Breaking the cache? Any function, callback filter etc.

I need to find a way to break the cache for a user systematically within either a javascript or php function. We are trying to provide an ad lite experience for subscribed users. The problem is when after they login to Piano (not wordpress), the cached page, with the ads, remains, at least until the cache naturally breaks. This has been extremely frustrating. This seems to be especially a problem on mobile devices.

Any help to figure this out would be very appreciated.