155 research outputs found
An overview on deep learning-based approximation methods for partial differential equations
It is one of the most challenging problems in applied mathematics to
approximatively solve high-dimensional partial differential equations (PDEs).
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. This has given rise to a lively field of research in
which deep learning-based methods and related Monte Carlo methods are applied
to the approximation of high-dimensional PDEs. In this article we offer an
introduction to this field of research, we review some of the main ideas of
deep learning-based approximation methods for PDEs, we revisit one of the
central mathematical results for deep neural network approximations for PDEs,
and we provide an overview of the recent literature in this area of research.Comment: 23 page
A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
Deep neural networks and other deep learning methods have very successfully
been applied to the numerical approximation of high-dimensional nonlinear
parabolic partial differential equations (PDEs), which are widely used in
finance, engineering, and natural sciences. In particular, simulations indicate
that algorithms based on deep learning overcome the curse of dimensionality in
the numerical approximation of solutions of semilinear PDEs. For certain linear
PDEs this has also been proved mathematically. The key contribution of this
article is to rigorously prove this for the first time for a class of nonlinear
PDEs. More precisely, we prove in the case of semilinear heat equations with
gradient-independent nonlinearities that the numbers of parameters of the
employed deep neural networks grow at most polynomially in both the PDE
dimension and the reciprocal of the prescribed approximation accuracy. Our
proof relies on recently introduced multilevel Picard approximations of
semilinear PDEs.Comment: 29 page
Deep splitting method for parabolic PDEs
In this paper we introduce a numerical method for nonlinear 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.Comment: 25 page
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