Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings
This work addresses the challenge of modeling nonlinear Hamiltonian systems for applications in physics and engineering, but it is incremental as it builds on existing lifting and symplectic methods.
The authors tackled the problem of learning Hamiltonian systems from data by proposing a framework that uses a lifting hypothesis to represent nonlinear Hamiltonian systems as quadratic dynamics with cubic Hamiltonians, resulting in long-term stability and low model complexity, as demonstrated on both low-dimensional and high-dimensional systems.
We present a framework for learning Hamiltonian systems using data. This work is based on a lifting hypothesis, which posits that nonlinear Hamiltonian systems can be written as nonlinear systems with cubic Hamiltonians. By leveraging this, we obtain quadratic dynamics that are Hamiltonian in a transformed coordinate system. To that end, for given generalized position and momentum data, we propose a methodology to learn quadratic dynamical systems, enforcing the Hamiltonian structure in combination with a weakly-enforced symplectic auto-encoder. The obtained Hamiltonian structure exhibits long-term stability of the system, while the cubic Hamiltonian function provides relatively low model complexity. For low-dimensional data, we determine a higher-dimensional transformed coordinate system, whereas for high-dimensional data, we find a lower-dimensional coordinate system with the desired properties. We demonstrate the proposed methodology by means of both low-dimensional and high-dimensional nonlinear Hamiltonian systems.