# Functional SDE approximation inspired by a deep operator network architecture

arXiv: https://arxiv.org/abs/2402.03028 · HTML: https://mlsanity.com/paper/2402.03028/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Solving the Kolmogorov PDE by means of deep learning](https://mlsanity.com/paper/1806.00421/) (2018)
- [Adaptive operator learning for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2310.17844/) (2023)
- [Pricing high-dimensional Bermudan options with hierarchical tensor formats](https://mlsanity.com/paper/2103.01934/) (2021)
- [Approximating Langevin Monte Carlo with ResNet-like Neural Network architectures](https://mlsanity.com/paper/2311.03242/) (2023)

## Built on by (most cited first, 3 of 3)

- [Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations](https://mlsanity.com/paper/2601.01021/) (2026)
- [Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments](https://mlsanity.com/paper/2507.17887/) (2025)
- [Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations](https://mlsanity.com/paper/2603.08219/) (2026)

## Abstract (arXiv)

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.