# A Comprehensive Survey on Nanophotonic Neural Networks: Architectures, Training Methods, Optimization, and Activations Functions

^{1}

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nature of Light

^{5}V/cm) where the presence of non-linear phenomena becomes significant, in the equation describing the polarization conditions of a higher rate appear, and the polarization is presented as an expansion of the Taylor sequence according to the following form [33,35]:

_{0}, with no laser beam present and the term n

_{2}I, where n

_{2}is the second-order non-linear refractive index and I is the intensity of the beam. The change in refractive index can be positive or negative.

- (1)
- |𝐸⟩: Quantum condition where, if power is calculated, the result will be E.
- (2)
- |𝑝⟩: Quantum condition where, if momentum is calculated, the result will be p.
- (3)
- |𝑥⟩: Quantum condition where, if position is calculated, the result will be x.

_{1}(𝐸) of calculating the value of energy as E, 𝑃

_{2}(𝑝) is the possibility of calculating the value of momentum as p and so on and so forth. In a |𝜓⟩, condition system, after the calculation, for example, of energy with an Ε

_{1}, result, the wave function is disrupted and collapses (transforms) into a new condition |𝛦

_{1}⟩, so that the repetition of the same calculation gives the same result. Respectively, in a |𝜓⟩, condition system, after calculating for example the momentum with the result p

_{1}, the wave function is disrupted and collapses (transforms) into a new condition |𝑝

_{1}⟩, so that the repetition of the same calculation gives the same result [37,38].

## 3. Photonic Neuromorphic Processors

- (1)
- Significant reduction of energy consumption in the applications of logical circuits as well as in data transfer.
- (2)
- Exceptionally high operating speeds with no energy consumption other than on the transmitters and the receivers.
- (3)
- Distribution of the computing power in the whole network, with each neuron performing simultaneously small parts of the whole computational activity.

## 4. Architectures

#### 4.1. Perceptron

_{ij}. The SLM is placed in the rear focal layer of the lenses, which apply Fourier transformation and sum up all the diffracted beams on the focal point as follows [49,51,54]:

^{85}Rb, in a magneto-optical trap (MOT) [55,56]. The materialization of this particular architecture is shown in Figure 3 [40,42,54,57,58]:

#### 4.2. Multilayer Perceptrons

^{th}rank into a matrix product is accomplished as shown in Equation (7) according to the singular-value decomposition (SVD) [18,49,64,65]:

_{i}of the ΝN is transferred to the optical circuit [22,66,67].

#### 4.3. Deep Photonic Neural Networks

#### 4.4. Convolutional Neural Networks

#### 4.5. Spiking Neural Networks

#### 4.6. Reservoir Computing

## 5. Training Methodologies

#### 5.1. Propagation

#### 5.2. Non-Linearity Inversion

## 6. Activation Functions

#### 6.1. z–Transform (Complex Non-Linearity)

#### 6.2. Electro-Optical Activation (Complex Non-Linearity)

- (1)
- α: the factor of input power transformation into an electric signal.
- (2)
- R: the response of the photodetector to the optical to electrical unit.
- (3)
- G: the gain of amplification rate.
- (4)
- V
_{b}: the biasing voltage (bias). - (5)
- V
_{π}: the required voltage for the π transformation of the phase.

#### 6.3. Sigmoid (Complex Non-Linearity)

#### 6.4. Softmax (Complex Non-Linearity)

#### 6.5. SPM Activation (Non-Linearity)

#### 6.6. zReLU (Non-Linearity)

#### 6.7. Cosine Activation Function (Non-Linearity)

_{i}(+) and λ

_{i}(−), are used, which, through the MZIs functioning as switches (frame sign of W

^{(1)}), are corresponded to positive and negative values of weights, respectively. Afterwards, the signals are led to the modulators (MOD, frame Input X

^{(1)}) so that the input signal can be “printed” on an optical signal of power P(

_{Xi}

^{(1)}). The next level (frame Weight |W

^{(1)}|) includes a variable optical attenuator (VOA) [104,105], which is responsible for the amplification of signal-weight as is shown in Relation (22) [18,40,49,117]:

_{1…9(+)}) and signals of negative weight (λ

_{1…9(−)}), and in the end are added up in photodiodes (blue color). In conclusion, the MZM modulator that follows (MOD) and receives the two signals operates in its non-linear area, materializing the transition function of cosine form. This particular architecture, where each neuron produces a signal that is led to the input of the next neuron, can be completed constructively and constitutes an independent photonic processor (chip) [20,36,84].

## 7. Conclusions

- (1)
- Most of the systems do not require energy for the processing of optical signals. As soon as the neural network is trained, the computations on the optical signals are conducted without any additional energy consumption, rendering this particular architecture completely passive.
- (2)
- The optical systems, in contrast to the conventional electronic ones, do not produce heat during their operation and, as a result, they can be enclosed in three-dimensional constructions.
- (3)
- The processing speed in the optical systems is restricted only by the operation frequency of the laser source of light, which reaches 1 THz.
- (4)
- The optical grids enable the multiplication of matrixes with vectors, something which is essential to NNs. The linear transformations (and some non-linear ones) can be performed at the speed of light and detected at a rate of over 100 GHz in photonic networks and, in some cases, with a minimum power consumption.
- (5)
- They are not particularly demanding as far as non-linearities are concerned, since many innate optical non-linearities can be used directly for the application of non-linear operations in PNNs, such as the activation functions.

- (1)
- The dimensions of optical devices are analogous to the light wavelength that they use (400 nm–800 nm).
- (2)
- The mass production of optical devices is limited compared to the electronic ones, since they lack at least 50 years of research and development.
- (3)
- The training of the optical grids is quite difficult because the controlled parameters are active in matrix elements deriving from powerful non-linear functions.
- (4)
- The application of matrix transformations with optical components of mass production (such as fibers and lenses) is a restriction to the spread of ONNs due to the need for stability in the signal phase and to the huge number of neurons, which are required in more complex applications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

ADC | Analog Digital Converter |

AI | Artificial Intelligence |

A-MZI | Asynchronous Mach–Zehnder Interferometer |

AONN | All Optical Neural Network |

AVM | Adjoint Variable Method |

CNN | Convolutional Neural Network |

CPU | Central Processing Unit |

CW | Continuous Wave |

DNN | Deep Neural Network |

DPNN | Deep Photonic Neural Network |

EIT | Electromagnetically Induced Transparency |

FDM | Finite Difference Method |

FM | Flip Mirror |

GPU | Graphics Processing Unit |

HL | Hyper-dimensional Learning |

MAC | Multiply Accumulate Operations |

MNIST | Modified National Institute of Standards and Technology |

MOD | Modulator |

MOT | Magneto-Optical Trap |

MP | Microprocessor |

MR | Micro Rings Resonator |

MUX | Multiplexor |

MZI | Mach–Zehnder Interferometer |

MZM | Mach–Zehnder Modulator |

NN | Neural Network |

NNN | Nanophotonic Neural Network |

NSoC | Neuromorphic Systems-on-Chip |

OCNN | Optical Convolutional Neural Network |

OIU | Optical Interference Unit |

OM | Optical Modulator |

ONN | Optical Neural Network |

ONU | Optical Non-Linear Unit |

PCC | Photonic Crystal Cavity |

PD | Photodetector |

PNN | Photonic Neural Network |

PRC | Photonic Reservoir Computing |

RC | Reservoir Computing |

RNN | Recurrent neural network |

ROC | Region Of Convergence |

SLM | Spatial Light Modulator |

SMF | Single-Mode Fiber |

SNN | Spiking Neural Networks |

SVD | Singular-Value Decomposition |

TPU | Tensor Processing Unit |

VOA | Variable Optical Attenuator |

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**Figure 1.**Photonic neural networks classification according to their architecture (stateless or stateful), their design (integrated or free-space optic) and their training ability, presented until 2019.

**Figure 2.**(

**a**) A neural network with two layers and a detailed view of one of its neurons. (

**b**) Implementation of an optical neuron with linear operation (SLM and lens units) and non-linear operation (activation function φ) [54].

**Figure 3.**Implementation of the all-optical neural network (AONN) based on free optics [54].

**Figure 4.**Average possibility of right (blue) and wrong (red) classification of this stage subject to temperature T (K) for 100 (

**a**) and 4000 (

**b**) settings [54].

**Figure 5.**Nanophotonic multilayer perceptron architecture: (

**a**) A typical NN with its input–output layers and n hidden layers. (

**b**) Hidden layers in optical implementation. (

**c**) The optical units in each hidden layer. (

**d**) The final arrangement in an integrated circuit [64].

**Figure 6.**The programmable phase shifter creates modifications in the phase, which, in turn, are converted to amplitude modifications in the directional coupler [64].

**Figure 7.**The architecture of a deep photonic neural network (DPNN) [69].

**Figure 8.**(

**a**) Schematic diagram ΝN of K-layers consisting of a multiplier (grey) and an element for the activation function (red). (

**b**) The multiplication performs a combination of inputs with the weight signals using homodyning [73].

**Figure 9.**The suggested architecture for a fully optical CNN. (

**a**) Logic Block Diagram and (

**b**) Schematic Illustration [75].

**Figure 10.**(

**a**) The circuit for the creation of repeated current peak. (

**b**) The waveforms of the implementation. One pulse of the output is led to the input via single-mode fiber (SMF), which acts as a delay element [82].

**Figure 11.**The reservoir structure in optical materialization (chip). It is consisted of interferometers for coupling and splitting between the nodes. Blue arrows represent the specific light flow, if for input is used the node indicated with black arrow. Nodes with yellow dots have output powers below the noise floor. Red ones have an amplitude above noise floor and were measured and used for offline training. For testing the device, an example waveform with sequences of bits with “1” and “0” were collected in the black square with a rounded red dot [50].

**Figure 12.**The reservoir with the 16 nodes made from silicon on insulator (SOI) MR [51].

**Figure 13.**Backpropagation ΡΝΝ. In stage (

**a**), the squares correspond to the OIUs, which materialize the linear operation (matrixes ${W}_{L}$). In blue color, we see the integrated phase shifters for the control of OIU and the training of the network. The red areas correspond to the non-linear activation functions ${f}_{L}$, which are performed through a computer. Respectively, in stage (

**b**), the presentation of the operation for the calculation of NN ranks. The route on top corresponds to the anterior propagation and the bottom to the backpropagation [96].

**Figure 14.**(

**a**) The mixed way for training: the optical signal from every node of the reservoir (blue) is transferred through a photodetector (PD) to the electric space (yellow) and through an A/D converter (ADC) to the microprocessor (MP). (

**b**) Non-linearity inversion method: the optical signals are modulated (OM) implementing the weights and summed (combiner structure), before converting to electric signal via PD. The states of the reservoir are estimated by setting the weights (red) according to a certain pattern [107].

**Figure 15.**The arrangement for the electro-optical activation function [110].

**Figure 16.**The operation principle of a neuron in an optical materialization [117].

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Demertzis, K.; Papadopoulos, G.D.; Iliadis, L.; Magafas, L.
A Comprehensive Survey on Nanophotonic Neural Networks: Architectures, Training Methods, Optimization, and Activations Functions. *Sensors* **2022**, *22*, 720.
https://doi.org/10.3390/s22030720

**AMA Style**

Demertzis K, Papadopoulos GD, Iliadis L, Magafas L.
A Comprehensive Survey on Nanophotonic Neural Networks: Architectures, Training Methods, Optimization, and Activations Functions. *Sensors*. 2022; 22(3):720.
https://doi.org/10.3390/s22030720

**Chicago/Turabian Style**

Demertzis, Konstantinos, Georgios D. Papadopoulos, Lazaros Iliadis, and Lykourgos Magafas.
2022. "A Comprehensive Survey on Nanophotonic Neural Networks: Architectures, Training Methods, Optimization, and Activations Functions" *Sensors* 22, no. 3: 720.
https://doi.org/10.3390/s22030720