The estranged couple, who divorced in 2004, have two children together -- daughter Maya, 15, and son Roan, 11, reported Us magazine.
"Gattaca was a great moment in my life and a great moment in Uma Thurman's life, and we fell madly in love and wanted to do all that. We tried to be as optimistic as we could be: we all want to believe in love and the possibility of love.
Though the 'Purge' actor never claimed to have cheated on Thurman, Hawke did go on to marry the estranged couple's nanny Ryan Shawhughes in 2008.
"My relationship with my present wife is thrilling to me and I'm committed to it. But neither she nor I know what shape the future will come in. Sexual fidelity can't be the whole thing you hang your relationship on. If you really love somebody you want them to grow, but you don't get to define how that happens. They do," he added.
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