Extracting conservative equations from nonconservative state data

Data-driven equation identification methods enable automated discovery of systems governing equations in differential, integral, or variational forms, provided with state data for autonomous systems while state and excitation data for forced systems. These methods for equation identification can undoubtedly outperform those for algebraic relation identification, however, they essentially belong to the paradigm of data fitting, viz., compulsorily establishing mathematical relations within data. In this work, we tackle a different problem of extracting hidden conservative equations only from nonconservative state data collected from randomly/deterministically excited, dissipative dynamical systems; that is, we attempt to distill intrinsic structures without any excitation information. A 3E framework is implemented, namely, embedding Euler-Lagrangian equations in systems, eliminating the influence of nonconservative factors by orthogonalization, while extracting Lagrangians (or Lagrangian densities) for discrete (or continuous) systems. Three illustrative examples, including the Duffing oscillator, the cart-pendulum system, and the Euler-Bernoulli beam, are investigated to show how this method can achieve simplicity in complexity.