Epigraphs are like little appetizers to the great entrée of a story. They illuminate important aspects of the story, and they get us headed in the right direction.
Out beyond ideas of wrongdoing and rightdoing, there is a field. I'll meet you there. –Rumi
I believe in nothing but the holiness of the heart's affections and the truth of the imagination. –John Keats
Where there is great love, there are always miracles. –Willa Cather
It takes courage to grow up and become who you really are. – e. e. cummings
What's up with the epigraph?
Some authors use one epigraph. Jandy Nelson used four. This relative excess at the very beginning of the novel sets the stage for its protagonists' outsized personalities. (Noah and Jude are a whole lotta character.) At the same time, the epigraphs introduce some of the book's most important ideas.
The Rumi quote suggests that the key to mending a relationship lies somewhere beyond blame. This bears directly on the troubled relationship of Noah and Jude, who've treated each other badly and have to figure out a way to get over it.
The Keats quote says that art and emotion are the most sacred and important things in the whole world. Yup, the book definitely seems to endorse that idea.
The Cather quote suggests that love has amazing—even magical—powers, another biggie in the universe of the book. (Hello, Grandma Sweetwine.)
Finally, the Cummings (sorry, cummings) quote recognizes the hardships involved with coming of age. Noah and Jude will face plenty of those. Trust.