Power law scaling for classification accuracy in physical neural networks
Physical neural networks (PNNs) harness the intrinsic complexity of physical systems to perform neural computation, potentially at speeds and energy efficiencies inaccessible to conventional digital hardware. Yet, a principled framework for quantifying and predicting their computing accuracy across diverse substrates has remained elusive. Here we introduce the Hotelling Trace Criterion (HTC), a task-conditioned measure of PNN- state separability that can be evaluated without training. We demonstrate that it predicts PNN classification performance with high fidelity across highly nonlinear optical fibres, vertical-cavity surface-emitting lasers, and coupled nonlinear oscillator networks, for benchmark tasks of different difficulty. Classification loss follows a power law in HTC, with Pearson correlation coefficients exceeding 0.99 for MNIST and $\approx$0.97 for Fashion-MNIST, noteworthy experimental and simulated data from physically distinct systems collapse onto a single scaling curve determined by the task rather than the substrate. Applying HTC layer-by-layer during training further reveals that gradient-based optimisation distributes representational capacity unevenly across PNN layers, providing a quantitative diagnostic of training and architecture efficiency invisible to standard loss monitoring. Crucially, once the scaling exponent is established from a small number of trained calibration systems, all further performance predictions require no training since performance can be derived from the much more efficient HTC measurement. These results establish HTC as a substrate-agnostic figure of merit for comparing and scaling PNNs, advancing the field further towards a complete theory connecting fundamental hardware parameters to task performance through universal scaling laws.