Digimon Adventure Movie

Same director, different plots - that is unless you happen to be watching the American version which manages to shove three Digimon movies into one. Since this is the first of the three movies shoved into one, this recommendation is being made based on the family elements that show up in both movies, something which is a strong point for both series, with the first Digimon movie focusing on the siblings Tai and Kari while Summer Wars - well, its one of the biggest families I've seen in Anime in such a short period of viewing, and yet the elements still end up working. The Digimon movie is also an earlier work, so those watching can see how the director has progressed over time if they so want. read more 

Both directed by Mamoru Hosoda, taking on the perspective of extremely young infants going through fantastical situations 

At first you may not think there are any similarities, but the art style is pretty similar in both, and they both have themes to do with the internet and the digital world, but Serial Experiments Lain has darker qualities 

Both series that have children with devices that can call forth creatures to fight other creatures, but something that keeps Onmyou Tai Senki different from Digimon is the effect of losing a monster has on it's child.