<p>Weighted essentially nonoscillatory (WENO) schemes are a popular class of numerical methods for solving hyperbolic conservation laws. Since WENO schemes are designed to deal with problems with both complicated solution structures and discontinuities / sharp gradient regions, their sophisticated nonlinear properties and high-order accuracy require more operations than many other schemes. The methodology of hybrid methods is an effective approach to decrease the computational costs and dissipation errors of WENO schemes and achieve better resolution. One of the key components for the success of hybrid WENO schemes is the application of a robust and efficient troubled-cell indicator, which detects the computational cells where the solution loses regularity. Recently, troubled-cell indicators based on artificial neural networks (ANNs) have been developed in the literature, which have the advantage of less dependence on tunable parameters and being more robust than many traditional troubled-cell indicators, and such ANN based troubled-cell indicators have been applied to hybrid finite difference WENO schemes effectively. Motivated by these works, in this paper we develop a hybrid finite volume WENO method with an ANN based troubled-cell indicator for solving hyperbolic conservation laws. While the finite difference WENO schemes are more efficient than the finite volume WENO schemes for multidimensional problems on uniform grids, the finite volume WENO schemes have the advantage such as being flexible and easy to apply on nonuniform grids. We introduce an ANN based troubled-cell indicator by constructing a multilayer perceptron (MLP) model, one of the most common ANN models. The third-order WENO scheme is focused in this paper. Extensive numerical experiments for solving various scalar equations with both convex and non-convex cases, and the Euler systems of equations on uniform and nonuniform grids of one-dimensional (1D) and two-dimensional (2D) domains, are performed to show the accuracy and nonlinear stability of the proposed hybrid finite volume WENO scheme with the MLP troubled-cell indicator. Significant accuracy improvement and computational-cost saving over the original WENO scheme are observed. Numerical experiments and comparisons with the widely-used KXRCF indicator also show the good performance of the MLP troubled-cell indicator. Although the MLP troubled-cell indicator is trained on uniform grids, it performs very well on nonuniform grids obtained by randomly perturbing uniform grids.</p>