Paper: arXiv 2511.01125

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.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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