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

Takashi Furuya, Anastasis Kratsios, Dylan Possamaï, Bogdan Raonić

TL;DR

The paper addresses solving infinite families of second-order backward SDEs with random terminal time by learning a coefficient-to-solution operator for the associated fully nonlinear elliptic PDEs. It introduces a neural-operator framework based on Kolmogorov–Arnol’d networks (KANO/Res--KANs) that learns the map $(f_0,g)\mapsto u$ and, via a non-linear Feynman–Kac adapter, yields stochastic processes $(Y,Z,\Upsilon,A)$ for the entire problem family. The main contributions are (i) a general approximability guarantee ensuring the operator $\Gamma^+$ can uniformly approximate the solution map on compact PDE-perturbation sets, (ii) a feasible-rate result showing exponential-in-depth rates for a structured semi-linear PDE with polynomial resource usage in $1/\varepsilon$, and (iii) an extension to the stochastic $2$BSDE setting yielding $\varepsilon$-accurate approximations of $(Y^x,Z^x,\Upsilon^x,A^x)$ in expectation. Empirically, the approach demonstrates accurate recovery of the solution $u$ and its derivatives on periodic semi-linear and linear-quadratic benchmarks in dimension $d=5$, while ablation on sample size exposes the data requirements for stable high-dimensional performance. This work provides a scalable, operator-level method to solve parametric families of $2$BSDEs and their PDE representations, enabling efficient, single-pass learning across problem instances.

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.

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

TL;DR

The paper addresses solving infinite families of second-order backward SDEs with random terminal time by learning a coefficient-to-solution operator for the associated fully nonlinear elliptic PDEs. It introduces a neural-operator framework based on Kolmogorov–Arnol’d networks (KANO/Res--KANs) that learns the map and, via a non-linear Feynman–Kac adapter, yields stochastic processes for the entire problem family. The main contributions are (i) a general approximability guarantee ensuring the operator can uniformly approximate the solution map on compact PDE-perturbation sets, (ii) a feasible-rate result showing exponential-in-depth rates for a structured semi-linear PDE with polynomial resource usage in , and (iii) an extension to the stochastic BSDE setting yielding -accurate approximations of in expectation. Empirically, the approach demonstrates accurate recovery of the solution and its derivatives on periodic semi-linear and linear-quadratic benchmarks in dimension , while ablation on sample size exposes the data requirements for stable high-dimensional performance. This work provides a scalable, operator-level method to solve parametric families of BSDEs and their PDE representations, enabling efficient, single-pass learning across problem instances.

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 (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 BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of 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.

Paper Structure

This paper contains 34 sections, 17 theorems, 163 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related literature
  3. Preliminaries
  4. Notation
  5. Deep learning
  6. Residual Kolmogorov--Arnold networks (Res--KANs)
  7. Neural operator architectures
  8. Main results
  9. Elliptic PDE representation of the 2BSDE system
  10. General approximability guarantee
  11. Feasible rates
  12. Semi-linear elliptic PDE
  13. Solutions to the family of second-order BSDEs
  14. Experimental results
  15. Periodic semi-linear case
  16. ...and 19 more sections

Key Result

Proposition 1

Let $u$ be a classical solution to the PDE eq:AssociatePDE_General, such that all the quantities below are defined and continuous in time where Then $(Y,Z,\Upsilon,A)$ is a solution to eq:FBSDE--eq:FBSDE_Martingale.

Figures (6)

  • Figure 1: The cardinal $B$-splines of orders $I=0,1$, and $2$.
  • Figure 2: The KANO (\ref{['def:neural-operator']}) pipeline.
  • Figure 3: Ground-truth and KANO-predicted solutions for the first randomly selected trajectory of the periodic semilinear example from chassagneux2023learning. Each panel shows the projection onto the $(x_1, x_2)$-plane with $u$, $\partial u / \partial x_1$, and $\partial^2 u / \partial x_1^2$ along this path.
  • Figure 4: Continuation of \ref{['fig:semilinear_samples_a']}, showing the second randomly selected trajectory for the same semi-linear example.
  • Figure 5: Comparison between the ground-truth and KANO-predicted solutions for the periodic linear--quadratic example of pham2021neural. The figure shows two randomly selected trajectories projected onto the $(x_1, x_2)$-plane, together with the corresponding values of $u$, $\partial u / \partial x_1$, and $\partial^2 u / \partial x_1^2$ along these paths.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Example 1: Haar wavelets and indicator function for discontinuous regularity
  • Definition 1: Residual KANs (Res--KANs)
  • Definition 2: Kolmogorov--Arnold neural operator (KANO)
  • Definition 3: $2$Generative neural operators (2FBNO)
  • Proposition 1: Non-linear Feynman--Kac's formula
  • proof
  • Definition 4: PDE perturbation space $\mathcal{X}_k(r)$
  • Example 2: Source perturbations only
  • Theorem 3.1: Approximability of the perturbation-to-solution map
  • Theorem 3.2: Exponential approximation rates: solution operator to the elliptic problem
  • ...and 28 more