Why does my Inductor Get Warm? (Inverter Output Filter)
Almost certainly your inductor core is saturating. The currents into the 6.5 uF will be quite high and this is the reason. You have basically a series tuned circuit formed by 2x 470uH and a 6.5uF capacitor. 

This will have a resonant frequency of: -

f = \$\dfrac{1}{2\pi\sqrt{LC}} = \dfrac{1}{2\pi\sqrt{0.001\times 6.5\times 10^{-6}}}\$ =1974 Hz.

Ideally, you want the resonant frequency to be logarithmically half way between 50 Hz and 16 kHz and this would be more like 894 Hz so maybe you are a little close to the switching frequency. You could probably go lower - maybe towards 300Hz to improve this but, all the time the inductor or the cap is getting bigger.

Basically if you could raise your PWM frequency by 3 or 4 times you'd see an improvement I reckon.