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Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations [article]

Weinan E and Jiequn Han and Arnulf Jentzen
2017 arXiv   pre-print
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and  ...  Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation,  ...  Acknowledgements Christian Beck and Sebastian Becker are gratefully acknowledged for useful suggestions regarding the implementation of the deep BSDE solver.  ... 
arXiv:1706.04702v1 fatcat:iz6iw4nrjjdqzfl5tbnjxsm7p4

Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations [article]

Maziar Raissi
2018 arXiv   pre-print
Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising  ...  To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations.  ...  Acknowledgements This work received support by the DARPA EQUiPS grant N66001-15-2-4055 and the AFOSR grant FA9550-17-1-0013.  ... 
arXiv:1804.07010v1 fatcat:xwtsic2p7bd2neqt7czg3kjjcm

Three Ways to Solve Partial Differential Equations with Neural Networks – A Review [article]

Jan Blechschmidt, Oliver G. Ernst
2021 arXiv   pre-print
, methods based on the Feynman-Kac formula and methods based on the solution of backward stochastic differential equations.  ...  Neural networks are increasingly used to construct numerical solution methods for partial differential equations.  ...  A general procedure based on data-driven machine learning to accelerate existing numerical methods for the solution of partial and ordinary differential equations is presented in [126] .  ... 
arXiv:2102.11802v2 fatcat:xc647il5q5f4baixs74arorbbm

Interpolating between BSDEs and PINNs – deep learning for elliptic and parabolic boundary value problems [article]

Nikolas Nüsken, Lorenz Richter
2021 arXiv   pre-print
For elliptic and parabolic problems, existing methods can broadly be classified into those resting on reformulations in terms of backward stochastic differential equations (BSDEs) and those aiming to minimize  ...  Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering.  ...  Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations.  ... 
arXiv:2112.03749v1 fatcat:i72753t6crf5borpb5zkqdmbuy

Neural Differential Equations as a Basis for Scientific Machine Learning (SciML) [article]

Christopher Rackauckas
2020 figshare.com  
Additionally, deep learning embedded within backwards stochastic differential equations has been shown to be an effective tool for solving high-dimensional partial differential equations, like the Hamilton-Jacobian-Bellman  ...  In this talk we will introduce the audience to these methods and show how these diverse methods are all instantiations of a neural differential equation, a differential equation where all or part of the  ...  Solving high-dimensional partial differential equations using deep learning, 2018, PNAS, Han, Jentzen, E 32 Stochastic RNN Formulation 33 But this method can be represented as a neural SDE  ... 
doi:10.6084/m9.figshare.12751955.v1 fatcat:zhwjvt23tfhmjljetsfobsv5q4

Solving high-dimensional partial differential equations using deep learning [article]

Jiequn Han, Arnulf Jentzen, Weinan E
2018 arXiv   pre-print
This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs.  ...  Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse  ...  Acknowledgement The work of Han and E is supported in part by Major Program of NNSFC under grant 91130005, DOE grant DE-SC0009248 and ONR grant N00014-13-1-0338.  ... 
arXiv:1707.02568v3 fatcat:mnogij2aovcbbpxsbhw3zuldui

Solving high-dimensional partial differential equations using deep learning

Jiequn Han, Arnulf Jentzen, Weinan E
2018 Proceedings of the National Academy of Sciences of the United States of America  
This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs.  ...  Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse  ...  Acknowledgement The work of Han and E is supported in part by Major Program of NNSFC under grant 91130005, DOE grant DE-SC0009248 and ONR grant N00014-13-1-0338.  ... 
doi:10.1073/pnas.1718942115 pmid:30082389 fatcat:cueasdqyqngyxn2qfkk22t235e

Recent Advancements in Differential Equation Solver Software [article]

Christopher Rackauckas
2020 figshare.com  
Extensions of these solver methods to adaptive high order methods for stochastic differential-algebraic and delay differential-algebraic equations will be demonstrated, and the potential use cases of these  ...  advancements in numerical analysis, computational methods, and hardware have accelerated computing.  ...  Solving high-dimensional partial differential equations using deep learning, 2018, PNAS, Han, Jentzen, E Represent 1000 dimensional PDEs as a 1000 dimensional neural SDE u Solving the PDE = training  ... 
doi:10.6084/m9.figshare.12751997.v1 fatcat:o7lbctk2evfnxha4b4wrxnt26e

Actor-Critic Algorithm for High-dimensional Partial Differential Equations [article]

Xiaohan Zhang
2020 arXiv   pre-print
We develop a deep learning model to effectively solve high-dimensional nonlinear parabolic partial differential equations (PDE).  ...  We follow Feynman-Kac formula to reformulate PDE into the equivalent stochastic control problem governed by a Backward Stochastic Differential Equation (BSDE) system.  ...  Conclusions We have developed a deep learning model to effectively solve high dimensional parabolic Partial Differential Equations.  ... 
arXiv:2010.03647v1 fatcat:fue6bj2k4zgjll63qgivzd3r7u

Handling Multiscale Stochastic Differential Equations in Julia(If You Give a Mathematician a Compiler) [article]

Christopher Rackauckas
2020 figshare.com  
Handling Multiscale Stochastic Differential Equations in Julia Chris Rackauckas, University of California, Irvine, U.S.  ...  Solve the forward problem 1,000,000 times to get a probability distribution, this distribution is the PDE's solution Solving high-dimensional partial differential equations using deep learning, 2018, PNAS  ...  Solve the resulting SDEs and learn ∇u via: u Diffusion-advection PDEs can be transformed into stochastic differential equations for high dimensional solving (Feynman-Kac, Forward Kolmogorov) u What about  ... 
doi:10.6084/m9.figshare.12752000.v1 fatcat:ijhnub2sc5ddrayowkiyoxpvdi

An overview on deep learning-based approximation methods for partial differential equations [article]

Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck
2021 arXiv   pre-print
Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs.  ...  It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs).  ...  work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics -Geometry -Structure and  ... 
arXiv:2012.12348v2 fatcat:3234pdzcmvdplg3py335en3fsa

Doing Scientific Machine Learning with Julia's SciML Ecosystem [article]

Christopher Rackauckas
2020 figshare.com  
Equations for Scientific Machine Learning](https://arxiv.org/abs/2001.04385)), Physics-Informed Neural Networks ([Physics-informed neural networks: A deep learning framework for solving forward and inverse  ...  problems involving nonlinear partial differential equations](https://www.sciencedirect.com/science/article/pii/S0021999118307125)), and Sparse Identification of Nonlinear Dynamics (SInDy, [Discovering  ...  state-of-the-art deep BSDE methods from the literature Solving high-dimensional partial differential equations using deep learning Jiequn Han, Arnulf Jentzen, and Weinan E Forward-Backward Stochastic  ... 
doi:10.6084/m9.figshare.12751949.v1 fatcat:3nodxm7ghzftflwbmlrtnhf5tu

Solving high-dimensional parabolic PDEs using the tensor train format [article]

Lorenz Richter, Leon Sallandt, Nikolas Nüsken
2021 arXiv   pre-print
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering.  ...  In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and  ...  We would like to thank Reinhold Schneider for giving valuable input and for sharing his broad insight in tensor methods and optimization.  ... 
arXiv:2102.11830v2 fatcat:m4hrv45wn5cq3ppic2qpkttkii

A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations [article]

Mohammad Amin Nabian, Hadi Meidani
2018 arXiv   pre-print
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality.  ...  We present a new solution framework for these problems based on a deep learning approach.  ...  and boundary conditions, e.g. by using Lagrange multipliers, or by developing algorithms for learning the functions C and G in Equation 11, and (4) investigating optimal sampling strategies in order to  ... 
arXiv:1806.02957v2 fatcat:appthxfrjba6pe24ewoazlho2u

Automated Discovery and Acceleration of Physical Equations with Julia's Scientific Machine Learning (SciML) Ecosystem [article]

Christopher Rackauckas
2020 figshare.com  
Then we will demonstrate how to reduce many methods, such as discrete-time physics-informed neural networks and deep forwards-backwards stochastic differential equation (FBSDE) solvers into the universal  ...  with the traditional numerical and symbolic differential equation solver software.  ...  for generating accelerated building models with transferred learning components Universal Differential Equations extend previous physics-informed neural network and deep BSDE algorithms UDE Methods  ... 
doi:10.6084/m9.figshare.12751952.v1 fatcat:kb2z4abogzc2nb3f4ylk57jj5m
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