One model to solve them all: 2BSDE families via neural operators

2025-11-02

AI Summary

This work addresses the unified and efficient solution of infinite families of second-order backward stochastic differential equations (2BSDEs) on regular bounded Euclidean domains with random terminal times. We propose a lightweight generative neural operator model based on Kolmogorov–Arnold networks. For the first time, we rigorously prove that the solution operator of such 2BSDEs can be uniformly approximated by neural operators. Moreover, we identify a structured subclass for which the required number of parameters grows only polynomially in dimension—overcoming the exponential parameter dependence inherent in conventional methods. The framework enables an end-to-end mapping from random terminal conditions to solutions in function space, delivering a scalable and parameter-efficient paradigm for high-dimensional stochastic control and mathematical finance.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Deep Generative Models & Autoencoders
• Multiagent Systems: Adversarial Agents

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov--Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees.

Problem

• Solving infinite families of 2BSDEs with neural operators
• Approximating solution operators for broad 2BSDE families
• Reducing parameter requirements for neural operator approximations

Innovation

• Neural operators solve 2BSDE families
• Kolmogorov-Arnold networks handle random terminal times
• Polynomial parameters replace exponential worst-case requirements