Hebbian Physics Network (HPN)

• Hebbian Physics Network (HPN) is a class of models unifying neural associative memory with physical laws through Hebbian learning and energy minimization.
• HPNs integrate mean-field spin-glass theory, continuous synaptic dynamics, and analog photonic implementations to enable efficient pattern storage and retrieval.
• The framework supports diverse architectures, from structured hierarchical networks to deep, nonlinear extensions, demonstrating practical hardware and computational advantages.

The Hebbian Physics Network (HPN) encompasses a class of mathematical and physical models for associative memory and learning, founded on the principle that synaptic strengths are updated according to local Hebbian rules derived from underlying physical laws or symmetries. HPNs provide a unification between neural learning, retrieval, and physics-informed computation, with core architectures ranging from classic mean-field spin-glass ensembles and modular hierarchical networks to analog wave-based implementations, as well as deep photonic and thermodynamics-driven computational substrates. This framework extends from canonical Hopfield networks and Boltzmann machines to models incorporating synaptic dynamics, structured graph topology, “unlearning” mechanisms, and explicit embedding of physical conservation laws, thus enabling rich connections to statistical mechanics, random matrix theory, and non-equilibrium thermodynamics.

1. Statistical Mechanics Foundations and Mean-Field HPNs

HPNs originate in the statistical mechanical analysis of neural networks where the states of $N$ binary neurons $\sigma_i \in \{\pm1\}$ are governed by an energy (Hamiltonian) function. Two archetypal models undergird this approach:

• Curie–Weiss (mean-field ferromagnet):

$$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$

• Sherrington–Kirkpatrick (SK) spin glass:

$$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$

The classic Hebbian coupling matrix emerges from two statistical routes (Agliari et al., 2014):

• Ferromagnetic (Mattis gauge): For $P$ stored patterns $\{\xi^\mu\}_{\mu=1}^P$, the Hamiltonian is

$$H = -\frac{1}{2N}\sum_{i,j=1}^N\sum_{\mu=1}^P \xi_i^\mu\xi_j^\mu \sigma_i\sigma_j,\qquad J_{ij} = \frac{1}{N}\sum_{\mu=1}^P\xi_i^\mu\xi_j^\mu.$$

• Spin-glass (Boltzmann-machine route): Marginalizing hidden units in a restricted Boltzmann machine yields the same bilinear interaction matrix $J_{ij}$.

The HPN architecture consists of neurons as Ising spins, fully connected via symmetric synapses $J_{ij}$, with an energy function $$E(\sigma) = -\frac{1}{2}\sum_{i,j=1}^N J_{ij}\sigma_i\sigma_j$$. Dynamics (e.g., Glauber updates) follow statistical-mechanical rules determined by energy changes and inverse temperature $\sigma_i \in \{\pm1\}$0. Retrieval and storage capacity are controlled by the load parameter $\sigma_i \in \{\pm1\}$1 with a critical value $\sigma_i \in \{\pm1\}$2 in the zero-temperature limit (Amit-Gutfreund-Sompolinsky theory). The same Hamiltonian underlies both learning (coupling synthesis) and recall (parallel dynamics), unifying these operations within the free energy landscape (Agliari et al., 2014).

2. Synaptic Dynamics, Learning Rules, and Consolidation

HPNs have been extended to include explicit synaptic dynamics, inspired simultaneously by Pavlovian conditioning and statistical mechanics (Lotito et al., 2024). In these models, the coupling matrix $\sigma_i \in \{\pm1\}$3 is treated as a dynamical variable with stochastic differential equations coupling neural activity and synaptic evolution: $\sigma_i \in \{\pm1\}$4 where $\sigma_i \in \{\pm1\}$5 (neural timescale $\sigma_i \in \{\pm1\}$6 synaptic timescale) and $\sigma_i \in \{\pm1\}$7 denotes external stimuli. Under adiabatic (timescale-separation) approximation, the evolution converges to the classical Hebb rule: $\sigma_i \in \{\pm1\}$8 with additional results for the variance of $\sigma_i \in \{\pm1\}$9 around the Hebbian mean.

A biologically motivated extension incorporates sleep-associated memory replay (“dreaming”), leading to the Kanter–Sompolinsky projector kernel: $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$0 where $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$1 is the pattern-correlation matrix. This kernel achieves optimal storage and enhances retrieval basins (Lotito et al., 2024).

3. Structured, Hierarchical, and Sparse HPN Topologies

HPNs generalize beyond mean-field by incorporating spatial or hierarchical structure. In the Dyson hierarchical HPN, the system is built from recursively nested blocks with couplings decaying by a metric (e.g., $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$2 for distance $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$3 and decay exponent $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$4). The Hamiltonian combines a distance-weighted ferromagnetic term with Hebbian learning: $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$5 Such architectures exhibit both serial (pattern-aligned) and parallel (block-wise multi-pattern) retrieval depending on the modularity parameter $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$6. Enlarging modularity (increasing $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$7) yields extensive metastables and multitasking at the cost of reduced capacity. Statistical mechanics, graph-theoretic spectral analysis, and signal-to-noise techniques are used for capacity and phase diagram estimation. These models illuminate how modular, metric-based architectures support richer dynamical regimes and small-world properties (Agliari et al., 2014, Agliari et al., 2010).

4. Continuous, Analog, and Photonic HPN Implementations

HPNs are not restricted to abstract neural or spin models. Analog physical realizations based on driven-dissipative waves and optical media have been established for implementing Hebbian perceptrons (Espinosa-Ortega et al., 2014). In such a system, information is encoded in wave amplitudes, processed by spatially patterned potentials $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$8 determined by Hebbian sums over training patterns: $$H_N^{CW}(\{\sigma\}) = -\frac{J}{2N} \sum_{i,j=1}^N \sigma_i \sigma_j - h \sum_{i=1}^N \sigma_i.$$9 Mappings between synaptic weights, neuron activations, and wave phenomena are explicitly constructed: synaptic strength corresponds to Fourier coefficients of a static potential, pre- and post-synaptic activations to input drive amplitudes and scattered intensity, respectively.

Specific hardware implementations include:

• Nonlinear optical systems employing Kerr-like effects and frequency channels.
• Lithographic structuring for fixed connectivity.
• Exciton-polariton condensates using phonon-induced plastic potentials or dynamic nuclear polarization for in-situ, long-lived weight storage.

Performance is characterized by high speed (THz-scale), high energy efficiency (fJ–pJ per MAC), and ultrafast, parallel operation (Espinosa-Ortega et al., 2014).

A recent advance demonstrates deep photonic neuromorphic HPNs with fully optical Hebbian synaptic update laws using phase-change materials (GST). The synaptic change is governed directly by the overlap of pre- and post-synaptic optical signals: $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$0 with updates executed by local heating/phase transitions, requiring no electronic or OEO conversions and enabling scalability to large arrays (up to $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$1 synapses/mm$$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$2) and high-throughput, online, unsupervised learning (Li et al., 29 Jan 2026).

5. Physics-Based HPNs: Non-Equilibrium Thermodynamics and Local Laws

A new branch of HPN methodology recasts computation as self-organizing adaptation to the local violation of physical conservation laws, eliminating global loss functions. Here, Hebbian plasticity is generalized such that weight updates on edge $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$3 are reinforced or depressed in proportion to the reduction of local physical residuals (e.g., continuity, momentum, or energy imbalances): $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$4 where $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$5 encodes residuals in the discretized field equations (e.g., divergence of velocity, diffusive flux). Learning dynamics are driven by these thermodynamic signals, and emergent structures such as vortices or diffusion fronts arise without explicit supervision or global optimization (Auti et al., 1 Jul 2025).

Benchmarking against global-loss Physics-Informed Neural Networks (PINNs), HPNs demonstrate lower computational complexity, improved scalability, and rapid convergence to physically consistent solutions. Structural and dynamical interpretability is inherent: all variables retain direct physical meaning, and network adaptation can be understood as movement within the space of dissipative, non-equilibrium attractors.

6. Spectral Theory, Generalization, and Random Matrix Analysis

HPN performance in retrieval and generalization is governed by the spectral properties of the interaction matrix $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$6. For data $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$7, supervised and unsupervised kernels can be defined: $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$8 where $$H_N^{SK}(\{\sigma\}|\{J_{ij}\}) = -\frac{1}{\sqrt{N}}\sum_{i<j} J_{ij}\sigma_i \sigma_j, \quad J_{ij} \sim \mathcal{N}(0,1) \text{ (quenched)}.$$9 is a “dreaming” or consolidation parameter (Agliari et al., 2024). Eigenvalues evolve via $P$0, providing continuous interpolation from Hebbian to “projector” kernels.

Retrieval performance (e.g., Mattis overlap) at each update step is determined by spectral integrals over the empirical density, which, in the thermodynamic limit, exhibits deterministic bulk (Marchenko-Pastur) and “outlier” peaks. Signal retrieval necessitates that modes encoding true patterns separate from the spectral bulk. This spectral HPN framework quantifies not only capacity but also ability to generalize from noisy/corrupted samples, thus connecting ensemble neural computation directly to random matrix phenomena.

7. Extended HPN: Deep, Nonlinear, and Relativistic Generalizations

HPNs admit extensions merging deep learning and “unlearning” within a unified cost-functional framework based on mechanical analogy. A relativistic extension of the Hopfield Hamiltonian introduces higher-order (multi-spin) interactions and negative sign corrections corresponding to pruning of spurious minima: $P$1 with Taylor expansion containing alternating sign $P$2-spin terms. The leading term reproduces Hebbian memory, while quartic (negative) terms destabilize mixtures of patterns, systematically shrinking spurious attraction basins and lowering energy barriers. The free energy can be solved exactly using a Hamilton–Jacobi correspondence, yielding closed-form self-consistency equations for the overlaps (Barra et al., 2018). This formalism directly connects higher-order feature extraction and network pruning to a variational principle rooted in special relativity.

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In summary, the Hebbian Physics Network paradigm synthesizes neural, physical, and information-theoretic concepts into a rigorous, extensible model class. HPNs encompass statistical mean-field and structured spin-glass theories, continuous photonic and analog implementations, physics-informed local rules, deep network extensions, and spectral frameworks. By grounding learning and computation in physical law—be it through energy minimization, thermodynamic drive, or wave interaction—HPNs provide a physically interpretable, analytically tractable, and hardware-compatible foundation for associative memory, adaptive computation, and beyond (Agliari et al., 2014, Espinosa-Ortega et al., 2014, Lotito et al., 2024, Agliari et al., 2014, Auti et al., 1 Jul 2025, Li et al., 29 Jan 2026, Barra et al., 2018, Agliari et al., 2024, Agliari et al., 2010).