This study demonstrates a rationally enriched Chebyshev (REC) trunk for deep operator network (DeepONet) surrogate models of singularly perturbed and high-Péclet transport problems whose solution profiles are characterized by thin localized boundary or wall layers. The REC trunk combines Chebyshev polynomial dictionary elements with rational dictionary elements constructed using the adaptive Antoulas-Anderson (AAA) algorithm. Over five independent training runs, the resulting REC-trunk DeepONet is evaluated against a vanilla DeepONet and a Chebyshev-trunk DeepONet whose prescribed dictionary consists only of Chebyshev polynomials across three problems whose singular perturbation parameters are diffusion-to-advection ratios: a singularly perturbed scalar boundary-value problem (BVP), the thermal entrance problem with a prescribed wall temperature, and the concentration entrance problem with an absorbing wall. Across the held-out test profiles, the REC-trunk DeepONet improves over the vanilla DeepONet and remains comparable to the Chebyshev-trunk DeepONet in predicting the scalar profile, with its clearest advantage over the Chebyshev-trunk DeepONet appearing when the perturbation parameter lies between $1.00\times10^{-4}$ and $1.78\times10^{-4}$, where it reduces the profile-error metrics by up to $19.5\,\%$ relative to the Chebyshev-trunk DeepONet. In predicting the wall-normal temperature and concentration profiles, the REC-trunk DeepONet reduces the profile-error metrics by up to $60.2\,\%$ and $32.2\,\%$ relative to the vanilla and Chebyshev-trunk DeepONets, respectively, while suppressing artificial near-wall oscillations as the Péclet or mass-transfer Péclet number ranges from $10^{2}$ to $10^{4}$.
The paper presents a new parameter-efficient adaptation method called ChebyMA (Chebyshev Manifold Adaptation). ChebyMA adopts weight matrices through a multi-surface superposition of Chebyshev polynomial bases evaluated on learnable coordinates and combined via trainable coefficient matrices, replacing standard linear projections with highly expressive continuous function approximation. Theoretically, we establish an Approximation Expressivity Theorem, proving from the perspective of function approximation theory that single-manifold ChebyMA guarantees convergence in Frobenius norm error of reconstruction. Besides, drawing on Kolmogorov $n$-width intuition, we demonstrate the expressive advantages of multi-manifold superposition ($S > 1$) in decoupling high-dimensional complex features. Experimental results on Computer Vision CIFAR datasets(CIFAR-10, CIFAR-100)\cite{CIFAR} and Natural Language Processing (AG News, SST-2) datasets demonstrate that ChebyMA consistently achieves a superior parameter-accuracy Pareto front compared to standard full-parameter fine-tuning, LoRA\cite{LoRA}, TLoRA\cite{TLoRA}, and StelLA\cite{StelLA}. ChebyMA significantly outperforms other tested methods in tested datasets, validating its solid theoretical foundation for generality with purely vectorized computations.