We discuss computational challenges that arise when training Fourier neural operators to predict wave fields in acoustic wave problems, in particular, full waveform inversion (FWI) for non-destructive testing [1]. FWI involves a typically large number of forward solutions for the wave equation, and so we originally intended to use operator learning to replace the classical numerical solution methods in this step. After an introduction to the main challenges in FWI and our issues with training the Fourier neural operators, we discuss an alternative, neural-network based method to solve the wave equation. In this alternative, the neural network activation functions are used as basis functions, and the weights and biases of the neural network are acquired via a backpropagation-free sampling algorithm we introduced in [2]. We then discuss how architectures like neural operators (DeepONet, Fourier neural operators) can be trained without backpropagation by using random features [2]. This avoids most of the challenges from iterative gradient descent, but introduces new obstacles to overcome. [1] L. Herrmann, T. Bürchner, F. Dietrich, and S. Kollmannsberger (2023). On the use of neural networks for full waveform inversion. Computer Methods in Applied Mechanics and Engineering, 415:116278. [2] E. L. Bolager, I. Burak, C. Datar, Q. Sun, and F. Dietrich (2023). Sampling weights of deep neural networks. Advances In Neural Information Processing Systems, NeurIPS 2023.