The common name for this power factor correction circuit is a Boost Converter. In most applications, it is used as a DC to DC step-up converter, to get a high-voltage DC source to use inside a device powered by low voltage DC (laptop LCD backlight, solar powered calculator, electric fence controller, etc.) without using up much space.
See
here for theory of operation. Most boost converters run at high frequency, making them more difficult to troubleshoot. This circuit, however, is synchronized with the power line, so it is easy to tell if everything is working as it should.
In the mini-split, it may or may not be being used to step up the power source. The main purpose of the circuit is to absorb all of the noise created by the drives for the compressor and fan motor, and to draw power from the AC line as efficiently as possible.The big electromagnet (reactor) acts as a shock absorber between the mains AC rectifier and the DC power supply circuit. The transistor is controlled so that it switches on when the AC line is close to its peak voltages (both positive and negative). This shorts out the reactor coil and builds a magnetic field in its core.
When the AC line passes its peak voltage values, the transistor switches off. The magnetic field in the coil collapses (due to the reduced current draw) and the inductor passes this magnetic energy (as counter-emf or back-emf) through the rectifier diode to the capacitor bank. When the energy stored in the capacitor bank is drawn off (by mainly the variable-speed compressor), the rectifier diode isolates all of the current spikes and high-frequency noise from the mains line, making the unit appear as a mainly resistive load to the power meter at the service entrance. Any small amount of noise that is passed while the diode is conducting is absorbed by the reactor before it makes its way to the power line.
Bottom line: apparent power (VA) becomes much closer to true power (W) and the unit costs less to run.
Hope this helps.