Functional SDE approximation inspired by a deep operator network architecture

Abstract

We present a novel approach to solve Stochastic Differential Equations (SDEs) with Deep Neural Networks by a Deep Operator Network (DeepONet) architecture. The notion of Deep-ONets relies on operator learning in terms of a reduced basis. We make use of a polynomial chaos expansion (PCE) of stochastic processes and call the corresponding architecture SDEONet. The PCE has been used extensively in the area of uncertainty quantification with parametric partial differential equations. This however is not the case with SDE, where classical sampling methods dominate and functional approaches are seen rarely. A main challenge with truncated PCEs occurs due to the drastic growth of the number of components with respect to the maximum polynomial degree and the number of basis elements. The proposed SDEONet architecture aims to alleviate the issue of exponential complexity by learning a sparse truncation of the Wiener chaos expansion. A complete convergence analysis is presented, making use of recent Neural Network approximation results. Numerical experiments illustrate the promising performance of the suggested approach in 1D and higher dimensions.