Hebbian Physics Networks: A Self-Organizing Computational Architecture Based on Local Physical Laws

TL;DR

The paper introduces Hebbian Physics Networks, a self-organizing computational framework that adapts local transport geometries based on physical residuals, ensuring conservation laws and emergent flow structures without global optimization.

Contribution

It presents a novel, local Hebbian learning-based approach for self-organizing physical transport networks that inherently satisfy conservation laws and adapt to complex flow conditions.

Findings

Emergence of physically consistent transport geometries from random initial conditions.

Convergence to symmetric, positive-definite operators near equilibrium.

Successful demonstration on scalar diffusion and incompressible flow problems.

Abstract

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 three-factor Hebbian rule, which we prove constitutes a strictly local gradient descent on the residual energy. This mechanism ensures thermodynamic stability: near equilibrium, the learned operator naturally converges to a symmetric, positive-definite form, rigorously reproducing Onsager\'s reciprocal relations without explicit enforcement. Far from equilibrium, the system undergoes a self-organizing search for a transport topology that restores global coercivity. 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 where the constitutive geometry and physical state co-evolve.