Page 80 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 80
FELIX KLEIN PRIZE WINNER
Overcoming the curse of dimensionality: from nonlinear Monte Carlo to
deep learning
Arnulf Jentzen, ajentzen@uni-muenster.de
University of Münster, Germany, and The Chinese University of Hong Kong, Shenzhen, China
Partial differential equations (PDEs) are among the most universal tools used in modelling prob-
lems in nature and man-made complex systems. For example, stochastic PDEs are a fundamen-
tal ingredient in models for nonlinear filtering problems in chemical engineering and weather
forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical
system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research
to describe optimal control problems where companys aim to minimise their costs, and deter-
ministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well
as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appear-
ing in such models are often high-dimensional as the number of dimensions, roughly speaking,
corresponds to the number of all involved interacting substances, particles, resources, agents,
or assets in the model. For instance, in the case of the above mentioned financial engineering
models the dimensionality of the PDE often corresponds to the number of financial assets in the
involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of
the most challenging tasks in applied mathematics to develop approximation algorithms which
are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approxi-
mation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality"
in the sense that the number of required computational operations of the approximation algo-
rithm to achieve a given approximation accuracy grows exponentially in the dimension of the
considered PDE. With such algorithms it is impossible to approximatively compute solutions
of high-dimensional PDEs even when the fastest currently available computers are used. In
the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of
dimensionality can be overcome by means of Monte Carlo approximation algorithms and the
Feynman-Kac formula. In this talk we prove that suitable deep neural network approximations
do indeed overcome the curse of dimensionality in the case of a general class of semilinear
parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic
PDE can be solved approximatively without the curse of dimensionality.
78
Overcoming the curse of dimensionality: from nonlinear Monte Carlo to
deep learning
Arnulf Jentzen, ajentzen@uni-muenster.de
University of Münster, Germany, and The Chinese University of Hong Kong, Shenzhen, China
Partial differential equations (PDEs) are among the most universal tools used in modelling prob-
lems in nature and man-made complex systems. For example, stochastic PDEs are a fundamen-
tal ingredient in models for nonlinear filtering problems in chemical engineering and weather
forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical
system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research
to describe optimal control problems where companys aim to minimise their costs, and deter-
ministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well
as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appear-
ing in such models are often high-dimensional as the number of dimensions, roughly speaking,
corresponds to the number of all involved interacting substances, particles, resources, agents,
or assets in the model. For instance, in the case of the above mentioned financial engineering
models the dimensionality of the PDE often corresponds to the number of financial assets in the
involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of
the most challenging tasks in applied mathematics to develop approximation algorithms which
are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approxi-
mation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality"
in the sense that the number of required computational operations of the approximation algo-
rithm to achieve a given approximation accuracy grows exponentially in the dimension of the
considered PDE. With such algorithms it is impossible to approximatively compute solutions
of high-dimensional PDEs even when the fastest currently available computers are used. In
the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of
dimensionality can be overcome by means of Monte Carlo approximation algorithms and the
Feynman-Kac formula. In this talk we prove that suitable deep neural network approximations
do indeed overcome the curse of dimensionality in the case of a general class of semilinear
parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic
PDE can be solved approximatively without the curse of dimensionality.
78
