Deep learning's success across various domains is undeniable, primarily due to training multi-layered artificial neural networks. The universal approximation theorem supports this, stating that a single-hidden-layer feed-forward network can approximate continuous functions. However, electronic devices face power and bandwidth limitations hindering deep learning's broader application in mobile and embedded systems. Optical neural networks (ONNs) offer a promising alternative due to their potential for high-bandwidth, low-latency, and power-efficient inference. However, building programmable ONNs using electronically controlled optical components presents significant challenges. These components deviate from perfect models due to inconsistencies in microstructures and materials, leading to unpredictable effects. Existing approaches struggle to simulate or train programmable ONNs effectively because of these unpredictable factors introduced by optics, electronics, and the physical environment. This work introduces a novel approach for practically training loose neuron arrays for various tasks, enabling the transition from handcrafted to non-handcrafted hardware designs. The approach uses a concept that connecting many artificial neurons in a non-handcrafted way significantly outperforms sophisticated handcrafted algorithms in many tasks. This concept is applied to create programmable incoherent ONNs, which are numerically verified using liquid-crystal (LC) hardware. The LC neurons are trained using a functional learning (FL) paradigm. A prototype, the Light Field Neural Network (LFNN), uses readily available optical components such as LCD panels, polarizers, and a camera without fine calibration. The LFNN can perform different tasks by switching parameters, functioning as a power-efficient light-speed compute unit or programmable lens/displays. The main challenge in training such a system stems from the lack of an accurate analytical model due to high dimensionality, non-differentiable parameters, and complex interactions within the system. Therefore, FL is proposed as the training paradigm.

Existing research in optical neural networks highlights the potential of ONNs for high-performance computation, but faces significant challenges in programmability and training. All-optical machine learning using diffractive deep neural networks and hybrid optical-electronic convolutional neural networks are notable examples. Deep learning with coherent nanophotonic circuits and in situ optical backpropagation training of diffractive optical neural networks also explore different aspects of optical computing. However, these methods often rely on idealized models and struggle to account for the complexities and imperfections of real-world optical components. The authors point to the difficulty of training ONNs with thousands of parameters using conventional methods like backpropagation due to high-dimensional, non-differentiable, nonlinear, and correlated physical systems. Existing alternatives such as finite difference and genetic algorithms are also found to be ineffective for large-scale optimization tasks like training LFNNs.

The core methodology centers on the Functional Learning (FL) paradigm, designed to train the parameters of a model-free system like the LFNN. The FL paradigm formulates the training problem as minimizing the difference between the system's output and the desired target output. Because the system's behavior (f(x;p)) is unknown, a Functional Neural Network (FNN) is used to approximate it. This introduces two sub-problems: minimizing the difference between the system's output and the FNN's output, and minimizing the difference between the FNN's output and the desired target output. These are solved iteratively using alternating minimization. The FNN consists of a physically-inspired function basis block reflecting the light propagation within the LFNN, and a convolutional neural network (CNN) for nonlinear combination. The training process involves two stages: z-learning and p-learning. Z-learning updates the FNN's parameters by capturing the actual output of the LFNN for randomly selected inputs. P-learning updates the LFNN's hardware parameters based on the training data and the FNN's approximation of the system. The LFNN prototype uses liquid crystal displays (LCDs) as controllable polarization rotators. The weights are modulated by an electronically controlled polarization field and polarizing filters. The architecture allows training of various neuron array configurations such as regular arrays and uniform arrays. For multilayer LFNNs, a cyclic feeding process is employed with X-activation, batch normalization, and an amplitude filter to introduce nonlinearities and handle the limitations of incoherent optical systems. Sparsification techniques are used to reduce the computational burden by pruning less significant connections in the FNN. The experiments use the MNIST and CIFAR-10 datasets for image classification and an RGB-D dataset for depth estimation. Comparison is made with digital neural network results and other training paradigms like finite difference and genetic algorithms.

The study demonstrates the successful training of the LFNN using the proposed FL paradigm. In object classification tasks using the MNIST dataset, the 1-layer LFNN achieved 91.02% accuracy, comparable to a digital dense layer (92.71%). A 2-layer LFNN achieved 94.35%, slightly lower than the digital counterpart (98.32%). With CIFAR-10, a 3-layer LFNN reached 45.62%, again comparable to a digital network (53.62%). The simulation of various neuron array configurations, including uniform arrays, demonstrated that non-handcrafted designs can outperform handcrafted ones. Object recognition experiments using MNIST and CIFAR-10 showed high recognition accuracy (98.12% for digit 0 recognition and 77.30% for plane recognition). The 4-layer LFNN was also successfully applied to depth estimation tasks. Experiments with Bernoulli arrays, where a certain percentage of LC neurons were randomly disabled, demonstrated the robustness of the FL paradigm, showing reasonable accuracy even with a significant number of malfunctioning neurons. The functional learning paradigm significantly outperformed the forward model, genetic algorithm, and finite difference approaches in terms of classification accuracy on the MNIST dataset. The study shows that increased neuron size in simulations leads to accuracy comparable to digital neural networks. The main limitations were training time due to the use of a low-speed camera and the relatively limited number of trainable parameters in the hardware compared to digital counterparts.

The results demonstrate that the functional learning paradigm enables the training of uncalibrated, low-precision optical hardware to achieve performance comparable to that of digital neural networks. This approach overcomes the limitations of traditional methods by implicitly training the system without relying on precise models of its internal behavior. The high accuracy achieved even with inexpensive and noisy components highlights the robustness and potential of the FL paradigm for building cost-effective, high-performance optical neural networks. The superior performance of the uniform neuron array in simulations underscores the potential of non-handcrafted hardware designs. The LFNN's ability to perform various tasks (classification, object recognition, depth estimation) demonstrates its versatility and potential as a general-purpose optical computing unit. The study opens new possibilities for hardware design, manufacturing, and system control by releasing these processes from the constraints of high-fidelity components and precise assembly. Future work should focus on improving the training speed and exploring advanced network architectures within the optical domain.

This research successfully demonstrates a new paradigm for training optical neural networks using functional learning. The light field neural network prototype achieved comparable performance to digital counterparts while requiring significantly fewer parameters. The functional learning approach proves robust, even with inexpensive and noisy components, paving the way for more affordable and efficient optical computing. Future directions include improving training speed, developing advanced network architectures, and integrating all-optical nonlinear activation mechanisms. The research contributes significantly to the field of optical computing by removing the limitations of handcrafted designs and achieving performance comparable to state-of-the-art digital neural networks.

The current LFNN prototype uses a low-speed camera, resulting in longer training times. Increasing the number of neurons and layers also requires greater computational resources. Noise from both optical and electronic components affects performance. While the FL paradigm handles some imperfections, further noise reduction is crucial. Additionally, current activation functions are implemented digitally, requiring all-optical or non-digital implementations for full speed-of-light operation. Lastly, while this work demonstrates the feasibility of the LFNN, testing it with real-world objects in a natural, unconstrained environment represents a significant future challenge.