Final answer:
A Koopman operator transforms non-linear dynamics into linear form in higher-dimensional space. The neural network is trained on time-series data to approximate this linear operator, aiding in predictions about system's behavior.
Step-by-step explanation:
The question you've asked involves finding a Koopman operator using a neural network for a given non-linear function. The Koopman operator is an infinite-dimensional linear operator that can be used to describe the evolution of observables in a dynamical system.
Essentially, it lifts the non-linear dynamics into a higher-dimensional space where the dynamics are linear. This approach of using a Koopman operator is particularly important in the field of dynamic mode decomposition (DMD) and is a key concept in modern dynamical systems theory.
To find a Koopman operator using a neural network, the first step would be to select or define a unique non-linear function that represents your dynamical system. Then, you would design a neural network capable of approximating the Koopman operator.
This usually involves setting up a deep learning framework where the network is trained on time-series data to learn the underlying linear dynamics that govern the evolution of the system in the lifted space. After training, the neural network can serve as a computational proxy for the Koopman operator, making predictions about the system's behavior.