We study operator learning for a nonlinear dynamical system describing symplastic plant leaf growth with multiple interacting cell files and stochastic cell division. The biomechanical model consists of coupled ordinary differential equations governing visible cell lengths, relaxed wall lengths, isosmotic lengths, and shared wall fragments. For $N$ cell files with $M$ cells per file, the state dimension scales as $D(N,M) \sim 3NM + K(N,M)$, where $K$ denotes the number of shared fragments. While the number of state variables grows linearly in $N$, fragment-based mechanical coupling induces a rapidly increasing interaction structure, leading to dense Jacobians and growing computational cost of numerical integration. In the multi-file regime, repeated simulation becomes computationally prohibitive for parameter exploration and inverse calibration. We formalize the simulator as a nonlinear operator $\mathcal{F} : \Theta \subset \mathbb{R}^p \to \mathbb{R}^B$ mapping mechanical parameters to the longitudinal cell length profile. We train multilayer perceptron (MLP) surrogates to approximate $\mathcal{F}$ using simulator-generated data. The learned surrogate replaces repeated ODE integration and enables fast prediction of spatial growth profiles. We evaluate generalization performance on held-out parameter configurations and demonstrate efficient parameter calibration to experimental profiles. We further analyze structural properties of the parameter-to-profile map, including local regularity and an intrinsic stochastic noise floor induced by random cell division. Our results show that neural operator approximation provides a scalable framework for accelerating analysis and inverse modeling of coupled high-dimensional biological growth dynamics.