Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems

Motivation

• DNN computational requirements have grown faster than Moore’s Law, making hardware energy efficiency an increasing constraint on scaling and broader use.DNN accelerators and alternative platforms address this problem but require engineering to suppress nonlinearities, noise, and device variations that violate idealized mathematical analogies.Source: §I · p. 1

Physical neural networks

• PNNs cascade controllable physical input-output transformations, encoding data and parameters into physical states and reading computed outputs through measurements.This architecture exploits physical operations such as convolutions, nonlinearities, and matrix-vector transformations while supporting more general layer functions than conventional DNNs.Source: §I · p. 1, Figure 1 · p. 2

Related work and limitations

• Earlier physical-learning proposals often used gradient-free algorithms, while backpropagation proposals imposed assumptions such as linearity or dissipation-free evolution or trained systems entirely in silico.Reservoir computing also leaves the physical transformations untrained and therefore provides only shallow learning.Source: §I · pp. 1–3

Contribution

• PAT directly trains arbitrary sequences of real physical input-output transformations in situ, enabling hierarchical PNNs that execute deep neural networks using backpropagation.The hybrid physical-digital algorithm trains physical transformations themselves rather than relying on numerical simulations or fixed reservoirs.Source: §I · pp. 2–3
• Three experimentally distinct systems—mechanical oscillations, nonlinear transistor dynamics, and optical second-harmonic generation—performed accurate hierarchical image classifiers.The classifiers use each system’s physical transformations while mitigating its characteristic noise processes and imperfections.Source: Figure 1 · p. 2, §I · p. 3

Implications

• PNNs can combine with conventional hardware in physical-digital hybrid architectures, supporting hardware-physics-software codesign and potentially improving machine-learning speed and energy efficiency by many orders of magnitude.The framework also suggests automatically designing functional devices such as nanoparticles, robots, and smart sensors.Source: §I · p. 3

Optical vowel classification

• The SHG system converts shaped near-infrared spectra near 800 nm into ultraviolet spectra near 400 nm, enabling five sequential trained nonlinear transformations for vowel prediction.Seven ultraviolet spectral bins determine the final vowel class, and the optical dynamics include second-harmonic, sum-frequency, and difference-frequency processes.Source: §II · p. 3, Figure 2 · p. 4

Physics-aware training

• PAT trained the five-layer SHG-PNN to 93% vowel-classification accuracy, whereas in-silico training reached only ∼40% and failed as physical depth increased.PAT executes physical forward passes while using a differentiable digital model for gradient estimation, limiting simulation-reality mismatch.Source: Figure 3 · p. 5, §II · p. 5

MNIST classification

• The mechanical PNN achieved 87% MNIST test accuracy, while replacing its physical transformation with identity reduced optimal performance to 10% random guessing.Its multimode plate oscillations enact a controllable convolution over 784-pixel inputs across three sequential physical transformations.Source: §II · p. 6
• The nonlinear analog-electronic PNN achieved 93% MNIST test accuracy by averaging predictions from seven independent three-layer PNNs.Its transistor circuit produces a noisy, highly nonlinear transient response that is more difficult to model accurately than the mechanical system.Source: §II · p. 6
• 97% MNIST accuracy was achieved by a hybrid SHG PNN, improving digital operations from roughly 90% by offloading computation to physical transformations.The architecture uses four channels with two physical layers each, whose outputs are concatenated for prediction.Source: §II · p. 6

• Final test accuracies reached 87%, 93% and 97% for mechanical, electrical, and optics-based PNNs, respectively.These results demonstrate image classification across three distinct physical systems.Source: Figure 4 · p. 7
• Realistic device implementations achieve self-simulation advantages over 106 (109) for speed (energy), with larger values possible through size-scaling.These advantages increase with physical connectivity, dimensionality, and interaction nonlinearity.Source: §III · pp. 6, 8
• Candidate PNNs can implement standard matrix-vector operations and exotic computations such as nonlinear convolutions and higher-order tensor contractions.Examples include multimode wave propagation, coupled spintronic oscillators, electrical networks, ultrafast optical SHG, and Kerr-media propagation.Source: §III · p. 8
• Physics-aware training and physics–algorithm co-design could translate self-simulation advantages into practical speed-ups and energy-efficiency gains despite imperfect simulations, device variation, and higher-order processes.For systems resembling neural networks or neural ordinary differential equations, PAT can learn hierarchical physical computations directly.Source: §III · p. 8
• Multidimensional layouts, in-place parameters, physical feedforward, and improved PAT can reduce practical costs while extending PNNs to smart sensors and other physical functionalities.PNNs may preprocess data within the physical domain before electronic conversion, while PAT broadly trains complex physical-system functionality.Source: §III · p. 8

• An expandable demonstration code for applying physics-aware training to physical neural networks is available at the Physics-Aware-Training GitHub repository.The passage also identifies the data generated and code used for this work.Source: DATA AND CODE AVAILABILITY · p. 8

• The paper addresses backpropagation for arbitrary physical systems.Source: Deep physical neural networks enabled… · p. 11
• The document is an arXiv version dated 27 April 2021.Source: Deep physical neural networks enabled… · p. 11

• PAT is presented as an algorithm for efficiently training physical neural networks, with its effectiveness explained through high-level intuition and rigorous subsequent analysis.The section motivates why PAT works and why alternatives fail before developing the method in detail.Source: §1 · pp. 11–12
• The section develops PAT mathematically, compares it with conventional backpropagation, and specifies its application to generic multilayer physical neural networks.Source: §1 · p. 11
• A numerical example trains a simulated feedforward physical neural network for vowel classification using PAT and alternative training algorithms.Source: §1 · pp. 11–12

A. Intuition for why physics-aware training works

• PAT applies backpropagation to sequences of real physical operations, enabling them to perform desired physical functions and physical neural-network computations.Backpropagation efficiently computes gradients for mathematical operations, whereas PAT extends this training objective to physical systems.Source: A. Intuition for why physics-aware… · p. 12
• Finite-difference gradients require n physical-system uses for n parameters, making them impractical when n reaches 10^6 or 10^9; simulations instead suffer an unavoidable simulation–reality gap.These are the two natural alternatives to PAT, but neither provides a practical route to accurately train real physical systems.Source: A. Intuition for why physics-aware… · p. 12
• A small simulation–reality gap can grow rapidly through physical layers: in the toy example, relative error rises from 0.5% at n = 1 to 3% at n = 5 and 30% at n = 20.The magnitude of this blow-up depends on the function, evaluation point, and model inaccuracies, but model error generally increases with depth.Source: A. Intuition for why physics-aware… · p. 12
• The training-step gap is more significant because parameter updates compose across 10^5 or more sequential steps, each depending on the full-depth network and previous updates.Consequently, even highly accurate simulations that predict deep-network behavior may be inadequate for training real physical systems.Source: A. Intuition for why physics-aware… · p. 12
• PAT uses the true physical system for outputs and intermediate layer inputs, preventing depth-error accumulation and estimating updates against the correct inputs and true loss.This keeps gradient directions aligned with whether the physical system is undershooting or overshooting the desired output, while allowing residual model gaps to develop more slowly.Source: A. Intuition for why physics-aware… · pp. 12–13
• PAT can tolerate forward-pass noise, heterogeneous devices, and model omissions by using physical noise directly and treating fluctuating or mis-modelled directions as noisy update dimensions.The passage relates these behaviors to quantization-aware training and feedback alignment.Source: A. Intuition for why physics-aware… · p. 13

Backpropagation basis

• The algorithm computes loss gradients with respect to network parameters and updates those parameters using reverse-mode automatic differentiation through forward and backward passes.Reverse-mode autodiff propagates error signals from outputs toward inputs and parameters through connected constituent functions.Source: B. General formulation of… · p. 13

Motivation

• Naively applying backpropagation to physical systems is impractical because analytic input-output derivatives are unavailable, while finite differences require n + p physical-system calls per backward pass.This overhead becomes significant for networks with thousands or millions of trainable parameters.Source: B. General formulation of… · p. 14

Core formulation

• Physics-aware training uses physical transformations in the forward pass and differentiable digital models in the backward pass to train arbitrary physical neural networks efficiently.The method replaces unavailable physical-system derivatives with gradient estimates computed through the digital model while retaining an iterative backpropagation workflow.Source: B. General formulation of… · p. 14, Figure S1 · p. 15

Implementation

• Custom autodiff functions let users define physics-aware forward and backward operations in PyTorch, then train physical neural networks with conventional machine-learning workflows.The implementation uses a physical-system computation for the forward function and PyTorch automatic differentiation for the backward function.Source: B. General formulation of… · pp. 14, 16

Gradient approximation

• Because the digital backward function differs from the physical forward function, physics-aware training approximates rather than exactly reproduces gradients at earlier layers.The resulting backward pass propagates estimators of gradients through the differentiable model.Source: B. General formulation of… · p. 14

Physical neural-network architecture

• In a feedforward physical neural network, each physical system passes its output to the next system, while a trainable parameter is supplied to every layer.The architecture applies sequential physical transformations to the input.Source: Figure S2 · p. 17

Ideal backpropagation

• Ideal backpropagation is impractical for real PNNs because finite-difference gradients require n+p repeated physical-system calls for every backward pass.Although it could in principle reach low-loss parameters, this repeated evaluation becomes prohibitively slow for networks with many parameters.Source: C. Motivating physics-aware training… · pp. 19–20

Feedforward PNN architecture

• The feedforward PNN chains N physical systems, feeding each layer’s output into the next and using the final layer output as the prediction.Training computes an output error vector, backpropagates gradients from the final layer, and updates each system’s parameters by gradient descent.Source: C. Motivating physics-aware training… · pp. 19–20

In silico training

• In silico training avoids repeated physical calls but can produce inaccurate updates because model errors distort the output error vector and intermediate Jacobians, increasingly with depth.It trains entirely on a differentiable digital model and transfers the resulting parameters to the physical system for evaluation.Source: C. Motivating physics-aware training… · pp. 20–21

Physics-aware training

• Physics-aware training combines physical forward passes with differentiable digital models in the backward pass, using actual outputs and intermediate states to improve gradient accuracy.The physical system supplies the true output and layer inputs, while digital models estimate otherwise unavailable Jacobians.Source: C. Motivating physics-aware training… · p. 21

D. Numerical example

• The simulated vowel-classification PNN used a differentiable autocorrelation-based physical transformation to emulate broadband optical second-harmonic generation and compared ideal backpropagation, in silico training, and physics-aware training.The example used 12-dimensional vowel formants, seven one-hot output classes, and a feedforward architecture with input duplication and output summation.Source: D. Numerical example · p. 22
• A three-hidden-layer digital model with 1000, 500, and 300 neurons was trained on 2000 physical input–output pairs to model the constituent transformation.The controls were clamped between 0 and 1, and the training loss combined cross-entropy with a regularization term using λ = 0.02.Source: D. Numerical example · p. 22
• ∼96% test classification accuracy was achieved by physics-aware training, close to ideal backpropagation, while in silico training failed after about ∼350 epochs as model mismatch increased.After ∼2000 epochs, loss and accuracy fluctuated, likely because small Jacobian differences caused parameter drift near an optimum.Source: D. Numerical example · p. 23, Figure S4 · p. 24
• Single-step and multi-step analyses showed that in silico gradient inaccuracies compound across training, whereas physics-aware training avoids this divergence by using experimentally aligned layer inputs.Figures S5–S7 compare gradients, layerwise propagation, and trajectories after 300 and 800 training steps.Source: D. Numerical example · p. 23, Figure S5 · p. 24, Figure S6 · p. 25
• Physics-aware training approximated ideal gradients because it used the physical transformation in the forward pass and estimated gradients at the correct layerwise parameter locations.In silico training evaluated gradients at increasingly mismatched locations, producing gradients with incorrect direction and magnitude.Source: D. Numerical example · p. 23, Figure S5 · p. 24
• At epoch 800, physics-aware training and ideal backpropagation classified all three example vowels correctly, whereas in silico training misclassified two of three.By 300 epochs, all algorithms showed some learning, but in silico training subsequently diverged rapidly from the other methods.Source: D. Numerical example · p. 23, Figure S7 · p. 25, Figure S6 · p. 25

2. SUPPLEMENTARY METHODS

• The supplementary methods describe experimental setups for three physical systems, data-driven differentiable and noise models, and PNN architectures for vowel and MNIST classification.These methods also present typical results illustrating how the physical neural networks perform learned classification tasks.Source: §2 · p. 26

1. Experimental setup

• A femtosecond SHG PNN shaped 780 nm pulses with a DMD, generated approximately 390 nm light in a BBO crystal, and measured its spectrum with a fast camera.The setup used a mode-locked Titanium:sapphire oscillator, a 0.5 mm BBO crystal, a ruled reflection grating, and a Basler ace acA800-510u camera.Source: §1 · p. 26
• The SHG-PNN operated reliably at 200 Hz for months, enabling the millions of physical evolutions required for training.The sustained operation also included remote use during the COVID-19 pandemic.Source: §1 · p. 27

2. Input-output transformation characterization

• For sharply peaked inputs, SHG produced half-wavelength outputs, while combining 772 nm and 777 nm inputs generated an additional central-wavelength output rather than the sum of separate outputs.This characterization demonstrates the nonlinear input-output behavior of the optical device.Source: §2 · p. 27
• Varying a small flat input-spectrum region substantially changed the output across a wide spectral range, with strongest nonlinearity at large amplitude (A ≈1).The response was measured for fixed-region amplitudes between 0 and 1.Source: §2 · p. 28, Figure S10 · p. 28

1. Experimental setup

• The electronic PNN used bulk components, a breadboard, and a 1 MS/s DAQ, with an RLC oscillator included in the final strongly nonlinear circuit for MNIST classification.The circuit was designed to remain as simple as possible while exhibiting strongly nonlinear dynamics.Source: §1 · p. 28

2. Input-output transformation characterization

• The analog transistor circuit produced highly nonlinear output time series and output-bin responses across input amplitudes from -10 to 10 V, with coupling after a changed input section.Its impulse response was characterized separately as that of a nonlinear oscillator.Source: §2 · pp. 28–29
• The electronic PNN intentionally operated where DAQ limiting added nonlinearity, whereas this regime was avoided for the oscillating plate because its response was determined to be very linear.The added DAQ nonlinearity was judged unlikely to substantially alter the already highly nonlinear circuit’s computational capabilities.Source: §2 · p. 29

1. Experimental setup

• The oscillating-plate setup couples a computer, audio amplifier, speaker-mounted titanium plate, and microphone to drive and record mechanical oscillations.The plate measures 3.2 cm × 3.2 cm × 1 mm and is mounted on the speaker voicecoil.Source: Figure S14 · p. 30, §1 · p. 30

2. Input and output encoding

• Inputs are converted to 192 kS/s analog pulses, amplified through the speaker and plate, recorded at 192 kS/s, and downsampled to the desired output dimension.Repeating trigger signals synchronize samples to about 5 µs.Source: §2 · p. 31, Figure S16 · p. 31

3. Characterization of input-output transformation

• The plate response is overwhelmingly linear, so its transient dynamics implement a convolution whose kernel size is controlled by input frequency while coefficients remain fixed.At 192 kHz, approximately N = 384 previous inputs are convolved; at 5 kHz, N = 10.Source: §3 · pp. 31–32, Figure S17 · p. 32

D. Data-driven differentiable models

• The differentiable-model framework can use analytic, data-driven, or physics-informed models, with neural architecture search providing the default models across the studied physical systems.Training data usually contained approximately N ∼ 10^4 input/parameter vectors sampled from a uniform distribution.Source: §1 · p. 32

1. Digital model for the mean

• For the oscillating plate, the realized linear dynamics motivated replacing a searched deep network with a fully linear model containing no hidden layers.The SHG and electronic-circuit systems instead used searched deep networks with system-specific layer dimensions.Source: §1 · p. 33

2. Noise digital model

• The noise model represents each output as y = fmean(x) + nnoise(x), learning input-dependent stochastic behavior from repeated physical measurements.The measured output distributions were reasonably described by multivariate normal distributions characterized by a mean vector and covariance matrix.Source: §2 · p. 33
• Noise predictions are compressed to dominant covariance eigenvectors, with a neural network predicting the matrix factors from inputs and generating noise samples during simulated forward passes.The model typically used 40 repeats per nominally fixed input to estimate the covariance factor.Source: §2 · pp. 33–34
• The noise model’s covariance and eigenvector predictions showed overall excellent agreement with measurements, improving simulation-based performance estimates but not matching PAT for training the physical system.All simulation-training results, including Figure 3, incorporated the physical-system noise model.Source: §2 · pp. 33–34, Figure S18 · p. 34, Figure S19 · p. 35

Common architectural features

• Across PNNs, PAT used Adadelta, identity-operation baselines isolated physical contributions, and Lagrangian penalties kept physical inputs and parameters within allowed ranges.Learning-rate scheduling was generally secondary, while digital pooling was used when needed to adjust dimensionality and contributed only minor computational power.Source: E. Descriptions of PNN architectures · p. 36

Femtosecond SHG-vowel PNN

• The SHG-vowel PNN rescales 12-dimensional normalized vowel formant frequencies with trainable parameters, repeats them to 48 dimensions, and concatenates them with 52 parameters to form a 100-dimensional physical input.Source: E. Descriptions of PNN architectures · p. 37

Analog transistor-MNIST PNN

• The electronic MNIST PNN reduced the circuit transformation to 70 input and output dimensions because accurate modeling was difficult, then used seven three-layer sub-PNNs, skip connections, and weighted averaging for robustness to noise.The circuit’s digital model had validation loss 0.09, while repeated identical inputs showed mean output deviations exceeding 5% of maximum amplitude and nearly 20% of mean amplitude.Source: E. Descriptions of PNN architectures · p. 39

Oscillating plate MNIST PNN

• The oscillating-plate MNIST PNN exploited a linear 784-D-to-784-D transformation, time-multiplexed whole images into voltage signals, and classified digits using the strongest of 10 microphone-output bins.Cascaded element-wise rescaling and speaker transmission approximately implemented a single-layer perceptron, with the readout window’s position and length trained as hyperparameters.Source: E. Descriptions of PNN architectures · p. 42

Oscillating-plate architecture

• The oscillating-plate MNIST architecture performs three sequential 784-D physical transformations separated by trainable element-wise rescaling, then averages selected outputs into 10 dimensions.Inputs are first rescaled, each physical transformation produces a 784-D vector, and outputs 724 to 774 are averaged in groups of five.Source: E. Descriptions of PNN architectures · pp. 42–43

Linear physical systems

• For shallow linear physical systems, in silico training transferred to experiment nearly matched PAT after about 3 epochs, but sequences of linear systems reduce to one matrix-vector multiplication.PAT still produced gains, whereas linear sequences cannot realize computations performed by deep neural networks.Source: E. Descriptions of PNN architectures · pp. 43–44

SHG-MNIST architecture

• The SHG input layer targets roughly 10–30 effective input modes because high-dimensional random inputs produce little output-spectrum variation.The implemented 50-dimensional middle layer was chosen to limit the number of slow 200 Hz SHG transformations and simplify testing configurations.Source: E. Descriptions of PNN architectures · p. 44
• ∼7% improvement on MNIST digit classification was achieved by using SHG nonlinear transformations instead of identity transformations.The model used a relatively large 50-dimensional input layer to reduce physical-system uses, although this made the SHG contribute less than in a more optimal design.Source: E. Descriptions of PNN architectures · p. 44
• The SHG architecture projects 196-dimensional MNIST inputs through 50 dimensions and back to 196, then processes them through four separate independent channels.Each SHG channel applies element-wise rescaling and adds the SHG input multiplied by a trained skip weight; one output is clipped from indices 70 to 140, producing a 70-dimensional vector.Source: E. Descriptions of PNN architectures · pp. 45–46

3. SIMULATION ANALYSIS OF CANDIDATE PNN SYSTEMS

• Candidate PNN systems approximate matrix-vector operations, convolutions, nonlinearities, and tensor contractions, usually within controllable subspaces constrained by sparsity, locality, and symmetry.Because these systems are rarely universally programmable, they may require cooperation with a universal digital computer.Source: §3 · p. 48, A. General case · p. 49
• Self-simulation advantages increase with physical complexity, especially higher-order coupled nonlinearities, heterogeneous long-range coupling, and higher dimensionality, although higher dimensionality can reduce nonlinearity.The simulations identify these features as the main design characteristics associated with larger advantages.Source: §3 · p. 48
• Self-simulation advantages significantly over 106 are common in candidate systems, indicating substantial potential for physical neural networks if key design and application challenges are solved.The estimate depends on how well self-simulation predicts useful-task advantages, how effectively PNN models exploit physical computations, and future device improvements.Source: §3 · p. 48

A. General case

• For a general N-mode nonlinear dynamical system, simulation complexity is dominated by the highest-order nonlinear term and depends on bandwidth, simulated duration, mode count, and heterogeneous coupling range.Sparsity, locality, and symmetry can reduce numerical evaluation cost, so coupling range must be considered for each nonlinear order.Source: A. General case · pp. 48–49
• The self-simulation advantage typically scales with the maximum coupling complexity, while increasing supplied energy can increase both nonlinearity and long-range coupling, making higher-dimensional physics harder to simulate.This scaling assumes the nonlinear order and coupling range remain independent of supplied power, which generally does not hold.Source: A. General case · p. 49
• High-dimensional tensor operations may provide the greatest physical advantages because higher-order nonlinear coupling enables computations not commonly used in modern deep neural networks.Efficiently using these operations for machine learning remains relatively unexplored.Source: A. General case · p. 49

B. Single-mode nonlinear optical pulse propagation: scaling

• Single-mode nonlinear optical pulse propagation is evaluated using split-step Fourier complexity, with spectral-point count set by bandwidth and time window and longitudinal steps set by nonlinear propagation rates.For nonlinear order n, the nonlinear rate scales generally as f(P) = P^(n−1)/2; the analysis assumes P0 = EpNω, which can overestimate nonlinearity.Source: B. Single-mode nonlinear optical pulse… · pp. 49–50
• The optical system is assumed capable of 30 GHz operation when pulse duration remains below 33 ps, while time and energy advantages compare digital simulation costs with physical evolution costs.The physical energy estimate includes pulse instantiation and measurement, not only the optical pulse itself.Source: B. Single-mode nonlinear optical pulse… · p. 50

1. Quantitative example

• The realistic nanophotonic nonlinear device requires approximately 50 nJ per optical pulse, while the near-future estimate is about 537 pJ per inference and both operate at 33 ps.The present-day and improved-device comparisons against conventional simulations are approximately 10^5–10^8 higher in energy and 10^4–10^5 longer in time.Source: §1 · p. 51

C. Self-simulation analysis of multimode wave propagation

• Multimode wave propagation can provide high-dimensional, potentially low-energy matrix-vector multiplication, while nonlinear propagation increases complexity and possible self-simulation advantages.Controlling the propagation medium would make these operations controllable and directly relevant to standard deep neural-network operations.Source: C. Self-simulation analysis of… · p. 52

1. Linear propagation

• Linear multimode propagation performs matrix-vector operations whose self-simulation advantage scales linearly with the number of modes.A 5 × 10^6-mode fiber example performs approximately 8 × 10^17 operations per second at about 10 aJ/operation, with controllable parameters offering a closer connection to standard neural networks.Source: §1 · p. 52

2. Nonlinear propagation

• The large advantages of nonlinear multimode systems should not be treated as general-purpose gains because their operations are difficult to control and exotic relative to modern neural networks.Practical benefits remain possible, but substantially more work is needed to exploit these systems for useful computations.Source: §2 · pp. 52–53, §1 · p. 52
• Extremely large self-simulation advantages require caution because more efficient or data-driven simulations may accelerate constrained-input or symmetry-limited systems.This caveat applies especially when simulations need not cover fully general inputs or when interactions obey symmetry constraints.Source: §2 · p. 53
• The standard multimode Kerr model scales as M^4, while general split-step simulations scale at least as M^2Nω/EpLb and effectively as M^αNω/Ep with α > 2.Symmetry can alleviate the M^4 scaling, but strong disorder breaks those assumptions; M denotes guided modes, Ep pulse energy, Nω frequency points, and Lb beat length.Source: §2 · p. 53
• 5×10−22 J/operation makes multimode nonlinear propagation 10^10 (10^11) more energy efficient than RTX GPU (DGX Superpod) simulation.The physical system executes 2 × 10^23 operations/s at a 31 kHz rate, based on 6.4 × 10^18 simulated operations per evolution.Source: §2 · p. 53

• The estimates assume locally nonlinear, controllably coupled oscillators, including spintronic devices projected at 1 µW per oscillator and frequencies into the 10s of GHz.Bulk multimode systems may instead require partial-differential-equation simulation, and active coupling power could become non-negligible across many connections.Source: D. Multimode nonlinear… · pp. 53–54
• Coupled oscillator dynamics closely resemble a deep residual neural network, and Euler integration produces its layer-by-layer propagation update.The local oscillator nonlinearity σ supplies the nonlinear transformation at each update.Source: D. Multimode nonlinear… · pp. 53–54
• 2 × 10^5 operations for 100 oscillators require about 4 nJ physically, producing an energy SSA of about 10^3 and a speed SSA between 0.1 and 10.The oscillator energy is 250 fJ, whereas DAC and ADC operations each cost 2 nJ at 20 GHz over 1 ns.Source: D. Multimode nonlinear… · p. 54
• Hidden oscillators can increase computational complexity while their trainable injection currents remain fixed during inference and avoid repeated DAC/ADC costs.For 10^5 oscillators, maintaining parameter oscillators at 10 µW requires about 1 nJ over 1 ns, still below conversion energy.Source: D. Multimode nonlinear… · p. 54
• 2 × 10^8 operations yield energy SSAs of about 10^6 to 10^7 and speed SSAs of 70 to 6000 for the larger oscillator network.The network uses N = 10^5 oscillators with M = 100 neighbours, while only 100 inputs and 100 outputs require conversion.Source: D. Multimode nonlinear… · p. 54
• Increasing oscillator count while holding N∆T constant increases the speed SSA linearly with N, but useful ∆T and exploitable spatial scaling depend on the physical transformation.As oscillator number grows, ∆T may need to increase so distant oscillators can interact meaningfully.Source: D. Multimode nonlinear… · p. 54

Simulation cost

• The standard nonlinear circuit-simulation approach scales as O(n1.3−1.8), mainly because Newton-Raphson iterations require linear-system solves.Topology-specific methods can improve scaling but sacrifice accuracy and depend on circuit topology.Source: E. Nonlinear electrical circuit… · pp. 54–55

Neural-network analogy

• Transistor-network dynamics resemble residual-layer propagation and conventional neural networks when coupling is controllable and nonlinearities are relatively local.The analysis estimates transistor networks may have a modest constant-factor self-simulation advantage of around 10-20 times over nonlinear oscillators.Source: E. Nonlinear electrical circuit… · p. 55

Concrete example

• For the example network, the physical transistors consume 1 pJ, DACs and ADCs consume 10 nJ, and digital simulation requires approximately 10^7 operations.The estimate uses NT = 100, Nd = 10, a 10 ns evolution time, 5 GHz input bandwidth, and coupling range M = 100.Source: E. Nonlinear electrical circuit… · p. 56
• A 100-transistor analog network has a self-simulation advantage of around 10^4 to 10^5 for energy and between 0.4 and 30 for speed versus the GTX GPU and DGX Superpod.The digital simulation takes 0.1 mJ (0.4 mJ) and 310 ns (3.6 ns), respectively.Source: E. Nonlinear electrical circuit… · p. 56

Scaling opportunities

• Adding 106 additional parameter transistors, each requiring 1 µW to maintain, could increase self-simulation advantages by nearly 106 before their cost becomes comparable to DAC/ADC costs.These transistors do not require fast input and output, avoiding DAC and ADC costs, but require power to compensate for their dissipation.Source: E. Nonlinear electrical circuit… · p. 56

Scalability constraints

• Connectivity requiring constant power could reduce energy advantages by up to M1 times, making passive or nearly passive coupling important for scalable physical neural networks.The predicted energy advantage can nevertheless exceed 10^10 for realistic transistor counts and M1 ≈ 10^3, while speed is unaffected by this connectivity cost.Source: E. Nonlinear electrical circuit… · p. 56

Computational bottlenecks

• Operational bottlenecking is necessary: most equivalent operations must occur physically, because read-in, read-out, and digital operations offer no physical advantage.The effective physical operation count is N_P,eff = N_C − N_D when digital operations contribute to matched performance; total PNN time and energy must be lower than digital alternatives.Source: §4 · p. 57
• For MNIST, the maximum possible gain is approximately 5, whereas CIFAR10 could permit an advantage of approximately 10^5 because its physical computation dwarfs data-transfer overhead.MNIST uses N_RO = 10, N_RI = 196, N_D ∼196, and at most 1960 equivalent operations; CIFAR10 high-accuracy classification is equivalent to 1–100 billion operations.Source: §4 · p. 58

Design strategies

• PNNs can reduce overhead by reusing few-parameter transformations, storing parameters in place, approximating desired operators, and directly feeding physical outputs into subsequent layers.PAT allows differentiable physical feature maps and digital preprocessing to be jointly trained, while physical feedforward can cascade layers without intermediate read-in or read-out costs.Source: §4 · pp. 58–59
• An optical Fourier transform is limited because N samples require read-in and read-out while the optics provide only N log N operation-equivalents, yielding at most (log N)/3 benefit.The benefit becomes significant only for extremely large N.Source: §4 · p. 58

Broader benefits

• Physical-domain inputs enable PNNs to avoid read-in and perform measurements or preprocessing unavailable to devices that first convert data electronically.PAT can train such physical “smart sensors,” potentially enabling higher-speed, higher-resolution, or lower-energy measurements, as in compressed-sensing single-pixel cameras.Source: §4 · p. 59
• Physical-output PNNs can be trained with PAT to produce a desired physical state or functionality, with measurements serving assessment while the physical object or state is the true output.Examples include output pulses, chemical effects, nanostructure functionality, and small-robot behavior.Source: §4 · p. 59