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

TL;DR

This paper presents a neural operator framework that efficiently approximates entire families of second-order backward stochastic differential equations ($2$BSDEs), enabling scalable solutions with polynomial complexity in certain structured cases.

Contribution

It introduces a neural operator model based on Kolmogorov--Arnold networks for solving infinite $2$BSDE families, with improved approximation efficiency for structured subclasses.

Findings

Neural operators can approximate $2$BSDE$ solutions across families.

Structured subclasses allow polynomial parameter complexity.

The approach extends neural operator applicability to stochastic differential equations.

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.