An autoencoder conjugacy network to learn one-dimensional chaotic maps

We introduce a data-driven method for learning one-dimensional chaotic maps using an improved autoencoder architecture that incorporates a conjugacy layer in the latent space. The network learns an unknown conjugacy mapping the original chaotic map into latent space, where a known transformation to the tent map is applied to constrain the representation. This approach enforces dynamical principles while improving the accuracy of predicted orbits for one-dimensional chaotic maps that share the topological entropy of, and the single unimodal shape of, the standard logistic map. We demonstrate its effectiveness within this scope on continuous and one-dimensional piecewise chaotic maps, showing improved performance over traditional and recent deep learning architectures, and identify possible failure modes when the target map lies outside this conjugacy class.