Photonic neural networks at the edge of spatiotemporal chaos in multimode fibers

3.1.1 Binary classification in numerical studies

First, we employ numerical studies to assess the effectiveness of our chaotic reservoir compared to known models with GRIN MMF and to determine the effect of peak power on the system. We use the Breast MNIST dataset, composed of breast ultrasound images, to characterize the type of imaged tumor, benign or malignant. For the simulations of the stable state, we modulated the phase of an input Gaussian beam only with the phase of the information as done previously. To simulate the chaotic state, we modulated the phase of an input Gaussian beam with the phase of the information and the orbital angular momentum (OAM) phase. Phase modulation by the OAM phase leads to a vortex beam shape, where intensity accumulates in the ring rather than accumulating in the center. As initial energy is around the ring, we excite high-order modes. Due to the initial energy of high-order modes being significantly high compared to fundamental modes being significantly high, we satisfy conditions for chaos in the modal energy flow attractor. In both simulations, we couple light into a multimode fiber after phase modulation and propagation through the lens. We then record output beam profiles and apply ridge classification to complete learning.

Simulations with 1 MW, 5 MW, 10 MW, and 20 MW of input peak powers are performed to assess system response in stable (without OAM phase) and chaotic regimes (with OAM phase). Here, we would like to emphasize that these peak powers are scaled to MW range to simulate less propagation distances thus to reduce simulation time. In our simulations we observe that the chaotic regime results in unorthodox outcomes compared to a stable regime. In 1 MW, due to low nonlinearity, neither encoding method is significant compared to the other. In 5 MW, the chaotic regime outputs the best accuracy of 83.34 %, while the stable model can achieve a maximum accuracy of 82.70 % at 10 MW. Results of the numerical simulations demonstrate that higher accuracies are feasible with less peak power compared to a stable state. After the optimal point, the accuracy of the chaotic model fluctuates due to over-processing. Similar effects on chaotic reservoirs have been demonstrated in the literature [22]. Simulation results and confusion matrices for the spatiotemporal chaos-based photonic neural network are illustrated in Figure 2.

Figure 2:

Numerical simulation results on breast MNIST dataset. (a) Recorded ridge classification accuracies of stable and chaotic regimes on different pulse peak powers. (b) Confusion matrices of ridge classification after chaotic transformation in different peak powers.

3.1.2 Analysis of spatiotemporal chaos in the system

When the ratio of initial energy coupled to high-order modes and initial energy coupled to fundamental modes is greater than 1/50, mode coupling becomes chaotic in graded-index multimode fibers [25]. To verify the previous study and the presence of spatiotemporal chaos in our system we have conducted another numerical study. In dynamical systems, the largest Lyapunov exponent is a widely used benchmark to characterize the presence of chaos in the system [26]. A largest Lyapunov exponent greater than zero indicates that the system is sensitive to initial conditions and is chaotic. Utilizing the same pipeline, we disclosed above, we scanned 20 different points with different peak powers. Then we estimated the largest Lyapunov exponent for each measurement using a method widely utilized in the literature [27]. We estimated the overall largest Lyapunov exponent by estimating the local largest Lyapunov exponent for each point in space and taking the average of all measured points since all these measured points are physically coupled to each other.

As demonstrated in Figure 3, although not as chaotic as known systems such as Lorenz attractor, our system has chaotic properties with a largest Lyapunov exponent greater than zero. The transition from order to chaos occurs around 4 MW peak power in the numerical simulations. In the experimental case, we observe similar effects around 2 kW peak power which is also demonstrated in Figure 3. The difference in peak powers of order to chaos between numerical and experimental studies is due to the power scaling method used in TBPM which is elaborately disclosed in the Methods section. In the literature, such weakly chaotic systems demonstrating the sensitivity of initial conditions while not compensating for a deterministic response to experimental noise were demonstrated on electrical hardware. Our system exhibits similar chaotic transformation and clustering effects [28], [29]. Also, in the literature polynomial reservoir computers are demonstrated to work best in the boundary between order and chaos [30]. Interpreting the literature with our results, our model also works at the edge of chaos but on the chaotic side of the border, thus harnessing the peculiarities of chaotic systems. In our experimental studies, we also empirically verify that the system does not converge to its stable state when using our methodology. In graded-index multimode fiber such a stable state corresponds to spatial beam self-cleaning.

Figure 3:

Numerical and experimental analysis of spatiotemporal chaos in our methodology. (a) Estimated largest Lyapunov exponent. The acronym HOM stands for high-order modes. (b) Encoded data in phase patterns. (c) Phase that was applied to the input Gaussian beam without OAM phase. (d) Output beam profile without high-order mode excitation. (e) The OAM phase that was applied to the input Gaussian beam. (f) Output beam profile with high-order mode excitation.

3.2.1 Experimental framework and selection of parameters

Similar to our numerical studies, we start the experiments by optimizing our pulse peak power for the reasons mentioned earlier that affect information processing and model performance. Again, we employ the Breast MNIST dataset to establish initial benchmarks in peak power. In the remainder of this study, we use the following encoding procedure. As the datasets we employ have small dimensionality, we up-sample all images to 800 × 800 and encode information to the input beam as a phase along with an orbital angular momentum phase using a spatial light modulator (SLM). After sending information to high-order modes and chaotic mode coupling within the multimode fiber, we obtain a speckle pattern on which we then employ ridge classification. After creating the pipeline of data encoding, we tested our model with various peak power settings.

With an input pulse peak power of 1 kW, our model performs at 78.84 %. Although the model surpasses the baseline accuracy of 75 %, data is under processed, and increasing peak power and nonlinearity is necessary. After increasing the pulse peak power to 2 kW, the model converges into its maximum recorded accuracy of 83.34 %. Finally, when we increase the pulse peak power to 2.5 kW, although beam cleaning is not visible like in numerical studies, data is over-processed, which harms overall learning accuracy. The results of the peak power test are illustrated in Figure 4.

Figure 4:

Binary classification results utilizing breast MNIST dataset. (a) Confusion matrix of baseline ridge classification accuracy without chaotic transformation. (b) Confusion matrices of ridge classification after chaotic transformation with different input pulse peak powers.

3.2.2 Breast MNIST dataset

To assess the initial performance of machine learning with spatiotemporally chaotic photonic reservoir computing, we employ a binary classification before moving on to complex multi-class datasets. Like the numerical studies, we use the BreastMNIST dataset for experimental classification. We employ the same methodology in data processing and process our data again with a ridge classifier. Without our chaotic reservoir, the linear ridge classification layer results in a test set accuracy of 75 %. As shown in Figure 4, we obtain a test set accuracy of 83.34 %, improving baseline by 8 % overall accuracy and by 14 % malignant tumor accuracy. Our model performs on par with contemporary convolutional neural networks while training approximately seven orders of magnitude fewer parameters, as shown in Table 1. Such performance demonstrates the effectiveness of chaotic reservoir computing utilizing multimode fibers in biomedical disease characterization settings.

3.2.3 Fashion MNIST dataset

Encouraged by the results in biomedical binary classification, we employ multi-class classification tasks with our computing method. The fashion MNIST dataset comprises 10 classes of various clothing types, such as T-shirts, bags, or boots. We use the official dataset split of 60,000 training and 10,000 test sets. We employ the same data encoding process to the SLM, record the corresponding output speckle patterns, and complete the data classification using linear readout. In this dataset, without chaotic transformation, the Ridge classification layer results with a complete test set accuracy of 75.82 %. After the chaotic reservoir, the same readout algorithm results with a complete test set accuracy of 78.62 %. As shown in the confusion matrices in Figure 5, we observe an increase in accuracy in most classes. Our framework performs better for this dataset than similar models implemented in multimode fibers, demonstrating the efficacy of chaotic elements in machine learning.

Figure 5:

Confusion matrices of ridge classification before and after chaotic transformation. (a, c) Confusion matrices before chaotic transformation for FashionMNIST and EuroSAT dataset. (b, d) Confusion matrices after chaotic transformation for FashionMNIST (b) and EuroSAT (d) dataset.

3.2.4 EuroSAT dataset

To demonstrate a wide range of implementation for our model, we decided to move on to a geospatial dataset that utilizes satellite images. For this task, we use the EuroSAT dataset, composed of 10 classes of satellite imagery, including industrial areas, residential areas, and different crop areas. After following the same data processing and collection procedure, we again employ ridge classification. For the EuroSAT dataset, baseline ridge classification without our chaotic reservoir outputs an accuracy of 26.46 %, demonstrating the task’s difficulty and linear inseparability of samples in feature space. After processing input data using our chaotic reservoir, we observe a 58.15 % increase in accuracy from 26.46 % to 84.61 %. Confusion matrices in Figure 5 also demonstrate the transition from scrambled data to linearly separable data, along with the significant increase in accuracy in all classes. Similar to the BreastMNIST dataset, we compare the performance of our model with modern convolutional neural networks.

Our method works on par with convolutional neural networks (CNN), again training many orders of magnitude fewer parameters. Our benchmark CNN models are not fine-tuned over many hours of training but are trained from scratch. While benchmarking our system with state-of-the-art neural networks, we also train smaller neural networks, LeNet-1 and LeNet-5 [33], with similar number of trainable parameters for fair comparison (see Supplementary Material for details). These results demonstrate the efficiency of the spatiotemporal chaotic information processing method on an experimental basis.

3.2.5 Clustering property of spatiotemporal chaotic reservoir

Previous works in the literature demonstrate pattern recognition and clustering by utilizing chaotic dynamical systems as reservoirs [28]. In this study, we further analyze the EuroSAT dataset’s results and observe our chaotic reservoir’s clustering properties. We perform linear discriminant analysis (LDA) on raw and processed data to visualize three-dimensional feature space. In the initial distribution illustrated in Figure 6, it can be seen that the classes cannot be separated with any linear function, demonstrating the complexity of the dataset. After chaotic transformation, data points cluster alongside other data points that belong to the same class, which results in easily separable data and significantly higher accuracies in all classes. While shedding light on classification results in Figure 5, these results also demonstrate the common effect of clustering among chaotic reservoir computers in our system.

Figure 6:

Impact of chaotic nonlinear transformation on EuroSAT data points to enhance classification performance. (a) Feature space before chaotic transformation. (b, c) Feature space after chaotic transformation.