A neural architecture termed GSPNN (Graph Shortest Path Neural Network) is introducedin which both inference and learning are carried out without backpropagation. Classificationis posed as a shortest-path problem over a layered directed acyclic graph (DAG). Each layerdefines a local softmax distribution over outgoing edges; per-edge cost is the (temperaturescaled) negative log-probability. Inference reduces to a Viterbi (min-sum) dynamic programover the graph. Learning proceeds by local, forward-only updates: for each training example,the shortest path to the true class is compared with the best competing class. If a margin isviolated, normalized per-node updates are applied that increase the probability of chosen (true)edges and decrease it for competing (wrong) edges. The updates require only forward signals(local probabilities and keys) and avoid gradients and backpropagation. On MNIST withPCA features, a compact configuration attains 94—96% test accuracy with sub-second epochtimes on a single Colab GPU due to fully vectorized updates and optional precomputation of per-layer keys. Algorithmic details, computational complexity, ablations (temperature and margin schedules, top-k negatives, EMA), limitations, and connections to Viterbi decoding, structured prediction, and local learning rules are discussed.