Neural operators like DeepONets [1] have recently emerged as powerful tools for approximating nonlinear mappings between function spaces, particularly the solution operators of PDEs. However, their reliance on deep neural architectures—whose training has been shown to be NP-hard, combined with extensive parameter and hyperparameter tuning demands substantial computational resources. This makes it challenging to achieve high accuracy under limited resources. To address these limitations, we have proposed RandONets [2]: shallow networks that first embed the input space via random projections exploiting tailor-made linear-algebraic numerical methods for training. Conceptually, RandONets’ use of random projections for input embeddings draws directly on the celebrated Johnson–Lindenstrauss lemma. We prove that RandONets retain the universal approximation capabilities of their deeper counterparts for linear and nonlinear operators of PDEs. In particular, the performance of RandONets is evaluated via a linear Diffusion-Advection PDE, the viscous Burgers, and the Allen-Cahn PDE. Our numerical experiments reveal that RandONets outperform deep-neural-networks by orders of magnitude in both approximation accuracy and computational cost. Additionally, when applied to linear operators, RandONets can attain up to machine-precision accuracy. In essence, our results demonstrate that numerical-analysis -informed “light” neural architectures can deliver faster and more precise operator approximations than traditional deep models. REFERENCES [1] Lu, L., Jin, P., Pang, G., Zhang, Z., & Karniadakis, G. E. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature machine intelligence (2021) 3(3): 218-229. [2] Fabiani, G., Kevrekidis, I. G., Siettos, C., & Yannacopoulos, A. N. RandONets: Shallow networks with random projections for learning linear and nonlinear operators. Journal of Computational Physics (2025) 520: 113433.