Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning
- Author(s)
- Julius Berner, Markus Dablander, Philipp Grohs
- Abstract
We present a deep learning algorithm for the numerical solution of parametric fam-
ilies of high-dimensional linear Kolmogorov partial differential equations (PDEs).
Our method is based on reformulating the numerical approximation of a whole
family of Kolmogorov PDEs as a single statistical learning problem using the
Feynman-Kac formula. Successful numerical experiments are presented, which
empirically confirm the functionality and efficiency of our proposed algorithm in
the case of heat equations and Black-Scholes option pricing models parametrized
by affine-linear coefficient functions. We show that a single deep neural network
trained on simulated data is capable of learning the solution functions of an entire
family of PDEs on a full space-time region. Most notably, our numerical observa-
tions and theoretical results also demonstrate that the proposed method does not
suffer from the curse of dimensionality, distinguishing it from almost all standard
numerical methods for PDEs.- Organisation(s)
- Department of Mathematics, Research Network Data Science
- External organisation(s)
- University of Oxford, Österreichische Akademie der Wissenschaften (ÖAW)
- No. of pages
- 13
- DOI
- https://doi.org/https://proceedings.neurips.cc/paper/2020/file/c1714160652ca6408774473810765950-Paper.pdf
- Publication date
- 2020
- Peer reviewed
- Yes
- Austrian Fields of Science 2012
- 101014 Numerical mathematics, 102018 Artificial neural networks
- Portal url
- https://ucris.univie.ac.at/portal/en/publications/numerically-solving-parametric-families-of-highdimensional-kolmogorov-partial-differential-equations-via-deep-learning(dcbb0b21-605f-4616-9592-358e7290f31c).html