3+1D Octonionic Field Theory: A Lagrangian PINN Approach to Non-Associative Spacetime Dynamics

This paper introduces a novel Physics-Informed Neural Network (PINN) framework designed to simulate octonionic scalar field dynamics within a 3+1 dimensional spacetime. The research primarily addresses the "associativity collapse" problem, a computational phenomenon where gradient descent naturally drives non-associative models toward simpler, associative quaternionic subspaces.To overcome this, the work presents several methodological breakthroughs:• Spatial Associator Loss: A key innovation that computes the associator across distinct, independently sampled spacetime points rather than at a single local point, providing a meaningful signal for smooth fields.• Asymmetric Penalty Functions: The implementation of a 100× penalty for fields falling below a target associativity threshold, effectively inverting the loss landscape to prevent algebraic collapse.• Learnable Octonionic Modulation: The use of modulation weights that favor pure octonionic components during the training process.The resulting model achieves a stable non-associative state where 94.7% of the field energy resides in pure octonionic components (e4–e7), with only 5.5% in the quaternionic subspace. The work provides a comprehensive Lagrangian formulation, derives the stress-energy tensor, and identifies 14 conserved G2 Noether currents corresponding to the automorphism group of the octonions.By demonstrating that neural networks can learn and preserve exotic, non-associative algebraic structures, this research opens new computational pathways for exploring M-theory compactifications, non-local field couplings, and the potential octonionic foundations of the Standard Model.