Modeling Dynamical Systems on Manifolds

If we know that a dynamics stays on a Manifold we can reduce the dimensions down to that manifold. If we don’t know anything about the problem we can use Autoencoders to reduce the dimensions automatically.

So we try to learn two transformations $g$ and $g^{-1}$ based on our data and we can learn the dynamical system on the transformed data $g(x)$.

We can then write $y_t=g^{-1}\circ f\circ g\left(y_{t-1}\right)$ for the high dimensional flow.

We can now combine this Autoencoder approach with ECNNs or HCNNs such that the network will learn the dynamics on a lower dimensional Manifold and the Autoencoder will transform the forecast back to the higher dimensional output.