Physical Transformer

Abstract: Digital AI systems spanning LLMs, vision models, and generative architectures that operate primarily in symbolic, linguistic, or pixel domains. They have achieved striking progress, but almost all of this progress lives in virtual spaces. These systems transform embeddings and tokens, yet do not themselves touch the world and rarely admit a physical interpretation. In this work we propose a physical transformer that couples modern transformer style computation with geometric representation and physical dynamics. At the micro level, attention heads, and feed-forward blocks are modeled as interacting spins governed by effective Hamiltonians plus non-Hamiltonian bath terms. At the meso level, their aggregated state evolves on a learned Neural Differential Manifold (NDM) under Hamiltonian flows and Hamilton, Jacobi, Bellman (HJB) optimal control, discretized by symplectic layers that approximately preserve geometric and energetic invariants. At the macro level, the model maintains a generative semantic workspace and a two-dimensional information-phase portrait that tracks uncertainty and information gain over a reasoning trajectory. Within this hierarchy, reasoning tasks are formulated as controlled information flows on the manifold, with solutions corresponding to low cost trajectories that satisfy geometric, energetic, and workspace-consistency constraints. On simple toy problems involving numerical integration and dynamical systems, the physical transformer outperforms naive baselines in stability and long-horizon accuracy, highlighting the benefits of respecting underlying geometric and Hamiltonian structure. More broadly, the framework suggests a path toward physical AI that unify digital reasoning with physically grounded manifolds, opening a route to more interpretable and potentially unified models of reasoning, control, and interaction with the real world.