The author uses combinations of the K-L decomposition and neural networks to obtain the intrinsic or true dimension of two PDEs, namely, the 1-d K-S equation and the 2-d N-S equations. For the 1-d K-S equation, he investigates one particular dynamical behavior which, in phase space, is represented by a heteroclinic connection. As for the 2-d N-S equation, a quasi-periodic behavior is examined. In both studies, the powers of neural networks in extracting the intrinsic dimension of both dynamics are presented.