where N hid denotes the prediction from the second network of DHPM that approximates f , the assumed right-hand side of Equation ( ) on the emergent space. We write the first loss term of Equation Let us now pause to recap our results: we transformed a "scrambled and shuffled" dataset with unknown physics into a generative model, while learning both emergent coordinates (for space and time) and the governing equations on those coordinates. Our technique differs from that of , which constructs a hidden physics-like model using DNNs but relies on a finite difference scheme. This requires an extra step of resampling the data on an equally spaced grid in emergent coordinates after interpolation. In our approach, we simultaneously train an interpolator (the first DNN) and learn the governing equation (the second DNN), computed employing the automatic differentiation to obtain ∂ Ĉ ∂ t from the first DNN. A practical benefit of using emergent coordinates is that the produced embeddings are inherently scaled, eliminating the need to scale the input data when building the NN.