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

ArXiv ID: 2511.01125 “View on arXiv”

Authors: Takashi Furuya, Anastasis Kratsios, Dylan Possamaï, Bogdan Raonić

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.

Keywords: Neural operators, Stochastic differential equations, Kolmogorov-Arnold networks, Mathematical finance, Derivatives pricing, General Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats
• Why: The paper is highly mathematically complex, featuring dense advanced mathematics including stochastic differential equations, elliptic PDEs, Sobolev spaces, and neural operator theory with polynomial/exponential approximation rates. It shows very low empirical rigor as it presents theoretical approximation theorems without any code, backtests, datasets, or implementation-heavy details; it focuses on pure mathematical theory rather than practical application.

  flowchart TD
    A["Research Goal:<br>Solve infinite families of 2BSDEs<br>using Neural Operators"] --> B["Methodology:<br>Neural Operator +<br>Kolmogorov-Arnold Networks KANs"]
    B --> C["Data/Input:<br>Regular bounded Euclidean domains<br>with random terminal time"]
    C --> D["Computational Process:<br>Approximation of the<br>solution operator for 2BSDEs"]
    D --> E["Key Finding 1:<br>Solution operator is approximable<br>by Neural Operators"]
    D --> F["Key Finding 2:<br>Structured subclass requires only<br>Polynomial parameters for convergence"]
    E --> G["Outcome:<br>One model to solve<br>infinite families of 2BSDEs"]
    F --> G