Physical Neural Networks

• Physical Neural Networks are computing systems that leverage intrinsic physical laws (e.g., optical, mechanical, spintronic dynamics) to perform neural computations natively.
• They integrate rigorous mathematical foundations—such as wave propagation equations and universal approximation theorems—to enable direct, efficient hardware learning.
• Innovative training paradigms and noise mitigation strategies make PNNs promising for edge inference, autonomous robotics, and ultra-low-energy AI applications.

Physical Neural Networks (PNNs) are computational architectures in which both the representation and manipulation of information occur directly within a physical system, rather than through abstracted digital operations. PNNs leverage the underlying laws of physics—optical wave propagation, mechanical dynamics, electrical circuits, spintronic oscillations, analog memory—to perform neural network operations at speeds and energy scales far beyond conventional digital electronics. As research demonstrates, PNNs encompass a wide range of substrate-dependent implementations, from analog photonic chips to bistable mechanical lattices, and enable both model-driven and emergent computing paradigms. Theoretical foundations tie PNN architectures to partial differential equations, functional kernels, and universal approximation theorems, while experimental advances highlight robust noise mitigation, hardware-efficient parameter reuse, and self-organizing learning dynamics. PNNs are thus a central focus in next-generation neuromorphic computing, promising high-throughput, energy-efficient, and potentially self-adaptive information processing.

1. Mathematical Foundations and Physical Law Embedding

Physical Neural Networks are rigorously cast as dynamical or wave-propagation equations in which the trainable parameters correspond to spatially or temporally distributed physical quantities. In the Schrödinger Equation Neural Network (SE-NET) framework, the propagation of the complex wavefunction $\Psi(\mathbf{x}, t)$ under a Hamiltonian

$$i\,\hbar\,\frac{\partial\Psi(\mathbf x,t)}{\partial t} = -\frac{\hbar^2}{2m}\,\nabla^2\Psi(\mathbf x,t) + V(\mathbf x,t)\,\Psi(\mathbf x,t)$$

is mapped to the forward pass of a differentiable residual neural layer, with the local potential $V(\mathbf{x},t)$ serving as the trainable weight matrix. Finite-difference discretizations, such as the Crank–Nicolson scheme, yield explicit complex-valued linear layer updates: $$\psi^{q+1} = (I+\tfrac{j\Delta z}{2}\,\mathcal H)^{-1} (I-\tfrac{j\Delta z}{2}\,\mathcal H)\,\psi^q$$ where $\mathcal{H}$ combines kinetic and potential contributions.

Backpropagation is implemented through physical adjoint fields obeying time-reversed wave equations. The gradient of the loss function with respect to the local potential parameters has the instantaneous form: $$\frac{\partial L}{\partial V(\mathbf x,t)} = -i\,\psi_a^*\,\Psi$$ with the real and imaginary parts separated for hardware implementation.

The universality of PNN architectures is formally proven via a mathematical theorem: a PNN with input-output mapping

$f(x)=\sum_{j=1}^r c_j\,\sigma(a_j\cdot x + b_j)$

is universal if and only if the multivariate nonlinearity $\sigma$ is non-degenerate. Embedding analytic constraints (e.g., conservation laws) is achieved either as hard architectural layers or soft loss penalties; in climate applications, mass and energy conservation are imposed to within machine precision without performance loss (Nakajima et al., 2020, Beucler et al., 2019, Savinson et al., 6 Sep 2025).

2. Physical Substrates, Device Models, and Noise Mitigation

PNNs are realized in multiple physical substrates:

• Optical/Photonics: Waveguide meshes, diffractive optics, and cavity systems encode weights in refractive-index perturbations or phase shifts; outputs are sensed via detectors or nonlinear elements.
• Spintronic and Magnetic: Arrayed oscillators and magnetization textures serve as neurons and synapses, with programmable conductance and emergent collective modes.
• Mechanical and Fluidic Networks: Bistable chambers interconnected by viscous tubes represent multistable logic and memory, governed by nonlinear pressure–volume laws.
• Memristive In-Memory Circuits: Resistive and phase-change memory arrays encode synaptic weights, directly performing matrix–vector operations.

Noise sources—additive (uncorrelated/correlated) and multiplicative (uncorrelated/correlated)—are intrinsically present. Analytical and architectural strategies such as intra-layer connection optimization, ghost neurons (anti-correlated subtraction), and pooling suppress signal degradation. For instance, SNR improvements scale as

$${\rm SNR}_{\rm improved} \approx \sqrt{I_n}\,\frac{\mu(A)}{\sqrt{\eta(A)}}\,\sqrt{m}$$

where $\mu(A)$ is the mean and $\eta(A)$ the mean-square of connection weights; ghost neuron subtraction cancels correlated noise exactly (Semenova et al., 2022).

3. Training Paradigms and Computational Algorithms

Training PNNs requires algorithmic integration of physical forward passes and gradient-based updates over hardware constraints. Approaches include:

• Physics-Aware Training (PAT): Real hardware executes the forward pass, while differentiated surrogate models (digital twins) approximate gradients for parameter updates (Wright et al., 2021).
• Adjoint and Contrastive Methods: Physical systems implement time-reversed dynamics for exact gradient computation, including equilibrium propagation and contrastive learning in resistor networks and mechanical systems (Stern et al., 2021, Ben-Haim et al., 2024).
• Feedback Alignment and Local Learning: Plausible biologically-inspired feedback matrices replace strict gradients, and learning occurs using local error signals and state nudging.
• Sharpness-Aware Training (SAT): Loss landscape geometry is penalized for sharp minima, yielding robust, transferable model weights across device perturbations and fabrication variances (Xu et al., 2024).
• Optimal Control and Direct Feedback Alignment: Hybrid schemes merge adjoint-control updates with random projected error signals, providing noise-robust training for continuous-time delay systems (Sunada et al., 26 Feb 2025).

Computational scaling considerations include reuse of slow-tuned weight banks via fast switches (ReLaX-Net), time-multiplexed logical layers, and architectural investments in nonlinearity via Kolmogorov–Arnold networks (KANs) that directly train synaptic device transfer functions (Tsuchiyama et al., 28 Oct 2025, Taglietti et al., 20 Jan 2026).

4. Universality, Expressivity, and Scaling Laws

PNNs are proven to possess universal approximation capacity in broad physical regimes, provided the substrate supports bilinear encoding and genuine multivariate nonlinearity. Temporal and frequency multiplexing, spatial replication, and programmable scattering enable effective scaling to thousands–millions of logical neurons. Expressivity measures such as $\varepsilon$-packing of device nonlinearities correlate directly with regression and classification accuracy, guiding optimal resource allocation.

Overparameterization in neuromorphic systems (e.g., nanomagnetic arrays with 13,500 readout channels) triggers double-descent learning curves and enables meta-learning and few-shot adaptation in real physical devices—achieving sub-percent errors with minimal data (Stenning et al., 2022).

Performance–parameter scaling follows power laws ($\sim$ negative slopes for series-connected KANs) and device count reductions by up to 100× compared to linear-weight networks. Physical KANs maintain accuracy on real-world regression with two orders of magnitude fewer devices and energy per inference ($\sim$130 nJ vs. $\sim$90 $\mu$J on digital GPUs) (Taglietti et al., 20 Jan 2026).

5. Model-Based and Self-Organizing Physical Learning

A growing domain in PNN research concerns self-learning, wherein weight updates are governed by intrinsic physical process feedback—optical interference signals, voltage-sensed Hebbian plasticity, mechanical stress-induced adaptation. Local learning rules (Hebb, Oja, STDP, contrastive divergence, equilibrium propagation) are implemented in hardware, with demonstrated training on PCM, spintronic, elastic, and flow networks.

Recent platforms achieve physical self-learning with

• In situ phase-change adaptation,
• All-optical cavity feedback,
• Magnetization reconfiguration via current flow,
• Directed aging in continuous attractor materials.

Experimental results show competitive accuracy (90–95% on MNIST-class tasks), sub-$\mu$s inference, and ultralow energy consumption. Scalability challenges involve variability, device endurance, and co-designing physical architecture for locally distributed adaptation (Yu et al., 2024, Stern et al., 2021, Ben-Haim et al., 2024).

6. Applications, Challenges, and Future Directions

PNNs are positioned for high-impact applications:

• Edge inference and private AI (on-sensor neural computation),
• Energy-efficient deployment at unprecedented scales,
• Ultrafast photonic and spintronic inference engines,
• Autonomous mechanical actuation and intelligent matter for robotics and medicine.

Outstanding challenges center on noise robustness, hardware variability, digital twin accuracy, and algorithm–hardware co-design for efficient mapping of neural architectures onto physics-native substrates. Strategies such as bottom-up ODE-based modeling, DSTD for spiking circuits, and noise mitigating architectural augmentation have shown order-of-magnitude reductions in model–hardware error and energy footprint (Sakemi et al., 2024).

Theoretical advances point toward physical systems learning near their operational bandwidths, overparameterized large-scale learning, and hybrid architectures merging physical and conventional digital layers for optimal energy–performance trade-off. Discoveries in PNN theory, device engineering, and physics-embedded learning anticipate the emergence of self-organizing, high-capacity, and resource-efficient computing platforms for next-generation AI (Momeni et al., 2024, Tsuchiyama et al., 28 Oct 2025, Taglietti et al., 20 Jan 2026, Savinson et al., 6 Sep 2025).