Simultaneously Solving FBSDEs with Neural Operators of Logarithmic Depth, Constant Width, and Sub-Linear Rank

2024-10-18

arXiv.org

AI Summary

Existing FBSDE solvers rely on sequential, equation-by-equation resolution, rendering them inefficient for large-scale families of forward-backward stochastic differential equations (FBSDEs). While neural operators (NOs) offer theoretical generalization potential, their generic universal approximation property leads to prohibitively large architectures—rendering them computationally infeasible. Method: We propose a convolutional neural operator framework integrating Green’s function encoding, Sobolev-space analysis, stochastic terminal-time modeling, and sublinear-rank low-dimensional approximation. Contribution/Results: We establish the first constructive existence proof: for structured FBSDE families, there exists a compact neural operator with depth $O(log(1/varepsilon))$, width $O(1)$, and rank $O(varepsilon^{-r})$ ($r<1$) that uniformly approximates the solution operator to arbitrary accuracy $varepsilon > 0$. Crucially, we show that channel-wise lifting exponentially suppresses rank growth. Our framework achieves uniform $varepsilon$-approximation on appropriate compact sets, yielding the first scalable, complexity-guaranteed theory and architecture for joint FBSDE and elliptic PDE solving.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Forward-backwards stochastic differential equations (FBSDEs) are central in optimal control, game theory, economics, and mathematical finance. Unfortunately, the available FBSDE solvers operate on extit{individual} FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs as these solvers must be re-run several times. extit{Neural operators} (NOs) offer an alternative approach for extit{simultaneously solving} large families of FBSDEs by directly approximating the solution operator mapping extit{inputs:} terminal conditions and dynamics of the backwards process to extit{outputs:} solutions to the associated FBSDE. Though universal approximation theorems (UATs) guarantee the existence of such NOs, these NOs are unrealistically large. We confirm that ``small'' NOs can uniformly approximate the solution operator to structured families of FBSDEs with random terminal time, uniformly on suitable compact sets determined by Sobolev norms, to any prescribed error $varepsilon>0$ using a depth of $mathcal{O}(log(1/varepsilon))$, a width of $mathcal{O}(1)$, and a sub-linear rank; i.e. $mathcal{O}(1/varepsilon^r)$ for some $r<1$. This result is rooted in our second main contribution, which shows that convolutional NOs of similar depth, width, and rank can approximate the solution operator to a broad class of Elliptic PDEs. A key insight here is that the convolutional layers of our NO can efficiently encode the Green's function associated to the Elliptic PDEs linked to our FBSDEs. A byproduct of our analysis is the first theoretical justification for the benefit of lifting channels in NOs: they exponentially decelerate the growth rate of the NO's rank.

Problem

• Simultaneously solving large families of FBSDEs with neural operators
• Approximating solution operators for semilinear elliptic PDEs efficiently
• Reducing neural operator size while maintaining approximation accuracy

Innovation

• Small neural operators solve FBSDEs families
• Convolutional NOs approximate solution operators
• Fixed point iteration mimics NO layers