I am preparing for an exam in neural networks. As an example for self-organizing maps they showed the inverted pendulum problem where you want to keep the pole vertical:
Now the part which I don't understand:
$$f(\theta) = \alpha \sin(\theta) + \beta \frac{\mathrm{d} \theta}{\mathrm{d} t}$$ Let $x= \theta$, $y=\frac{\mathrm{d} \theta}{\mathrm{d} t}$, $z=f$.
Solution with SOM:
- three-dimensional surface in $(x,y,z)$
- adapt two-dimensional SOM to surface
- Method of control
- For a given $(x,y)$ find neuron $k$ for wich $w_k = [w_{k1}, w_{k2}, w_{k2}, w_{k3}]$
- $f(\theta)$ is then $w_{k3}$
I guess we use the SOM to learn the function $f$. However, I would like to understand where $f$ comes from / what it means in this model.