Functional SDE approximation inspired by a deep operator network
  architecture
Martin Eigel Charles Miranda
A novel approach to approximate solutions of Stochastic Differential
Equations (SDEs) by Deep Neural Networks is derived and analysed. The
architecture is inspired by the notion of Deep Operator Networks (DeepONets),
which is based on operator learning in function spaces in terms of a reduced
basis also represented in the network. In our setting, 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 (UQ) 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 an optimal sparse truncation of the
Wiener chaos expansion. A complete convergence and complexity 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.