I'm so sorry. What a horribly painful thing to be dealing with.
Please don't assume you will lose the house. His name is on it, and he is still responsible for that, and his children. He cannot just walk away and leave you with whatever you have to take care of everything.
I know you want him back, but right now you need to see an attorney. You need to get some child support in place, get legal custody of your kids (yes, right now he can take them anywhere if he wants), and find out what your rights are, and what you should do. Again, if he changes his mind and wants to come home, you can pull the plug on all of this. Until then, you need to protect you and your children.
If you have joint bank accounts, go take half of the money out, now, and put it in an account in your name only. If you have direct deposit at work, have your pay moved to the account that is only in your name. Do NOT assume he won't wipe you out. It happens, a lot. You always thought he wouldn't cheat, or leave. He has done both. Please don't make the mistake of thinking he won't do other things if those things are in his best interest. Right now he is all about him - completely selfish. So please, protect yourself.