由深度运算网络架构启发的功能微分方程近似
Functional SDE approximation inspired by a deep operator network architecture
AI总结：
本文提出SDEONet架构，通过深度运算网络思想，利用多项式混沌展开解决SDE的近似问题，缓解指数复杂性，展示在1D和高维中的有效性。
摘要：
一种新的方法被推导和分析，用于通过深度神经网络近似随机微分方程（SDE）的解。该架构受深度运算网络（DeepONets）概念的启发，其基于函数空间中的算子学习，以减少基底形式也表示在网络中。在我们的设定中，我们利用随机过程的多项式混沌展开（PCE）并称相应的架构为SDEONet。PCE已在参数偏微分方程的不确定性量化（UQ）领域中被广泛使用。然而，SDE却并非如此，经典采样方法占主导地位，而功能方法则很少见。截断PCE的主要挑战在于，随着最大多项式次数和基底元素数量的增加，组件数量急剧增长。所提出的SDEONet架构旨在通过学习Wiener chaos展开的最优稀疏截断来缓解指数复杂性的问题。本文给出了完整的收敛性和复杂度分析，利用了最近的神经网络近似结果。数值实验展示了所提出方法在1维和更高维度中的有希望的表现。
英文摘要：
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.
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