Simultaneously Solving FBSDEs and their Associated Semilinear Elliptic PDEs with Small Neural Operators
Takashi Furuya, Anastasis Kratsios
TL;DR
This work addresses the challenge of solving infinite families of forward-backwards stochastic differential equations (FBSDEs) by learning a single solution-operator map with neural operators. The authors introduce forward-backwards neural operators (FBNOs) that approximate the mapping from terminal/boundary data $(g,f_0)$ to the FBSDE solution $(Y_{ullet},Z_{ullet})$ across a large family, leveraging a PDE-FBSDE correspondence. A key technical advance is a convolutional neural operator with domain lifting that captures both the singular and smooth parts of the Green’s function, enabling uniform approximation guarantees with sublinear complexity in the reciprocal error. The paper proves two main results: (i) the FBNO can approximate the entire FBSDE family with controlled error and computational depth, and (ii) the corresponding class of semilinear elliptic PDEs can be uniformly approximated by a NO, with explicit depth, width, and rank bounds. Domain lifting channels are shown to be crucial for achieving favorable convergence rates, linking operator learning theory with deep PDE structure to produce practically scalable solvers for large model families.
Abstract
Forward-backwards stochastic differential equations (FBSDEs) play an important role in optimal control, game theory, economics, mathematical finance, and in reinforcement learning. Unfortunately, the available FBSDE solvers operate on \textit{individual} FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs, as these solvers must be re-run several times. \textit{Neural operators} (NOs) offer an alternative approach for \textit{simultaneously solving} large families of decoupled FBSDEs by directly approximating the solution operator mapping \textit{inputs:} terminal conditions and dynamics of the backwards process to \textit{outputs:} solutions to the associated FBSDE. Though universal approximation theorems (UATs) guarantee the existence of such NOs, these NOs are unrealistically large. Upon making only a few simple theoretically-guided tweaks to the standard convolutional NO build, we confirm that ``small'' NOs can uniformly approximate the solution operator to structured families of FBSDEs with random terminal time, uniformly on suitable compact sets determined by Sobolev norms using a logarithmic depth, a constant width, and a polynomial rank in the reciprocal approximation error. This result is rooted in our second result, and main contribution to the NOs for PDE literature, showing that our convolutional NOs of similar depth and width but grow only \textit{quadratically} (at a dimension-free rate) when uniformly approximating the solution operator of the associated class of semilinear Elliptic PDEs to these families of FBSDEs. A key insight into how NOs work we uncover is that the convolutional layers of our NO can approximately implement the fixed point iteration used to prove the existence of a unique solution to these semilinear Elliptic PDEs.