Most interesting systems, such as robots, fluids, and power grids, are nonlinear, which makes them hard to predict and control. Koopman theory offers a workaround: lift the system into a larger space where its behavior becomes linear, so the well-understood tools for linear systems apply. The difficulty is finding that space, since exact Koopman operators are rarely possible to derive by hand.
Existing approaches each have a weak spot. Programmatic methods solve for the operator directly, but they depend on system-specific choices and become expensive as problems grow. Learning methods train a neural network to find the lifted space, but they are sensitive to hyperparameters, most of all the number of dimensions the lifted space should have, which usually has to be guessed.
This paper combines the two. Semidefinite programming first finds a representation of the linear operator, and a network then learns only the mappings into and out of the space that operator acts on. The optimization handles the structure, and learning handles the parts that are hard to write down. This reduces the effort needed to learn the operator and removes the need to fix unknown structure, such as the lifted dimension, in advance.
