One model to solve them all: 2BSDE families via neural operators
This work addresses computational challenges in solving complex stochastic differential equation families for applications in finance and physics, representing an incremental advance in neural operator methods.
The paper tackles solving infinite families of second-order backward stochastic differential equations (2BSDEs) using neural operators, showing that the solution operator is approximable by these models and identifying a subclass where parameter requirements scale polynomially rather than exponentially.
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.