Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning

Jiequn Han, Arnulf Jentzen, E Weinan · arXiv.org · 2017

Summary

A deep learning-based approach that can handle general high-dimensional parabolic PDEs is presented, reformulated as a control theory problem and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function.

Type: article
Published: 2017-07-09
Cited by: 83
References: 33
OpenAlex: https://openalex.org/W2734689136
Semantic Scholar: https://api.semanticscholar.org/CorpusID:126123995

Keywords

Curse of dimensionality, Partial differential equation, Mathematical finance, Applied mathematics, Nonlinear system

References

Approximate Dynamic ProgrammingWarren B. Powell — 2007
Recursive valuation of defaultable securities and the timing of resolution of uncertaintyDarrell Duffie, Mark Schröder, Costis Skiadas — 1996
The randomized information complexity of elliptic PDEStefan Heinrich — 2006
Pricing and hedging derivative securities in markets with uncertain volatilitiesM. Avellaneda, Arnon Levy, Antonio Paras — 1995
Option Pricing with Differential Interest RatesYaacov Z. Bergman — 1995
Implicit solution of uncertain volatility/transaction cost option pricing models with discretely observed barriersPeter Forsyth, K.R. Vetzal — 2001
The Pricing of Options and Corporate LiabilitiesFischer Black, Myron S. Scholes — 1973
Forward-backward stochastic differential equations and quasilinear parabolic PDEsÉtienne Pardoux, Shanjian Tang — 1999
Counterparty Risk Valuation: A Marked Branching Diffusion ApproachPierre Henry‐Labordère — 2012
A PRIMAL–DUAL ALGORITHM FOR BSDESChristian Bender, Nikolaus Schweizer, Jia Zhuo — 2013
Counterparty Risk and Funding: The Four Wings of the TVAStéphane Crépey, Rémi Gerboud, Zorana Grbac — 2012
Backward Stochastic Differential Equations in FinanceNicole El Karoui, Shigē Péng, M.C. Quenez — 1997
Approximation theory of the MLP model in neural networksAllan Pinkus — 1999
Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research GroupsGeoffrey E. Hinton, Li Deng, Dong Yu — 2012
ImageNet classification with deep convolutional neural networksAlex Krizhevsky, Ilya Sutskever, Geoffrey E. Hinton — 2012
Mastering the game of Go with deep neural networks and tree searchDavid Silver, Aja Huang, Chris J. Maddison — 2016
Dynamic ProgrammingRichard Bellman — 1957
On full history recursive multilevel Picard approximations and numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equationsE Weinan, Martin Hutzenthaler, Arnulf Jentzen — 2016
On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential EquationsE Weinan, Martin Hutzenthaler, Arnulf Jentzen — 2017
Monte-Carlo Methods and Stochastic Processes: From Linear to Non-LinearEmmanuel Gobet — 2016

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Deep Residual Learning and PDEs on ManifoldZhen Li, Zuoqiang Shi — 2017
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential EquationsChristian Beck, E Weinan, Arnulf Jentzen — 2017
The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational ProblemsE Weinan, Bing Yu — 2017
A unified deep artificial neural network approach to partial differential equations in complex geometriesJens Berg, Kaj Nyström — 2017
Optimal Controller Synthesis for Nonlinear SystemsYoke Peng Leong — 2018
An Unbiased Itô Type Stochastic Representation for Transport PDEs: A Toy ExampleGonçalo dos Reis, Greig Smith — 2017
Nesting Monte Carlo for high-dimensional non-linear PDEsXavier Warin — 2018
Monte Carlo for high-dimensional degenerated Semi Linear and Full Non Linear PDEsXavier Warin — 2018
A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential EquationsMohammad Amin Nabian, Hadi Meidani — 2018
Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equationsMartin Hutzenthaler, Arnulf Jentzen, Thomas Kruse — 2018
Relu Deep Neural Networks and Linear Finite ElementsJuncai He, Lin Li, Jinchao Xu — 2018
Deep Learning-Based BSDE Solver for Libor Market Model with Application to Bermudan Swaption Pricing and Hedging.Haojie Wang, Han Chen, Agus Sudjianto — 2018
Weakly-Supervised Deep Learning of Heat Transport via Physics Informed LossRishi Sharma, Amir Barati Farimani, Joe Gomes — 2018
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A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationsPhilipp Grohs, Fabian Hornung, Arnulf Jentzen — 2018
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Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning

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