Dirichlet parabolic problem with a repulsive gradient term

Stochastic Gradient Descent in Continuous Time

Reynolds averaged turbulence modelling using deep neural networks with embedded invariance

Accelerating Eulerian Fluid Simulation With Convolutional Networks

Training Very Deep Networks

Adam: A Method for Stochastic Optimization

HJB equations in infinite dimensions under weak regularizing properties

Large Scale Distributed Deep Networks

Option pricing with a direct adaptive sparse grid approach

A highly parallel Black–Scholes solver based on adaptive sparse grids

EXISTENCE RESULT FOR NONLINEAR PARABOLIC EQUATIONS WITH LOWER ORDER TERMS

Understanding the difficulty of training deep feedforward neural networks

Summability and existence results for nonlinear parabolic equations

Analysis of linear difference schemes in the sparse grid combination technique

Stochastic Simulation: Algorithms and Analysis

Numerical solution for high order differential equations using a hybrid neural network - Optimization method

Feedforward sigmoidal networks - equicontinuity and fault-tolerance properties

Sparse grids

Monte Carlo valuation of American options

Pricing American Options: A Duality Approach

Valuing American Options by Simulation: A Simple Least-Squares Approach

Stationary Hamilton--Jacobi Equations in Hilbert Spaces and Applications to a Stochastic Optimal Control Problem

Neural-network methods for boundary value problems with irregular boundaries

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II

Gradient Convergence in Gradient methods with Errors

Existence of solutions for some "noncoercive" parabolic equations

Approximation theory of the MLP model in neural networks

Long Short-Term Memory

Nonlinear Parabolic Equations with Measure Data

Artificial neural networks for solving ordinary and partial differential equations

Linear and Quasi-linear Equations of Parabolic Type

Approximation capabilities of multilayer feedforward networks

Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks

Neural algorithm for solving differential equations

Approximation by superpositions of a sigmoidal function

Compact sets in the spaceLp(O,T; B)

Deep Learning Models in Finance, SIAM News, 2017

Solving Partial Differential Equations Using Artificial Neural Networks

Optimal control of a stochastic heat equation with boundary-noise and boundary-control

Efficient Hierarchical Approximation of High-Dimensional Option Pricing Problems

Compact Sets in the Space L~(O,

Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White Noise I

Numerical Experiments with SPDEs

Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type (Translations of Mathematical Monographs Reprint)

Elliptic Partial Differential Equa-tions of Second Order

A Real Variable Lemma and the Continuity of Paths of Some Gaussian Processes

and M

Annali di Matematica pura ed applicata

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