In estimation theory, the Fisher information matrix (FIM) is a fundamental concept from which we can infer the well-known Cramér-Rao bound. A closed-form expression of the FIM is often intractable due to the lack or sophistication of statistical models. In this paper, we propose a Fisher Information Neural Estimator (FINE) based on neural networks and a relation between the f-divergence and the Fisher information. The proposed method produces an estimate of the FIM directly from observed data. It does not require knowledge or an estimate of the probability density function (pdf), and is therefore universally applicable. The proposed FINE is applicable for not only deterministic parameters but also random parameters. We show via numerical results that the proposed FINE can provide a highly-accurate FIM estimate with a low-computational complexity. Furthermore, we also propose an accelerated FINE version which can be used for scenarios with a high parameter dimension. Finally, we develop an algorithm to choose an appropriate size of the employed neural networks.