"We gave everything last night to make our fans proud and I'm sorry it wasn't enough to bring the trophy home for you," Walker wrote on social media.
"We will be back stronger. Thanks for all your support."
Ilkay Gundogan also posted: "It damn sucks to post after a big final defeat, we've been so close.
"It hurts so much, today even more than yesterday to be honest. But I don't want to miss the chance to say thank you for all your support throughout the season."