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The discovery of physical laws consistent with empirical observations lies atthe heart of (applied) science and engineering. These laws typically take theform of nonlinear differential equations depending on parameters, dynamicalsystems theory provides, through the appropriate normal forms, an "intrinsic",prototypical characterization of the types of dynamical regimes accessible to agiven model. Using an implementation of data-informed geometry learning wedirectly reconstruct the relevant "normal forms": a quantitative mapping fromempirical observations to prototypical realizations of the underlying dynamics.Interestingly, the state variables and the parameters of these realizations areinferred from the empirical observations, without prior knowledge orunderstanding, they parametrize the dynamics {\em intrinsically}, withoutexplicit reference to fundamental physical quantities.

No equations, no parameters, no variables: data, and the reconstruction of normal forms by learning informed observation geometries