  In this article we introduce and study a deep learning based approximation
algorithm for solutions of stochastic partial differential equations (SPDEs).
In the proposed approximation algorithm we employ a deep neural network for
every realization of the driving noise process of the SPDE to approximate the
solution process of the SPDE under consideration. We test the performance of
the proposed approximation algorithm in the case of stochastic heat equations
with additive noise, stochastic heat equations with multiplicative noise,
stochastic Black--Scholes equations with multiplicative noise, and Zakai
equations from nonlinear filtering. In each of these SPDEs the proposed
approximation algorithm produces accurate results with short run times in up to
50 space dimensions.
