Functional SDE approximation inspired by a deep operator network architecture
This work addresses the challenge of functional approximation for SDEs, which is less common than classical sampling methods, offering a potential improvement for uncertainty quantification in domains like finance or physics.
The authors tackled the problem of approximating solutions to Stochastic Differential Equations (SDEs) by proposing a novel neural network architecture called SDEONet, which learns an optimal sparse truncation of the Wiener chaos expansion to alleviate exponential complexity, with numerical experiments showing promising performance in 1D and higher dimensions.
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.