We first reconstruct the scrambled advection-diffusion dataset from Figure using the Questionnaire algorithm. This process yields several eigenvectors (diffusion coordinates) for each tensor axis: space, time, and system parameters. In this illustrative example, the first eigenvector of each axis accurately parameterizes space, time, and system parameters, as demonstrated in Figure . Although the dataset was generated by varying both v and D, our data-driven algorithm successfully identifies the single governing parameter: the Péclet number. Given that we sampled space and time along a 1D curve, we anticipate and confirm that the algorithm produces a 1D embedding for space and time, each of them one-to-one with the corresponding (true) coordinates. Figure D.11 presents a comparison of true spacetime (a) with the reorganized data obtained using emergent embeddings (b) for a single experiment with a Péclet number of 2. From the reorganized data on the emergent space, we make the following observations: • The embedding, as a nonlinear function of the true coordinate, introduces deformations in the space-time plots. This implies that one cannot simply rewrite the known physics in terms of the emergent coordinates as demonstrated in the PINNs section of Appendix C.3.