I have been looking into reducing the dimensionality of (nonlinear ordinary differential) equation systems in order to reduce online computation time. I have interpreted 'dimensions' as the number of equations or unknown variables in the systems, but it seems that this is not always the true meaning.
Now this lecture shows multiple examples of single equations or small systems of a couple of equations that are said to have hundreds or thousands of dimensions, so what does 'dimensionality' actually mean here?
For example the FitzHugh-Nagumo system supposedly has 1024 dimensions:
$$ \epsilon v_t (x,t) = \epsilon ^2 v_xx (x,t) + f(v(x,t))-w(x,t)+c \\ w_t(x,t) = bv(x,t)-\gamma w(x,t) + c $$
This just leads to so much confusion since I am not sure if any of these 'dimensionality 'reduction' methods can actually be used for reducing simulation time. Thanks for your time.