Hamiltonian Operator Inference

[shanemcq18]

New feature: Hamiltonian Operator Inference from the paper Hamiltonian operator inference: Physics-preserving learning of reduced-order models for canonical Hamiltonian systems by Harsh Sharma (@harsh5332392), Zhu Wang, and Boris Kramer (@bokramer). The goal is to use Operator Inference for a canonical Hamiltonian system to learn a ROM that

1. is a canonical Hamiltonian system;
2. retains the physical interpretation of the state variables and preserves the coupling structure; and
3. respects the symmetric property of structure-preserving space discretizations.

@harsh5332392 will take the lead on this. To begin, the main steps will be creating a SymplecticBasis class (cotangent lift algorithm) and a HamiltonianModel class that does the constrained optimization in fit() and symplectic integration in predict().

Suggested implementation steps:

Basis

• Create a new file, /src/opinf/basis/_symplectic.py
• Write a SymplecticBasis class that inherits from opinf.basis.LinearBasis and implement fit(). Or, you might be able to do this quickly by inheriting from the PODBasisMulti class, which represents a block diagonal POD (one POD for each variable).
• Import the new class in /src/opinf/basis/__init__.py.
• Write and run tests for the new classes in a new file /tests/basis/test_symplectic.py.
• Compile the docs (make docs) and check that the automatically generated documentation page looks good.
• If possible, write a short section about this class in docs/source/guides/reduction.md. We should probably turn this into a notebook that shows the different kinds of basis functions you get from POD and the symplectic approach.

Model Class

• Create a new file, /src/opinf/models/multi/_hamiltonian.py.
• Write a HamiltonianModel class in the new file.
  • The fit() method should take in the data matrices, do the constrained optimizations, and initialize the operators of the ROM.
  • Implement the predict() method with a symplectic integrator.
• Write and run tests for the new class in a new file /tests/models/multi/test_hamiltonian.py
• Compile the docs and check that the automatically generated documentation page looks good.
• Write a tutorial as a new Jupyter Notebook, docs/source/tutorials/hamiltonian.ipynb with the linear wave equation example.