Solving high-dimensional partial differential equations using deep learning

Keywords

Partial differential equation, Deep learning, Computer science, Applied mathematics, Mathematics

References

• The Diffuse Interface Approach in Materials Science: Thermodynamic Concepts and Applications of Phase-Field ModelsHeike Emmerich — 2003
• 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
• Option Pricing and Replication with Transactions CostsHayne E. Leland — 1985
• 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
• TensorFlow: a system for large-scale machine learningMartı́n Abadi, Paul Barham, Jianmin Chen — 2016
• Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equationsEmmanuel Gobet, Plamen Turkedjiev — 2017
• 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

Cited by

• Solving parametric PDE problems with artificial neural networksYUEHAW KHOO, JIANFENG LU, LEXING YING — 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
• Neural networks catching up with finite differences in solving partial differential equations in higher dimensionsVsevolod I. Avrutskiy — 2017
• Strong error analysis for stochastic gradient descent optimization algorithmsArnulf Jentzen, Benno Kuckuck, Ariel Neufeld — 2018
• Enforcing constraints for interpolation and extrapolation in Generative Adversarial NetworksPanos Stinis, Tobias Hagge, Alexandre M. Tartakovsky — 2018
• Solving linear parabolic rough partial differential equationsChristian Bayer, Denis Belomestny, Martin Redmann — 2018
• Nesting Monte Carlo for high-dimensional non-linear PDEsXavier Warin — 2018
• A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential EquationsMohammad Amin Nabian, Hadi Meidani — 2018
• Neural Networks Trained to Solve Differential Equations Learn General RepresentationsMartin Magill, Faisal Z. Qureshi, Hendrick W. de Haan — 2018
• Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equationsMartin Hutzenthaler, Arnulf Jentzen, Thomas Kruse — 2018
• A Multiscale Neural Network Based on Hierarchical MatricesYuwei Fan, Lin Lin, Lexing Ying — 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
• Machine-learning solver for modified diffusion equationsQianshi Wei, Ying Jiang, Jeff Z. Y. Chen — 2018
• Machine Learning for Semi Linear PDEsQuentin Chan-Wai-Nam, Joseph Mikael, Xavier Warin — 2018
• DNN Expression Rate Analysis of High-Dimensional PDEs: Application to Option PricingDennis Elbrächter, Philipp Grohs, Arnulf Jentzen — 2018
• 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
• A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientsArnulf Jentzen, Diyora Salimova, Timo Welti — 2018

Related papers

• ИСПОЛЬЗОВAНИЕ ПОТЕНЦИAЛA СОЦИAЛЬНЫХ ПAРТНЕРОВ В ПОДГОТОВКЕ БУДУЩИХ ПЕДAГОГОВБ.Б. Атaбекова — 2024
• Primärzerlegung in Steinschen AlgebrenOtto Förster — 1964
• Über unirationale Scharen auf algebraischen MannigfaltigkeitenLudwig Reich — 1966
• Produkttreue Klassen universeller AlgebrenJürgen Timm — 1969
• Approximation of fixed points of multifunctions in partial metric spacesDaniela Paesano — 2015
• The World of Jagdish N. SrivastavaSubir Ghosh, Sudhir Gupta, Rahul Mukerjee — 2013
• Kennzeichnung derp-adischen und der endlichen algebraischen ZahlkörperJ�rgen Neukirch — 1969
• Algebraic varieties with automorphism groups of maximal rankDe‐Qi Zhang — 2012
• Kohomologie vonp-Lie-AlgebrenBodo Pareigis — 1968