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Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena, in natural sciences, engineering or even finance. For example, in physical systems, the Allen-Cahn equation describes pattern formation associated to phase transitions. In finance, instead, the Black-Scholes equation describes the evolution of the price of derivative investment instruments. Such modern applications often require to solve these equations in high-dimensional regimes in which classical approaches are ineffective. Recently, an interesting new approach based on deep learning has been introduced. The main idea is to construct a deep network which is trained from the samples of a discrete stochastic differential equation underlying Kolmogorov's equation. The network is able to approximate the solutions of the Kolmogorov equation in polynomial time, therefore avoiding the curse of dimensionality. In this contribution we study variants of these deep networks by using different discretizations schemes of the stochastic differential equation. We compare the performance of the associated networks, on benchmarked examples, and show that, for some discretization schemes, improvements in the accuracy are possible without affecting the computational complexity.
This talk will be given on zoom https://kth-se.zoom.us/j/68365874335