Can a collapsed lung improve on its own?

Yes. A healthy normal person with no particular risk factors, can get a spontaneous pneumothorax (collapsed lung). If the amount of air that leaked out was small, and the leak had sealed itself off, the doctor may decide to observe the patient and let the leaked air get reabsorbed by the body.
Potentially. A small spontaneous pneumothorax is often treated with observation. Your body can absorb the air forming the pneumothorax. If your pneumothorax is larger, symptomatic or increasing in size, it will require more invasive treatment.