Learning Nonlinear Hamiltonain Systems in a Suitable Quadratic Embeddings

  • Yildiz, Suleyman (Max Planck Institute DCTS)
  • Goyal, Pawan (Max Planck Institute DCTS)
  • Bendokat, Thomas (Max Planck Institute DCTS)
  • Benner, Peter (Max Planck Institute DCTS)

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Hamiltonian systems intrinsically have several properties; coupling between position and momenta variables, symplectic structure, and energy preservation are essential properties. In this talk, we present data-driven modeling for nonlinear Hamiltonian systems. One of the key ingredients of our approach is the lifting principle. That is, sufficiently smooth nonlinear dynamics can be written as quadratic systems in a lifted coordinate system. We extend this principle of nonlinear Hamiltonian systems with a hypothesis that nonlinear Hamiltonian systems can be transformed in another space via Symplectic transformation so that the Hamiltonian of the transformed system can be a cubic function; this would then result in a quadratic system. Given data, we learn such a transformation using auto-encoder, which mildly enforces symplectic as well. Hence, we do not require any second layer of approximation, for example, using (D)EIM, and yet preserve the Hamiltonian structure in the learned coordinate or space. Moreover, we discuss its extension to high-dimensional data. We illustrate the proposed methodology using various examples.