Neural networks are a prominent tool for identifying and modeling complex patterns, which are otherwise hard to detect and analyze. While machine learning and neural networks have been finding applications across many areas of science and technology, their use in decoding ultrafast dynamics of quantum systems driven by strong laser fields has been limited so far. Here we use standard deep neural networks to analyze simulated noisy spectra of highly nonlinear optical response of a 2-dimensional gapped graphene crystal to intense few-cycle laser pulses. We show that a computationally simple 1-dimensional system provides a useful "nursery school" for our neural network, allowing it to be retrained to treat more complex 2D systems, recovering the parametrized band structure and spectral phases of the incident few-cycle pulse with high accuracy, in spite of significant amplitude noise and phase jitter. Our results offer a route for attosecond high harmonic spectroscopy of quantum dynamics in solids with a simultaneous, all-optical, solid-state based complete characterization of few-cycle pulses, including their nonlinear spectral phase and the carrier envelope phase.