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

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 Doléans–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.

💡 Research Summary

The paper addresses a fundamental gap in neural operator (NO) theory: while universal approximation results guarantee that NOs can represent any continuous operator between infinite‑dimensional Banach spaces, information‑theoretic lower bounds show that, for generic operators defined only by regularity (e.g., uniform continuity or Sobolev smoothness), the number of trainable parameters required to achieve a uniform error ε must grow exponentially in 1/ε. Consequently, most existing NO guarantees fall into an “exponential scaling” regime, limiting their practical applicability to high‑accuracy tasks.

To escape this barrier, the authors identify a class of structured, non‑Markovian backward stochastic differential equations (BSDEs) whose solution operators possess special algebraic and analytic features that can be aligned with a tailored NO architecture. The BSDE family is parameterized by a Sobolev‑regular terminal condition g and a Sobolev‑regular additive perturbation f₀ of the generator. The dynamics also involve a predictable process β_t satisfying a strong Novikov condition; its Doléans‑Dade exponential Υ_t appears multiplicatively in both the forward SDE and the BSDE driver, introducing a common non‑Markovian factor.

The key methodological insight is twofold:

1. Green‑function factorization: The associated semilinear elliptic PDE L u + α(x,u)=f₀, u|_{∂D}=g has a Green’s function whose singular part can be analytically isolated. By incorporating a convolution with this singular kernel directly into the NO’s encoder, the network inherits the exact spatial smoothing properties of the PDE, reducing the effective dimensionality of the approximation problem.

2. Doléans‑Dade exponential embedding: The common non‑Markovian factor Υ_t is inserted into the decoder layers. Since Υ_t is a stochastic exponential, its logarithm is linear in the underlying Brownian motion, allowing the decoder to treat the time‑dependent, path‑dependent component of the BSDE as a linear transformation of learned features.

With these design choices, the authors construct a structure‑informed NO whose hypothesis class matches the intrinsic structure of the BSDE solution map Γ*. They prove two main theorems:

• Theorem 1 (BSDE operator approximation): For any ε∈(0,1), there exists a NO with at most C ε^{-k} trainable parameters (C, k depend only on the spatial dimension d, Sobolev indices, and regularity constants) that uniformly approximates Γ* over the prescribed Sobolev balls of (f₀,g). The error bound holds in the natural norm of the solution pair (Y₀,Z₀).

• Theorem 2 (Semilinear PDE operator approximation): An analogous polynomial‑complexity guarantee holds for the deterministic semilinear elliptic PDE solution operator Γ⁺, extending recent polynomial‑scaling results from linear elliptic PDEs to the semilinear setting.

The proofs combine several advanced tools: (i) precise estimates of the Green’s function singularity, (ii) stochastic calculus for the Doléans‑Dade exponential, (iii) wavelet‑based Besov–Sobolev approximation theory to control the approximation error of the convolution and the decoder, and (iv) a domain‑lifting trick that maps the original problem onto a higher‑dimensional product space where the NO architecture can be applied more naturally.

Beyond the theoretical contributions, the paper discusses practical implications for fields where BSDEs are central—mathematical finance (e.g., option pricing under stochastic volatility), stochastic control, game theory, and macro‑economic modeling. By demonstrating that the solution operator can be learned with polynomial parameter growth, the work suggests that NO‑based pipelines can be deployed for high‑precision, data‑driven simulation and inference in these domains, overcoming the previously prohibitive exponential cost.

In summary, the authors provide the first rigorous polynomial‑scaling approximation results for neural operators acting on families of non‑Markovian BSDEs, bridging a critical gap between universal expressivity and computational tractability in stochastic analysis. Their approach showcases how problem‑specific analytical structure—here, Green’s function factorization and stochastic exponentials—can be encoded into deep learning architectures to achieve provably efficient learning in infinite‑dimensional settings.