PRONE: Petrov-Galerkin Operator Learning Unifies DMD, SINDy & Koopmanism
Data-driven dynamics often asks how to linearize a nonlinear system. We ask instead: which observables should be advanced, and where should their futures live? This leads to Petrov Regression Of Nonlinear Evolution (PRONE), a Petrov--Galerkin regression framework based on $ \Psi(\mathbf{X})K \approx \Phi(\mathbf{Y}), $ with distinct trial and test dictionaries. In this form, DMD, EDMD, SINDy, Koopman regression, sparse regression, and low-rank regression become variants of one construction: different dictionaries, weights, and constraints. We keep the linear algebra of Koopman learning, but drop the artificial requirement that a finite model map a dictionary into itself. With this asymmetry, eigenmodes are no longer the right objects. Instead, we use singular modes, which identify the observable combinations captured by the data, their projected futures, and the strength of the coupling between the two spaces. We identify the limiting projected operator and prove $L^2$ convergence of the resulting nonlinear predictor. We give examples from chaotic maps, the double gyre, a pitching-airfoil wake, and Lorenz--63, where PRONE outperforms DeepONets, Fourier neural operators, and reservoir computers with considerably fewer parameters. These examples show the same message: lift once, regress once, and let the singular structure reveal statistics, transport, prediction, and dimension.