Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

E Weinan, J Han, A Jentzen - Nonlinearity, 2021 - iopscience.iop.org
In recent years, tremendous progress has been made on numerical algorithms for solving
partial differential equations (PDEs) in a very high dimension, using ideas from either …

Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential …

C Beck, WE, A Jentzen - Journal of Nonlinear Science, 2019 - Springer
High-dimensional partial differential equations (PDEs) appear in a number of models from
the financial industry, such as in derivative pricing models, credit valuation adjustment …

Machine learning for semi linear PDEs

Q Chan-Wai-Nam, J Mikael, X Warin - Journal of scientific computing, 2019 - Springer
Recent machine learning algorithms dedicated to solving semi-linear PDEs are improved by
using different neural network architectures and different parameterizations. These …

[图书][B] Backward stochastic differential equations

J Zhang, J Zhang - 2017 - Springer
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Neural networks-based algorithms for stochastic control and PDEs in finance

M Germain, H Pham, X Warin - arXiv preprint arXiv:2101.08068, 2021 - cambridge.org
This chapter presents machine learning techniques and deep reinforcement learning-based
algorithms for the efficient resolution of nonlinear partial differential equations and dynamic …

Neural networks-based backward scheme for fully nonlinear PDEs

H Pham, X Warin, M Germain - SN Partial Differential Equations and …, 2021 - Springer
We propose a numerical method for solving high dimensional fully nonlinear partial
differential equations (PDEs). Our algorithm estimates simultaneously by backward time …

Optimal transportation under controlled stochastic dynamics

X Tan, N Touzi - The annals of probability, 2013 - JSTOR
We consider an extension of the Monge-Kantorovitch optimal transportation problem. The
mass is transported along a continuous semimartingale, and the cost of transportation …

Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

P Henry-Labordere, N Oudjane, X Tan, N Touzi… - 2019 - projecteuclid.org
We provide a representation result of parabolic semi-linear PDEs, with polynomial
nonlinearity, by branching diffusion processes. We extend the classical representation for …

On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward …

M Hutzenthaler, A Jentzen, T Kruse - Journal of Scientific Computing, 2019 - Springer
Parabolic partial differential equations (PDEs) and backward stochastic differential
equations (BSDEs) are key ingredients in a number of models in physics and financial …

Approximation error analysis of some deep backward schemes for nonlinear PDEs

M Germain, H Pham, X Warin - SIAM Journal on Scientific Computing, 2022 - SIAM
Recently proposed numerical algorithms for solving high-dimensional nonlinear partial
differential equations (PDEs) based on neural networks have shown their remarkable …