Yes, I had the same problem on my 2013 mini, Core I5/2, 500Gb SSD. Some things that got me past this that may help you:
- Recovery Mode: First power up (down first if it is on) and then hold down Command-R as soon as you hear the <bong!> at boot-up. The Recovery Menu should come up, which will include options to reinstall Catalina or restore from Time Machine Backup. I tried reinstalling on the SSD which kept crashing during install and powering off, followed by hanging during power on. So, after that, I tried reinstalling to an old USB hard drive I had. Slow, but it worked. Still, that was slo-o-ow compared to SSD.
- Restore from Time Machine Backup to go back to Mojave: Selecting this option from the Recovery Menu will show you the dates and now OS version, which is great and allowed me to select the last backup under Mojave. Bonus that now my 32-bit apps (MS Office) now work again. Mojave was so great... think maybe I'll just stop here.
- If you happen to have a different disk that you can boot from, simply hold down the Option key when you here the <bong!>. This will bring up a list of possible boot drives and allow you to select another drive or partition to boot from.
Honestly, I've been so disappointed in Catalina's install and poor regression testing (Mojave wasn't awesome either, but at least I could get it to work, eventually) that I will no longer install on my boot drive but always install first to a test drive, so that I can switch back if it's buggy.