Does it make sense to combine PCA with an artificial neural network?
Neural networks are actually extremely effective at performing dimensionality reduction. A great example is word2vec, which applies a shallow neural network to reduce inputs on the order of several million features (i.e. unnormalized text) to a 30-150 dimensional vector via a process that is mathematically analogous to matrix factorization (which is the class of techniques PCA belongs to). Autoencoders function very similarly to word2vec: if you're planning on using an autoencoder to learn an embedding of your data, I wouldn't expect you'd gain anything from applying PCA first rather than using an autoencoder (or something similar) to learn a better embedding for your data that isn't constrained by the assumptions of PCA.