Functional SDE approximation inspired by a deep operator network architecture

2024-02-05

arXiv.org

AI Summary

Approximating functional solutions of stochastic differential equations (SDEs) remains challenging due to the “curse of dimensionality” inherent in traditional polynomial chaos expansions (PCE). Method: We propose SDEONet—the first framework integrating Wiener chaos expansion (WCE) with deep operator networks (DeepONets)—to enable end-to-end functional approximation of the SDE solution operator. SDEONet employs sparse learning to automatically identify optimal low-dimensional truncations of WCE, circumventing ad hoc truncation and preserving statistical fidelity. Contributions/Results: (i) First systematic incorporation of WCE into SDE solvers; (ii) rigorous proof of convergence and subexponential computational complexity; (iii) scalability to high-dimensional, nonlinear SDEs beyond sampling-based methods. Numerical experiments demonstrate substantial accuracy and efficiency gains over Monte Carlo—especially for 1D to high-dimensional SDEs—with exponential speedup in computation and built-in uncertainty quantification.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

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.

Problem

• Approximating SDE solutions using neural networks
• Reducing complexity of polynomial chaos expansion
• Learning optimal sparse truncation for Wiener chaos

Innovation

• DeepONet-inspired SDE approximation via neural networks
• Polynomial chaos expansion reduces SDE complexity
• SDEONet learns optimal sparse Wiener chaos truncation