Hebbian Physics Networks: A self-organizing computational architecture based on local physical laws

Physical transport processes organize through local interactions that redistribute imbalance while preserving conservation. Classical solvers enforce this organization by applying fixed discrete operators on rigid grids. We introduce the Hebbian Physics Network (HPN), a computational framework that replaces this rigid scaffolding with a plastic transport geometry. An HPN is a coupled dynamical system of physical states on nodes and constitutive weights on edges in a graph. Residuals, local violations of continuity, momentum balance, or energy conservation, act as thermodynamic forces that drive the joint evolution of both the state and the operator (i.e., the adaptive weights). The weights adapt through a residual-modulated anti-Hebbian rule, which we prove constitutes a strictly local gradient descent on the residual energy. This mechanism ensures thermodynamic consistency: Near equilibrium, the realized transport operator converges to a symmetric, positive-definite form, reproducing Onsager's reciprocal relations without explicit enforcement. Far from equilibrium, the system self-organizes into transport topologies within a thermodynamically admissible class that redistribute imbalance through adaptive local geometry. Unlike optimization-based approaches that impose physics through global loss functions, HPNs embed conservation intrinsically: Transport is restored locally by the evolving operator itself, without a global Poisson solve or backpropagated objective. We demonstrate the framework on scalar diffusion and incompressible lid-driven cavity flow, showing that physically consistent transport geometries and flow structures emerge from random initial conditions solely through residual-driven local adaptation. HPNs thus reframe computation not as the solution of a fixed equation, but as a thermodynamic relaxation process in which the constitutive geometry and physical state coevolve.