Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs

TL;DR

This paper demonstrates that neural operator architectures can achieve polynomial scaling in approximation accuracy for certain structured families of non-Markovian BSDEs, advancing the understanding of their efficiency in stochastic analysis.

Contribution

The paper introduces the first polynomial-scaling regime for neural operator approximations of solution operators in stochastic analysis, leveraging problem-specific structure and tailored architectures.

Findings

Neural operators can approximate solution operators with polynomial complexity for structured BSDE families.

Incorporating problem-specific structure into neural architectures enables polynomial scaling.

Extends polynomial-scaling guarantees from linear to semilinear PDEs.

Abstract

Neural operator (NO) architectures learn nonlinear maps between infinite-dimensional function spaces and are widely used to accelerate simulation and enable data-driven model discovery. While universality results ensure expressivity, they do not address \emph{complexity}: for broad operator classes described only through regularity (e.g.\ uniform continuity or $C^r$-regularity), information-theoretic lower bounds imply that minimax-optimal NO approximation rates scale \emph{exponentially} in the reciprocal accuracy $1/\varepsilon$. This has shifted the focus of NO theory toward identifying additional problem-specific structure, beyond regularity, under which suitably tailored NO architectures can leverage to unlock polynomial scaling in $1/\varepsilon$. We exhibit the first polynomial-scaling regime for NO approximations of solution operators in stochastic analysis; by identifying structured families of \emph{non-Markovian} BSDEs with randomized terminal condition parameterized by the Sobolev-regular terminal condition and by Sobolev-regular additive nonlinear perturbations of the generator. We prove that their solution operator can be approximated (uniformly over the family) by a tailored NO whose number of trainable parameters grows \emph{polynomially} in $1/\varepsilon$. We unlock this polynomial scaling regime by \emph{informing the NO's inductive bias} by factoring out the singular part of the associated semilinear elliptic PDE Green's function and by incorporating the Dol\'{e}ans--Dade exponential of the BSDE's common non-Markovian factor into the NO's decoding layers. As a byproduct, we extend polynomial-scaling guarantees from families of linear elliptic PDEs on regular domains to the semilinear setting.