Look at your court order/judgment. What does it say about removal of the children from the U.S.? Who's got the children's passports? Have you notified the passport service regarding issuance of new passports for the children? You may wish to speak to an experienced family law attorney NOW, before the trip, so as to get some new orders in place if necessary.
The prevention of a parent (and child) from LEAVING the U.S. is on the airline companies and their respective policies and less about federal law; it's coming IN to the U.S. where the Federal government has stringent procedures. I agree that you should consult with a family attorney, and since you're in LA, it shouldn't be hard to find a good one who handles international aspects of family law. If you were divorced in California (or other state in the United States), review that last order (decree) where your parental rights and duties, parenting time, and any restrictions are described. And don't forget to take that (and any other paperwork from the court, your ex-wife, etc.) to the consultation with the attorney. They will be able to explain your rights, and what options that you have to protect the safety of your children. Best of Luck!
If you do not have appropriate restraining orders, hire an attorney in Los Angeles. Although supposedly for a child to travel outside the US it must be with permission of both parents.
My name is Stephen R. Cohen and have practiced since 1974. I practice in Los Angeles and Orange County, CA. These answers do not create an attorney client relationship. My answers may offend I believe in telling the truth, I use common sense as well as the law. Other state's laws may differ.. There are a lot of really good attorneys on this site, I will do limited appearances which are preparation of court documents it is , less expensive. However generally I believe an attorney is better than none.