Deep splitting method for parabolic PDEs

In this paper we introduce a numerical method for parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control, and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

[1]  Cornelis W. Oosterlee,et al.  A Fourier-Cosine Method for an Efficient Computation of Solutions to BSDEs , 2013, SIAM J. Sci. Comput..

[2]  J. Douglas,et al.  Numerical methods for forward-backward stochastic differential equations , 1996 .

[3]  Larry L. Schumaker,et al.  Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory , 2007 .

[4]  G. N. Milstein,et al.  Numerical Algorithms for Forward-Backward Stochastic Differential Equations , 2006, SIAM J. Sci. Comput..

[5]  G. Pagès,et al.  A quantization algorithm for solving multidimensional discrete-time optimal stopping problems , 2003 .

[6]  Xavier Warin,et al.  Machine Learning for Semi Linear PDEs , 2018, Journal of Scientific Computing.

[7]  N. Krylov,et al.  On the Rate of Convergence of Splitting-up Approximations for SPDEs , 2003 .

[8]  E. Gobet,et al.  Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations , 2006 .

[9]  Justin A. Sirignano,et al.  DGM: A deep learning algorithm for solving partial differential equations , 2017, J. Comput. Phys..

[10]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[11]  Geoffrey E. Hinton,et al.  Deep Learning , 2015, Nature.

[12]  N. Krylov,et al.  Lectures on Elliptic and Parabolic Equations in Holder Spaces , 1996 .

[13]  Thomas Kruse,et al.  Multilevel Picard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities , 2017, SIAM J. Numer. Anal..

[14]  Emmanuel Gobet,et al.  Numerical simulation of BSDEs using empirical regression methods: theory and practice , 2005 .

[15]  Michael V. Tretyakov,et al.  Discretization of forward–backward stochastic differential equations and related quasi-linear parabolic equations , 2007 .

[16]  Xiaolu Tan,et al.  A numerical algorithm for a class of BSDEs via branching process , 2013 .

[17]  S. Kruse,et al.  Parabolic Differential Equations with Unbounded Coefficients – A Generalization of the Parametrix Method , 2002 .

[18]  H. Lisei,et al.  Approximation of Stochastic Nonlinear Equations of Schrödinger Type by the Splitting Method , 2013 .

[19]  J. Ma,et al.  Forward-Backward Stochastic Differential Equations and their Applications , 2007 .

[20]  Andrea Molent,et al.  Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models , 2019, Quantitative Finance.

[21]  Sergey Ioffe,et al.  Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift , 2015, ICML.

[22]  G. Maruyama.  Continuous Markov processes and stochastic equations , 1955 .

[23]  Dan Crisan,et al.  Solving Backward Stochastic Differential Equations Using the Cubature Method: Application to Nonlinear Pricing , 2010, SIAM J. Financial Math..

[24]  Gitta Kutyniok,et al.  A Theoretical Analysis of Deep Neural Networks and Parametric PDEs , 2019, Constructive Approximation.

[25]  E Weinan,et al.  Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations , 2017, Communications in Mathematics and Statistics.

[26]  P. Protter,et al.  Numberical Method for Backward Stochastic Differential Equations , 2002 .

[27]  Cornelis W. Oosterlee,et al.  Computational and Applied , 2015 .

[28]  N. Touzi,et al.  On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights , 2010 .

[29]  N. Krylov.  On Kolmogorov’s equations for finite dimensional diffusions , 1999 .

[30]  Philippe von Wurstemberger,et al.  Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks , 2019, Electronic Journal of Probability.

[31]  A Dual Method For Backward Stochastic Differential Equations with Application to Risk Valuation , 2017, 1701.06234.

[32]  Arnulf Jentzen,et al.  Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations , 2018, Proceedings of the Royal Society A.

[33]  E Weinan,et al.  Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations , 2017, J. Nonlinear Sci..

[34]  Akihiko Takahashi,et al.  Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs , 2017, Asia-Pacific Financial Markets.

[35]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[36]  Christian Bender,et al.  A PRIMAL–DUAL ALGORITHM FOR BSDES , 2013, 1310.3694.

[37]  Arnulf Jentzen,et al.  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 coefficients , 2018, Communications in Mathematical Sciences.

[38]  Cornelis W. Oosterlee,et al.  Numerical Fourier Method and Second-Order Taylor Scheme for Backward SDEs in Finance , 2014 .

[39]  Kaj Nyström,et al.  A unified deep artificial neural network approach to partial differential equations in complex geometries , 2017, Neurocomputing.

[40]  Bin Dong,et al.  PDE-Net: Learning PDEs from Data , 2017, ICML.

[41]  Arnulf Jentzen,et al.  A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations , 2018, Memoirs of the American Mathematical Society.

[42]  Huyên Pham,et al.  Some machine learning schemes for high-dimensional nonlinear PDEs , 2019, ArXiv.

[43]  I. Gyöngy,et al.  On the splitting-up method and stochastic partial differential equations , 2003 .

[44]  Shinzo Watanabe,et al.  On the branching process for Brownian particles with an absorbing boundary , 1965 .

[45]  Arnulf Jentzen,et al.  Solving high-dimensional partial differential equations using deep learning , 2017, Proceedings of the National Academy of Sciences.

[46]  D. Crisan,et al.  Second order discretization of backward SDEs and simulation with the cubature method , 2014 .

[47]  Arnulf Jentzen,et al.  Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations , 2020, ArXiv.

[48]  J. Chassagneux,et al.  Numerical simulation of quadratic BSDEs , 2013, 1307.5741.

[49]  H. McKean.  Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov , 1975 .

[50]  H. Pham.  Feynman-Kac Representation of Fully Nonlinear PDEs and Applications , 2014, 1409.0625.

[51]  A. Skorokhod.  Branching Diffusion Processes , 1964 .

[52]  Christian Beck,et al.  An overview on deep learning-based approximation methods for partial differential equations , 2020, ArXiv.

[53]  Jean-François Chassagneux,et al.  Numerical Stability Analysis of the Euler Scheme for BSDEs , 2014, SIAM J. Numer. Anal..

[54]  Arnulf Jentzen,et al.  DNN Expression Rate Analysis of High-Dimensional PDEs: Application to Option Pricing , 2018, Constructive Approximation.

[55]  Pierre Henry-Labordere,et al.  Deep Primal-Dual Algorithm for BSDEs: Applications of Machine Learning to CVA and IM , 2017 .

[56]  S. Menozzi,et al.  A Forward-Backward Stochastic Algorithm For Quasi-Linear PDEs , 2006, math/0603250.

[57]  E Weinan,et al.  On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations , 2017, Journal of Scientific Computing.

[58]  Dan Crisan,et al.  RUNGE-KUTTA SCHEMES FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS , 2014 .

[59]  Jean-François Chassagneux,et al.  Linear Multistep Schemes for BSDEs , 2014, SIAM J. Numer. Anal..

[60]  Philipp Dörsek,et al.  Semigroup Splitting and Cubature Approximations for the Stochastic Navier-Stokes Equations , 2011, SIAM J. Numer. Anal..

[61]  E Weinan,et al.  The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems , 2017, Communications in Mathematics and Statistics.

[62]  Arnulf Jentzen,et al.  Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations , 2016, 1604.02053.

[63]  Robert Denk,et al.  A forward scheme for backward SDEs , 2007 .

[64]  T. Faniran.  Numerical Solution of Stochastic Differential Equations , 2015 .

[65]  E. Gobet,et al.  A regression-based Monte Carlo method to solve backward stochastic differential equations , 2005, math/0508491.

[66]  J. Yong,et al.  Solving forward-backward stochastic differential equations explicitly — a four step scheme , 1994 .

[67]  D. Duffie,et al.  Recursive valuation of defaultable securities and the timing of resolution of uncertainty , 1996 .

[68]  Sebastian Becker,et al.  Deep Optimal Stopping , 2018, J. Mach. Learn. Res..

[69]  Martin Magill,et al.  Neural Networks Trained to Solve Differential Equations Learn General Representations , 2018, NeurIPS.

[70]  Plamen Turkedjiev,et al.  Two algorithms for the discrete time approximation of Markovian backward stochastic differential equations under local conditions , 2013, 1309.4378.

[71]  Pierre Henry-Labordere,et al.  Counterparty Risk Valuation: A Marked Branching Diffusion Approach , 2012, 1203.2369.

[72]  Carlos Vázquez,et al.  Stratified Regression Monte-Carlo Scheme for Semilinear PDEs and BSDEs with Large Scale Parallelization on GPUs , 2016, SIAM J. Sci. Comput..

[73]  Tuan Anh Nguyen,et al.  A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations , 2019, SN Partial Differential Equations and Applications.

[74]  Sebastian Becker,et al.  Solving stochastic differential equations and Kolmogorov equations by means of deep learning , 2018, ArXiv.

[75]  G. N. Mil’shtejn.  Approximate Integration of Stochastic Differential Equations , 1975 .

[76]  Claudio Muñoz,et al.  Decay in the one dimensional generalized Improved Boussinesq equation , 2019, SN Partial Differential Equations and Applications.

[77]  Yoshua Bengio,et al.  Understanding the difficulty of training deep feedforward neural networks , 2010, AISTATS.

[78]  Daniel W. Stroock,et al.  Lectures on Topics in Stochastic Differential Equations , 1983 .

[79]  Arnulf Jentzen,et al.  Solving the Kolmogorov PDE by Means of Deep Learning , 2018, Journal of Scientific Computing.

[80]  Martin Hairer,et al.  Loss of regularity for Kolmogorov equations , 2012, 1209.6035.

[81]  Deep Ray,et al.  Deep learning observables in computational fluid dynamics , 2019, J. Comput. Phys..

[82]  Céline Labart,et al.  A parallel algorithm for solving BSDEs , 2013, Monte Carlo Methods Appl..

[83]  Arnulf Jentzen,et al.  Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations , 2018, SIAM J. Math. Data Sci..

[84]  Emmanuel Gobet,et al.  Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions , 2015, Math. Comput..

[85]  Daniel Nikovski,et al.  Deep reinforcement learning for partial differential equation control , 2017, 2017 American Control Conference (ACC).

[86]  Céline Labart,et al.  Solving BSDE with Adaptive Control Variate , 2010, SIAM J. Numer. Anal..

[87]  Arnaud Lionnet,et al.  Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs , 2013, 1309.2865.

[88]  Maziar Raissi,et al.  Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations , 2018, J. Mach. Learn. Res..

[89]  B. Bouchard,et al.  Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations , 2004 .

[90]  Nadia Oudjane,et al.  Branching diffusion representation of semilinear PDEs and Monte Carlo approximation , 2016, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[91]  Jianfeng Zhang.  A numerical scheme for BSDEs , 2004 .

[92]  Dan Crisan,et al.  Probabilistic methods for semilinear partial differential equations. Applications to finance , 2010 .

[93]  Antoine Jacquier,et al.  Deep PPDEs for Rough Local Stochastic Volatility , 2019, SSRN Electronic Journal.

[94]  Emmanuel Gobet,et al.  Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression , 2016, 1601.01186.

[95]  Arnulf Jentzen,et al.  Multilevel Picard iterations for solving smooth semilinear parabolic heat equations , 2021, Partial Differential Equations and Applications.

[96]  G. N. Milstein,et al.  Solving parabolic stochastic partial differential equations via averaging over characteristics , 2009, Math. Comput..

[97]  Jiequn Han,et al.  Convergence of the deep BSDE method for coupled FBSDEs , 2018, Probability, Uncertainty and Quantitative Risk.

[98]  Yoshua. Bengio,et al.  Learning Deep Architectures for AI , 2007, Found. Trends Mach. Learn..