Schematic of the autoencoder and latent ODE system. The upper part of the sketch represents the autoencoder, with both encoder and decoder deep neural networks. Each rectangle represents a layer of the deep neural network, linked together by weights W and biases b (omitted for the sake of clarity). The input to the encoder is , with N nodes (dimensions), connected to a sequence of hidden layers h_i with decreasing dimensionality/number of nodes, until reaching the layer with M nodes (dimensions), where the maximum compression is obtained. The decoder is symmetric w.r.t. the encoder, with layers of increasing dimensionality, ending with an output layer of N nodes (dimensions). We note that, in our case, we have six hidden layers instead of the 4 shown in this sketch. In the lower part of the sketch, we show the latent ODE system that uses as inputs and produces as output, both with M dimensions. This additional neural network is controlled by the parameters p (one for each latent reaction), and has an analytical representation . The obtained latent space derivatives are decoded to the target derivatives with the same procedure as of Eq. (5).