Many physical systems exhibit limit-cycle oscillations that can typically be modeled as stochastically driven self-oscillators. In this work, we focus on a self-oscillator model where the nonlinearity is on the damping term. In various applications, it is crucial to determine the nonlinear damping term and the noise intensity of the driving force. This article presents a novel approach that employs a deep operator network (DeepONet) for parameter identification of self-oscillators. We build our work upon a system identification methodology based on the adjoint Fokker–Planck formulation, which is robust to the finite sampling interval effects. We employ DeepONet as a surrogate model for the operator that maps the first Kramers–Moyal (KM) coefficient to the first and second finite-time KM coefficients. The proposed approach can directly predict the finite-time KM coefficients, eliminating the intermediate computation of the solution field of the adjoint Fokker–Planck equation. Additionally, the differentiability of the neural network readily facilitates the use of gradient-based optimizers, further accelerating the identification process. The numerical experiments demonstrate that the proposed methodology can recover desired parameters with a significant reduction in time while maintaining an accuracy comparable to that of the classical finite-difference approach. The low computational time of the forward path enables Bayesian inference of the parameters. Metropolis-adjusted Langevin algorithm is employed to obtain the posterior distribution of the parameters. The proposed method is validated against numerical simulations and experimental data obtained from a linearly unstable turbulent combustor.