We investigate a machine learning based classification of noise acting on a small quantum network with the aim of detecting spatial or multilevel correlations, and the interplay with Markovianity. We control a three-level system by inducing coherent population transfer exploiting different pulse amplitude combinations as inputs to train a feedforward neural network. We show that supervised learning can classify different types of classical dephasing noise affecting the system. Three non-Markovian (quasi-static correlated, anti-correlated and uncorrelated) and Markovian noises are classified with more than 99% accuracy. On the contrary, correlations of Markovian noise cannot be discriminated with our method. Our approach is robust to statistical measurement errors and retains its effectiveness for physical measurements where only a limited number of samples is available making it very experimental-friendly. Our result paves the way for classifying spatial correlations of noise in quantum architectures.