Non-invasive detection of hidden structural defects is a crucial task in civil engineering. One of the most successful applied techniques is the Full Waveform Inversion (FWI, [1]), which is an inverse method that retrieves subsurface material information using wave signals measured from the sensors mounted on the structure. In FWI, the subsurface material distribution is initialized at random, and then refined iteratively. In each iteration, the wave equation must be solved with the current guess for the material distribution. The result is compared to the given wave data on the object surface, and the guess is updated towards a better match to the true underlying material distribution. As part of FWI involves a typically large number of forward solutions for the wave equation---two in each iteration---efficient numerical solution methods in this step are crucial. In this contribution, we discuss a method to solve the wave equation using a neural network ansatz. Density scaling is employed to model voids. The solution to this wave equation is approximated by a finite linear combination of basis functions. The neural network activation functions themselves serve as basis functions, and the weights and biases of the neural network are acquired via a new sampling algorithm we introduced in [2]. Then, the only unknown components of this model are the time-dependent coefficients, which we determine by solving a system of linear second order Ordinary Differential Equations (ODE). We obtain accurate results up to 10,000 time steps, which is comparable to the accuracy of classical Finite Element Methods. Major benefits from our machine learning-based approach are that our method exhibits spectral convergence, i.e., an exponentially fast decrease of the error with linear increase of the number of basis functions. We can also easily deal with complicated geometries, because the approach is mesh-free: the activation functions of the neural network only require randomly sampled points over the geometry, not a structured mesh.