<p>This paper presents a deep learning-based numerical algorithm named Deep Rayleigh Quotient Iteration (DRQI), which extends the classical, high order convergent scheme in matrix eigenvalue problems, Rayleigh quotient iteration (RQI), to large scale discretization and further continuous linear differential operator through neural network parametrization. By embedding the RQI structure into a semi-implicit relaxation loss, which couples successive Rayleigh updates, preserves the convergence behavior of RQI in the extended functional setting. And the mean squared residual of the governing equation serves as a stopping criterion to avoid overfitting. The performance of DRQI is systematically analyzed from both theoretical and experimental perspective. Theoretical analysis proves that the DRQI inherits the higher-order convergence property of RQI and provides the guidelines for network design and standards for sampling density, establishing the consistency between discrete RQI and its continuous operator counterpart. While the benchmark experiments across the Laplace and Fokker-Plank operators demonstrate that DRQI is more robust in lower dimensional cases, outperforming the deep Ritz method and the inverse power method neural network in accuracy of eigenpairs estimation. The results also indicate that the relaxation factor enables control over the early training stages, enhancing adaptability in practical applications. Finally, DRQI is applied to the 1D neutron transport equation, achieving errors below 10^–5 within 1500 training epochs when estimating the effective multiplication factor. These results highlight DRQI as a general, higher-order convergent, and practically robust eigenvalue problem solver with strong potential for engineering applications.</p>