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

TL;DR

This work introduces Hebbian Physics Networks (HPNs), a framework where the transport geometry on a graph dynamically co-evolves with the physical state through a local, three-factor Hebbian update to edge weights. By interpreting residuals as local thermodynamic forces and constructing a residual energy, the authors prove that the weight updates perform local gradient descent and drive the system toward thermodynamically consistent transport, recovering Onsager reciprocity near equilibrium. Demonstrations on diffusion and incompressible lid-driven cavity flow show emergent, physically plausible transport pathways and flow structures arising from random initialization without global optimization or solves. The approach reframes computation as a thermodynamic relaxation process with a learnable constitutive geometry, offering a scalable, physically grounded alternative to fixed-operator solvers and suggesting new avenues for multiscale and turbulence applications.

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 Onsagerś 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.

Figures (3)

• Figure 1: Schematic of the graph $\mathcal{G}$: each node $i$ interacts with its neighbors $j\in\mathcal{N}(i)$ through directed couplings $W_{ij}$; reciprocal edges $j\to i$ carry distinct couplings $W_{ji}$. $U_i$ and $U_j$ are the state variables at nodes $i$ and $j$, respectively.
• Figure 2: Residual-driven diffusion in a HPN (a) Evolution of the concentration field on an unstructured graph. (b) Spatiotemporal trajectory of the diffusion front obtained by tracking the $y$-coordinates of nodes crossing a threshold concentration. Red dashed lines correspond to the iteration indices of the snapshots in (a). (c) Mean residual $\langle R_c\rangle$ during early relaxation. Fluctuations decay as local imbalances are removed by the Hebbian update. (d) Lyapunov function $\mathcal{L}$ showing monotonic decrease with iteration. (e) Root-mean-square change in weights, $\Delta W_{\mathrm{RMS}}$, indicating rapid early adaptation followed by asymptotic stabilization of the constitutive operator. (f) Mean weight magnitude $\langle W\rangle$ approaching a steady value as the learned operator converges. (g) Histograms of the weight distribution $W_{ij}$ at selected iterations.
• Figure 3: Residual-driven flow formation in a HPN. (a) Streamline snapshots for the lid–driven cavity. From a nearly quiescent state, the network organizes the velocity into the characteristic recirculating pattern; pressure is shown as a background colormap, centered at zero. (b) Mean continuity, streamwise, and cross-stream residuals $\langle|R_c|\rangle$, $\langle|R_u|\rangle$, and $\langle|R_v|\rangle$, showing an early surge followed by rapid decay as imbalances are redistributed through the learned transport operator. (c) Lyapunov function $\mathcal{L}=\tfrac12\sum_i (R_{c,i}^2+R_{u,i}^2+R_{v,i}^2)$, increases until the networks "discovers" the vertical degree of freedom and bifurcates after reaching a maximum. (d) Root-mean-square weight changes $\Delta W_{u,\mathrm{RMS}}$ and $\Delta W_{v,\mathrm{RMS}}$, indicating an initial phase of strong plasticity followed by stabilization. (e) Mean weights $\langle W_u\rangle$ and $\langle W_v\rangle$ approaching steady values as the transport geometry settles. (f) Histograms of $W^{(u)}_{ij}$ and $W^{(v)}_{ij}$ at selected iterations, initially narrow, then broadening during adaptation and developing heavy-tailed structure associated with shear, before converging as the flow reaches steady state.

Theorems & Definitions (4)

• Lemma 1: State Dissipation / Onsager Process
• proof
• Lemma 2: Constitutive Dissipation
• proof