Training Neural Networks is ER-complete

Training neural networks is a fundamental problem in machine learning. An (artificial) neural network is a brain-inspired computing system. For an example consider Figure 1. Neural networks are modelled by directed acyclic graphs where the vertices are called neurons. The source nodes are called the input neurons and the sinks are called output neurons, and all other neurons are said to be hidden. A network computes in the following way: Each input neuron s receives an input signal (a real number) which is sent through all out-going edges to the neurons that s points to. A non-input neuron v receives signals through the incoming edges, and v then processes the signals and transmits a single output signal to all neurons that v points to. The values computed by the output neurons are the result of the computation of the network.