This talk explores machine learning heuristics for complex algebraic decision problems, focusing on the Word Problem and its role in post-quantum cryptography. Within the Baumslag-Solitar group BS(1,2) and Artin groups, unreduced words are mapped into graphs using their generators and defining relations. A Graph Neural Network is trained on contrastive triplets (W, P, N) to embed these graphs into a 128-dimensional unit sphere. The model learns to cluster algebraically equivalent words (W and P) while strictly separating non-equivalent decoys (N). This learned embedding space is then leveraged to successfully attack the Wagner-Magyarik cryptosystem. Additionally, a variant graph neural network architecture accurately predicts the reduced geodesic length of randomly generated sequences in both groups. The talk concludes by extending these graph-based architectures to Right-Angled Artin Groups (RAAGs) and random Artin groups to evaluate the Nielsen equivalence of generating sets of subgroups.