Quantum assisted data-driven simulation methods and systems

Abstract

Implementations involve generating quantum hybrid neural networks on a computing platform, by leveraging the computational advantages of both quantum and classical neural networks to enhance data processing capabilities. A hybrid neural network may include a quantum neural network and a classical neural network. The quantum neural network may process input data using a quantum processor to produce intermediate data. Subsequently, the classical neural network may process this intermediate data as its own inputs to generate output data. The hybrid neural network may then be stored back onto the electronic storage device for immediate or later use or training.

Description

QUANTUM ASSISTED DATA-DRIVEN SIMULATION METHODS AND SYSTEMSCROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Application No. 63 / 613,731 , filed on 21 December 2023, the entire disclosure of which is hereby incorporated by reference.BACKGROUND

[0002] Engineering simulations have undergone a dramatic transformation since their beginnings in the 1940s and 1950s. The widespread adoption of computing systems for testing designs fostered a growing reliance on computer-based simulations.

[0003] This interdependence fueled the advancement of computing power, making it an indispensable tool in the product development cycle. The recent surge in machine learning and deep learning methods further exemplifies the exponential growth of computing capabilities in engineering simulations.

[0004] The demand for high-performance computing with graphical processing units (GPUs) is rising in tandem with the growing interest in machine learning. Scientific computing, particularly engineering simulations, is a major driving force behind both advancements. This synergy positions engineering simulations as a candidate for leveraging data-driven methods. Across various industries, data-driven modeling techniques may be implemented in product development. Material science, for example, may adopt deep learning for atomistic simulations, material imaging, spectral analysis, and even natural language processing of vast material databases. Similarly, chemical engineering is exploring data-driven models for thermodynamics, statistical mechanics, and molecular simulations.

[0005] In heavy-engineering industries like aerospace, automotive, and others, the design space is massive and requires a combination of high-fidelity studies and techniques like surrogate modeling. Machine learning, especially deep learning, is being utilized for surrogate-based modeling. However, the needs of such industries exceed the availability of computing resources using conventional models and simulations.

[0006] One conventional approach to performing defect detection using image processing is classical computer vision. Computer vision simulates human vision by processing and analyzing images and videos, allowing machines to extract meaningful information from visual inputs and make decisions or perform actions based on thatinformation. Computer vision algorithms may be trained to identify objects, scenes, and activities in images and videos, such as defects. This recognition process often involves detecting and identifying patterns, shapes, and colors to understand the content of an image.

[0007] Computer vision employs object detection, or identifying specific objects within an image, such as features understood to be defects, or potential defects. Object detection algorithms may be machine learning algorithms trained using large datasets of labeled images, where each object of interest is binned and categorized. These algorithms can use features like edges, textures, and shapes to distinguish different objects and features. Some object detection models can even detect multiple objects in complex scenes.

[0008] Computer vision techniques are conventionally followed by classification ( / .e., binning) of the detected features (e.g., defects). The classification process can categorize images into different classes based on their contents or features. For example, an image classification model can distinguish between images of actual defects and non-defects. Using machine learning, the model can form a set of training images for each category and then use this knowledge to classify new images.

[0009] Deep learning algorithms, such as convolutional neural networks (CNNs), can improve the accuracy of image classification, enabling machines to categorize images with a high degree of accuracy. However, this process is computing-power- intensive, and as such it is difficult to scale due to the linearity of classical computing systems.

[0010] Therefore, improved simulation and defect detection capabilities are needed.SUMMARY

[0011] This Summary is intended to introduce, in an abbreviated form, various topics to be elaborated upon below in the Detailed Description. This Summary is not intended to identify key or essential aspects of the claimed invention. This Summary is similarly not intended for use as an aid in determining the scope of the claims.

[0012] In some aspects, the techniques described herein relate to a method, including: receiving, at a processor of a computing platform from an electronic storage device in electronic communication with the processor, a classical neural network; generating, using the processor, a hybrid neural network including: a quantum neural network configured for processing input data using a quantum processor andyield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and storing, using the processor, the hybrid neural network to the electronic storage device.

[0013] In some aspects, the techniques described herein relate to a system, including: a processor of a computing platform; and an electronic storage device in electronic communication with the processor; wherein the processor is configured to: receive a classical neural network; generate a hybrid neural network including: a quantum neural network configured for processing input data using a quantum processor and yield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and store the hybrid neural network to the electronic storage device.

[0014] In some aspects, the techniques described herein relate to a tangible, non- transitory, computer-readable medium having instructions thereupon which when implemented by a processor cause the processor to perform a method including: receiving, from an electronic storage device in electronic communication with the processor, a classical neural network; generating a hybrid neural network including: a quantum neural network configured for processing input data using a quantum processor and yield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and storing the hybrid neural network to the electronic storage device.BRIEF DESCRIPTION OF THE FIGURES

[0015] For a fuller understanding of the nature and objects of the disclosure, reference should be made to the following detailed description taken in conjunction with the accompanying drawings, in which:

[0016] FIG. 1A illustrates a method of performing QTL and QML, according to one or more implementations herein.

[0017] FIG. 1B illustrates an example of performing image recognition for defect detection and classification QML of an image, according to one or more implementations herein.

[0018] FIG. 2 illustrates an architecture of quantum-assisted PINN, according to one or more implementations herein.

[0019] Fig. 3 illustrates the results of QA-PINN to simulate the 1-D Burger equation at four different time steps (t = 0.25, 0.50, 0.75, 0.99) and with three different qubit architectures.

[0020] Fig. 4 illustrates an L2-norm of QA-PINN, c-PINN and GAAF-PINN with respect to ground truth over the four time steps.

[0021] FIG. 5 Illustrates an operational environment, according to one or more of the implementations herein.

[0022] FIG. 6 is a diagram of example components of a device, according to one or more implementations herein.

[0023] FIG. 7 is a diagram of example components of a device, according to one or more implementations herein.

[0024] FIG. 8 is a flowchart illustrating an example method, according to one or more implementations herein.

[0025] FIG. 9 illustrates an artificial neural network (ANN), according to an implementation.

[0026] FIG. 10 illustrates a node, according to an implementation.

[0027] FIG. 11 illustrates a method of training a machine learning model of a machine learning module, according to an implementation.

[0028] FIG. 12 illustrates a method of analyzing input data using a machine learning module, according to an implementation.DETAILED DESCRIPTION

[0029] Implementations involve generating quantum hybrid neural networks on a computing platform, by leveraging the computational advantages of both quantum and classical neural networks to enhance data processing capabilities. A hybrid neural network may include a quantum neural network and a classical neural network. The quantum neural network may process input data using a quantum processor to produce intermediate data. Subsequently, the classical neural network may process this intermediate data as its own inputs to generate output data. The hybrid neural network may then be stored back onto the electronic storage device for immediate or later use or training.

[0030] Quantum machine learning (QML) may refer to the integration of quantum computers into the operational systems and processes that implement machine learning. Quantum computers, unlike classical computers, leverage the principles of quantum mechanics to process information. The fundamental units of quantum computation are qubits, which, unlike classical bits that are either 0 or 1 , can exist in a superposition of states. This allows quantum computers to process vast amounts of data simultaneously, potentially solving complex problems much fasterthan traditional computers. Quantum algorithms can perform linear algebra operations with exponentially greater efficiency than classical computers.

[0031] Quantum machine learning algorithms exploit the properties of quantum computers by adapting classical machine learning algorithms to the quantum context.“Quantum” versions of linear algebra techniques and optimization procedures can be used.

[0032] In implementations herein, quantum convolutional neural networks (QCNNs) can be used to solve optimization problems inherent in training machine learning models, such as image processing for defect detection. Implementations may employ QMLs such as QCNNs for identifying surface defects in an image. The feature extraction capabilities of such deep learning models using convolutional neural networks may provide for enhanced identification of defects.

[0033] Implementations may employ quantum transfer learning (QTL) models for training the model. Use of QTL models according to implementations herein may significantly reduce the time and computing power required for training. QTL may begin with a quantum model trained on a large and diverse dataset. This pretrained model may serve as the starting point for the transfer learning process.

[0034] Transfer learning includes machine learning techniques whereby a model trained on one task is repurposed or adapted for use on a second related task. Instead of training a model from scratch for a new task, transfer learning leverages knowledge learned from solving one problem and applies it to a different but related problem.

[0035] Conventional classical methods of transfer learning may be based on localization, classification, or segmentation methods. However, the complexity of the tasks is increased greatly while processing large images, high resolution images, or many images due to high memory consumption and computational cost, in addition to practical limitations arising from noisy and unclear images. Thus, conventional classical methods of transfer learning may be unsuitable for use with such large data sets.

[0036] Various implementations herein employ quantum computational processing to solve the shortcomings of conventional classical AI / ML methods through one or more of the following: feature extraction using quantum circuits; hybrid quantum-classical preprocessing; hybrid quantum-classical transfer learning methods; generic plug- and-play module for custom solution; problem-specific feature identification and classification; efficient quantum-enhanced low-level feature extraction on potentially noisy images; exploring hybrid quantum-classical transfer learning methods to addquantum enhancements to existing state-of-the-art-classical AI / ML methods. Scaling up problems for larger images with higher resolutions may employ processing on advanced quantum computers, implementing error mitigation and / or correction techniques to realize one or more quantum advantages on a larger scale.

[0037] QTL may involve feature extraction and fine-tuning. In a feature extraction stage, the pre-trained quantum model may be used to transform new data into a quantum representation, through quantum embeddings, where classical data is encoded into quantum states. The quantum model may process these states, extracting relevant features. This process may leverage the inherent parallelism and high-dimensional state space of quantum systems, allowing for complex and nuanced feature extraction.

[0038] In a fine-tuning stage, the extracted features may then be used to train a new, often smaller, quantum or classical model specific to a target task. This stage may involve adjusting or retraining some of the layers of the quantum model while keeping others fixed. The model may thus be adapted to the specifics of the new task using a potentially much smaller dataset than what was used for the initial training. This adaptation is facilitated by the rich feature representation extracted in the first stage, allowing the model to quickly learn the new task with fewer resources and data.

[0039] Implementations herein may use QTL to transfer a general model to a model that employs QML (e.g., a QCNN) to perform a specific task on a specific dataset. The QML employed may include a trainable quantum circuit which can be integrated with the pre-trained classical model.

[0040] FIG. 1A illustrates a method 100 of performing QTL and QML, according to one or more implementations herein. A generic network 110 (e.g. machine learning model) may be provided, which may be configured to perform a generic task 118 using generic data 116. The generic network 110 may be pre-trained, or may be trained using the generic data 116 to perform the generic task 118. Using QTL, the generic network 110 may be transformed into a QML model 120 comprising a classical pretrained network 122 and a trainable quantum circuit 124. The QML model 120 may be configured to receive input of a specific data set 126 and perform a specific task 128 using the specific data set 126. For example, the specific data set 126 input to the QML model 120 may be image data, which may or may not contain possible defects, and the specific task may include image processing for defect detection and classification.

[0041] Unlike conventional transfer learning methods that solely rely on classical computational processes, the disclosed methods and systems employ adaptable quantum circuits that work in conjunction with pre-trained classical models. This synergy allows for the leveraging of quantum parallelism and entanglement to potentially solve complex problems more efficiently than classical methods alone. Furthermore, unlike ensemble learning, which combines multiple models to make a prediction, the disclosed approach refines a single model with quantum enhancements, potentially reducing computational overhead and improving performance on specific tasks. In contrast to sparse representation techniques that focus on representing data with a minimal number of basis elements, the quantum- enhanced method focuses on exploiting the high-dimensional space of quantum states to achieve more sophisticated data representation and processing.

[0042] FIG. 1B illustrates an example of performing image recognition for defect detection and classification QML of an image 131 , according to one or more implementations herein. A quantum convolution layer 132 (e.g., a QCNN sliding window with a filter size 2x2 and a stride 2) may utilize a sliding window concept for filtering a part of the image 131 to identify features. Following this, a classical filtering processing 133 (e.g., a classical convolution of filter size 2x2, stride 1 and implementing ReLU activation) may be performed implementing rectified linear unit (ReLU) activation. Following this, the data may be normalized at 134 (e.g., a fully connected layer of 30 units using ReLU activation) to compare a fully connected network 135 (e.g., a fully connected layer using sigmoid activation) with the partial image. Such a sliding window in a quantum convolutional neural network may process the 4x4 pixel part from the image for identifying defects. Such identification may occur using / within a CNN.

[0043] Surface defect detection is used in many industries for quality control in both design and manufacturing. In a design context, it may involve identifying imperfections and / or design trouble spots (e.g., stress / strain / flow trouble spots) in a designed part. In a manufacturing context, it may involve identifying imperfections such as scratches, dents, or inconsistencies on the surfaces of materials like metals, plastics, or ceramics. In both contexts, the process may be used to enhance reliability and functionality of a design or produced part.

[0044] With advancements in technology, automated systems using image processing have become prevalent. These systems may employ cameras to capture images of the product surfaces. The images may then be analyzed using algorithms to detectdefects. This approach may enhance accuracy and efficiency, significantly reducing the time required for inspection.

[0045] Implementations herein solve at least some of the shortcomings of conventional image processing for defect detection by leveraging quantum computing to conduct processing-intensive tasks. Implementations herein may perform image processing defect detection and classification using a system of QML and QTL models. As such, implementations herein may increase accuracy, as well as reduce the time and computing power for effectively identifying and classifying surface defects. Implementations may further reduce false positives (e.g., non-defect features or nuisance identified or classified as defects), and reduce training time (e.g., fewer epochs to train). By providing these advantages, implementations herein may contribute to improved quality and reliability of finished products.

[0046] An illustrative implementation of a QML model within the context of a quantum- assisted Physics-Informed Neural Network (PINN) will now be provided.

[0047] PINNs correspond to a method in computer-aided engineering (CAE), which may be used for design space exploration studies (e.g., surrogate modeling and design of experiments) by combining deep learning with physics-based modeling. Among others, PINNs provide advantages of generalizability and training efficiency, with one aspect of these factors being trainable parameters. Some implementations herein improve these two factors by introducing QML in a classical PINN framework. Such implementations may represent a QML model implemented as a Quantum-Assisted PINN (QA-PINN). A QA-PINN may reduce the trainable parameters while maintaining accuracy.

[0048] An example implementation utilized a standard forward problem of simulating partial differential equations (PDEs) from the field of Computational Fluid Dynamics (CFD), a well-understood area within classical PINN. A comparative analysis was performed between QA-PINN according to implementations herein, classical vanilla PINN (c-PINN), and state-of-the-art classical PINN (e.g., Globally Adaptive Activation Function PINN (GAAF-PINN)).

[0049] All methods were benchmarked against a ground-truth analytical solution for 1 D Burger’s equation. The analysis is conducted for 3, 4, and 5-qubit QA-PINN architectures. In comparison to c-PINN and GAAF-PINN, 4 and 5-qubit QA-PINN according to implementations herein was shown to reduce the number of trainable parameters of the network by 20%, all while maintaining the accuracy.

[0050] This represents a dramatic reduction in needed computing power to solve a CFD or engineering design optimization problem. The reduced number of trainable parameters further indicate the reduced complexity of the model and reduced risk of overfitting, enabling further generalization. Thus, implementations can significantly improve uses of PINNs by enhancing generalization and improving training efficiency for deploying the model for real-world applications.

[0051] For surrogate-based modeling, one of the deep learning methods that have gained traction in recent years is the PINN, a data-driven approach to solving forward problems (such as partial differential equations) and inverse problems (such as design optimization). Fluid mechanics is one of the most challenging and popular applications of PINN. Though PINNs are not meant to replace high-fidelity CFD studies, they alleviate the challenges associated with mesh-based approximate methods for design space exploration studies.

[0052] Despite its popularity, a major issue with PINN and most of its variants is the lack of generalization due to overfitting issues caused because of high trainable parameters. The downstream effect of a high number of trainable parameters is an increase in training time. Together these reasons lead to a situation where a PINN trained for a particular problem requires retraining to solve another similar problem, leading to inefficient training performance. Hence, the usefulness of PINN has mainly been restricted to inverse problems.

[0053] To address these inherent challenges of classical PINN, implementations herein include a QA-PINN approach to solve a forward problem of solving a nonlinear and time-dependent PDE - 1 D viscid Burgers equation.

[0054] This approach represents an improvement over conventional PINNs or quantum simulation proposals, as it minimally alters classical PINN architecture and requires a low number of qubits. Implementations may reduce the trainable parameters, thereby improving the generalizability and training efficiency of classical PINNs.

[0055] The investigation of QA-PINN was performed on the 1-D viscid Burger’s equation, a non-linear time-dependent partial differential equation (as shown in Eq. 1). The equation is a fundamental model in computational sciences, widely used to study various phenomena across disciplines. It is a fundamental partial differential equation and can be derived from the Navier-Stokes equations for the velocity field by dropping the pressure gradient term. Though simplified, it retains the time dependency and nonlinearity, making it an ideal test case for assessing the accuracy and efficiency of computational approaches, including PINN.du du d2u~ o^t+ u~ oTx = ox£[Eq. 1]

[0056] Here u is velocity, x is displacement, t is time, and v is viscosity. The PDE is solved as a forward problem with v = 0.01 / TT. For the choice of v « 1, Burgers’ equation leads to a shock formation, a phenomenon that is challenging to solve by traditional numerical methods and serves as a good test case for benchmarking PINNs.

[0057] The initial and boundary conditions applied to the problem are as follows: u(x, t = 0) = — sin(7rx) [Eq. 2] u x = —1, t) = 0 [Eq. 3] u(x = +1, t) = 0 [Eq. 4]

[0058] The problem may be formulated as a conventional PINN problem, with the PDE and the corresponding initial and boundary conditions imposed on the network as the loss function. The lower the loss function value, the better the network obeys the PDE and the associated initial and boundary conditions. The dataset values for x and t are randomly generated in the range [-1,1] and [0,1], respectively.

[0059] FIG. 2 illustrates an architecture 200 of quantum-assisted PINN, according to one or more implementations herein. Implementations of a QA-PINN architecture may introduce a quantum hidden layer before the classical hidden layer in the classical PINN architecture. Unlike classical hidden layers, the quantum hidden layer uses the rotation values of the rotational gates as trainable parameters.

[0060] Introduction of the quantum hidden layer may enhance the performance of systems performing data-driven simulations by reducing the number of trainable parameters without a tradeoff in accuracy. QA-PINN may assist PINN by overcoming the challenges of generalizability and training efficiency. In an example implementation, the accuracy and performance of the QA-PINN for simulating the 1 D Burger equation with 3, 4, and 5 qubit architectures may be investigated. The total number of trainable parameters for 3-qubit, 4-qubit, and 5-qubit QAPINNs corresponding to this example is given in Table 1.

[0061] In this example, for each QA-PINN architecture, 8 layers of quantum rotational gates (RX gates) along with alternative blocks of full entanglement were used. The architecture begins with an input layer 210 that takes in the value for x and t, followed by a quantum hidden layer 220, a classical hidden layer 230, and then an output layer 240 that predicts the value for the function u. The output from thequantum hidden layer 220 may be extracted as a probability state vector and then may be passed to the following classical hidden layer 230.

[0062] The classical PINN conventionally uses the hyperbolic tangent (tanh) activation function inside its architecture. Unlike popular activation functions like ReLU, the hyperbolic tangent activation function does not vanish for the second or higher- order gradient terms required to calculate the PINN loss function. However, using tanh activation function inside the neural network architecture results in the ’’vanishing gradient” problem. An activation function 222 of the quantum hidden layer 220 an activation function 232 of the classical hidden layer 230, may each be, for example, tanh.

[0063] The architecture of the c-PINN replaced quantum hidden layer 220 with a traditional classical hidden layer. The performance of QA-PINN was also tested against the Globally Adaptive Activation Function PINN (GAAF-PINN), considered a state-of- the-art PINN. GAAF-PINN is a modern variant of the classical PINN. GAAF-PINN may implement trainable parameter (per hidden layer) alongside the activation function. This may enable the network to change the shape of the activation function making the activation function learnable. By adapting activation function, GAAF-PINN can deal with the vanishing gradient and poor weight initialization issues. These extra trainable parameters distinguish GAAF-PINN from the c-PINN.

[0064] In testing, QA-PINN was benchmarked against c-PINN and GAAF-PINN. These equivalent classical PINNs were obtained by replacing the quantum hidden layer of QA-PINN with a classical hidden layer while ensuring that the number of features coming into the layer and the number of features leaving the layer are the same. It was observed that replacing the quantum hidden layer with a classical hidden layer results in a higher number of trainable parameters. The difference in the number of trainable parameters between QA-PINN and the classical PINNs increased with the number of qubits of the QA-PINN as it is observed in Table 1.

[0065] TABLE 1 : NUMBER OF TRAINABLE PARAMETERS FOR QA-PINN, c-PINN, AND GAAF- PINN

[0066] The governing PDE 250, and the initial and boundary conditions 260 may be imposed as the network’s loss function 270, which may be used to update the weights 280 of either or both of the quantum hidden layer 220 or the classical hidden layer 230. The individual loss functions that were used in this example are described below:lossic= (u(x, t = 0) + sin(7rx))2[Eq. 6] lossbcl= u(x = +1, t)2[Eq. 7] lossbc2= u(x = — 1, t)2[Eq. 8] loss = losspde+ lossic+ lossbcl+ lossbc2[Eq. 9]

[0067] In the above equations, u represents the network’s prediction of the underlying velocity u. The QA-PINN, c-PINN and GAAF-PINN models were implemented using the PennyLane and PyTorch frameworks. The QA-PINN was executed in a cloud computing environment with PennyLane’s local simulator. The differential terms in Eq. 5 were calculated using the autograd functionality provided by PyTorch. The loss functions in Eqs. 5-8 contribute to the overall loss function as described in Eq. 9. The Adam optimizer with a learning rate of 0.001 was used for training all the QA-PINN, c-PINN and GAAF-PINN models.

[0068] A dataset of size 25,600 was used to train each variation of QA-PINN, c-PINN and GAAF-PINN. The dataset consisted of randomly generated values of x and t in the range [-1,1] with 256 data points and [0,1] with 100 data points, respectively.

[0069] With respect to results of the example implementation, the example implementation of QA-PINN exhibited a dramatic improvement over c-PINN and GAAF-PINN in terms of the performance, accuracy, and efficiency of solving a 1-D Burgers equation. While the QA-PINN showed a significant reduction in the number oftrainable parameters, especially for 4 and 5-qubit architectures, accuracy of the results was also benchmarked.

[0070] Fig. 3 illustrates the results of QA-PINN to simulate the 1-D Burger equation at four different time steps (t = 0.25, 0.50, 0.75, 0.99) and with three different qubit architectures. The plots also showcase the benchmarking of QA-PINN with respect to c-PINN, GAAF-PINN and reference analytical solution (or ground truth).

[0071] Fig. 4 illustrates an L2-norm (||L2||) of QA-PINN, c-PINN and GAAF-PINN with respect to ground truth over the four time steps. As observed from FIG. 3 and FIG. 4, 3-qubit QA-PINN performed poorly with respect to ground truth for all time steps both at the boundaries and near the shock, though the accuracy improved for the latter time steps. The 3-qubit equivalent c-PINN and GAAF-PINN did perform better than QA-PINN for later time steps with c-PINN predicting the shock the best and at the correct x-location than GAAF-PINN.

[0072] For 4-qubit architecture, the performance of QA-PINN improves slightly for the first three time steps and significantly for t = 0.99. The reduction in ||L2II for QA-PINN may correspond to the first and last 50 points from the boundary, though still failing to capture the shock. On the other hand, the c-PINN shows an improvement, while the GAAF-PINN, even though predicting the behavior well, did not capture the shock at the correct x-location.

[0073] Accuracy drastically improved for QA-PINN for 5-qubit architecture. For t = 0.25 and t = 0.50, most of the spatial points lie on top of the ground truth except at the peaks, which were also seen for the c-PINN and GAAF-PINN. At t = 0.75 and t = 0.99, the QA-PINN showed the highest accuracy with ||L21| of 0.8 x 10-3and 0.6 x 10-3, respectively, even at the shock where there is a sharp change.

[0074] As a result, QA-PINN, even with fewer trainable parameters, performs with high accuracy, overcame the challenges of overfitting and improved training performance. This may significantly decrease computing resources needed for accurately solving a simulation problem. The accuracy of c-PINN was poorer than the previous architecture for the last two time steps, for example, because of convergence behavior or due to shock being predicted at a different x-location than ground truth. QA-PINN and GAAF-PINN matched well at t = 0.75 and t = 0.99.

[0075] Scaling up the QA-PINN to utilize more qubits may require GPU acceleration since an increase in qubit count increases the training time when the model is trained on a CPU. Reduction of trainable parameters can also be employed for other deep learning architectures like Convolutional Neural Networks and Transformers.

[0076] In a further illustrative example, implementations herein were used in an image processing use case to detect defects. A convolutional neural network (CNN) was compared the performance of an implementation of the present disclosure in the detection of defects in concrete surfaces via images of those surfaces.

[0077] In this example, a QML model implemented as a hybrid quantum convolutional neural network (HQCNN) was used, combining a quantum neural network with a classical neural network. A pre-trained classical model was used to extract features. These extracted features were then transferred to a quantum layer for further processing, and aligned with current noisy intermediate-scale quantum (NISQ) hardware limitations. In turn, they offered immense potential for their applications in surface defect (e.g., cracks) detection. The model achieved improved accuracy and efficiency over the classical CNN compared against.

[0078] The dataset included 40,000 images of concrete surfaces, divided into 'negative' (without cracks) and 'positive' (with cracks) classes. Each image was a 227x227- pixel RGB image. The dataset was comprised of high-resolution images, exhibiting significant variance in surface finish and lighting conditions. To maintain data consistency, no augmentations were applied.

[0079] The HQCNN was applied to analyze the dataset of high-resolution RGB high- accuracy images, which captured the detailed textures and contours essential for differentiating cracks from other similar patterns in construction materials for identifying these cracks. The comparison of the HQCNN was with the VGG16 and Low-Rank Approximation (LoRA) classical network architecture models.

[0080] LoRA is a technique that is used to efficiently fine-tune neural networks by introducing small and low-rank matrices to the neural network, rather than modifying the entire model’s weight. This approach significantly reduced the number of trainable parameters, making the fine-tuning process more efficient.

[0081] With respect to performance, the classical model used 14714688 trainable parameters while attaining an accuracy of 93.44%. It struggled to accurately predict positive cases.

[0082] The HQCNN outperformed the classical approach across all evaluation metrics with only 2137 trainable parameters, while achieving 98% accuracy.

[0083] In a first dataset (10:90 Crack:Non-Crack), the HQCNN excelled in handling the highly imbalanced dataset, delivering a remarkable 99.8% accuracy compared to the classical model's 98.5%. The F1 -score, a harmonic mean of precision andrecall, further emphasizes the quantum model's superiority with a value of 0.9921 compared to the classical model's 0.9590. These results indicate that the quantum model consistently performs better across different class imbalance levels.

[0084] In a second dataset (70:30 Crack:Non-Crack), while the class imbalance was less severe, the HQCNN consistently outperformed the classical approach, achieving 99.55% accuracy versus 97.87%. The F1-score also favors the quantum model with a value of 0.9970 compared to the classical model's 0.9846. This reinforces the hybrid model's adaptability to varying imbalance levels.

[0085] The HQCNN demonstrated exceptional performance even with a relatively small dataset of 1000 images, consistently surpassing the classical model in accuracy. The HQCNN exhibited rapid convergence, reaching near-perfect accuracy (99% after 10 epochs, 100% after 20 epochs). In contrast, the classical model peaked at 98.73% accuracy (after 20 epochs) and displayed instability at earlier stages (95.67% after 10 epochs). The classical model obtained an F1 score of 0.9876, which is very good but not perfect. The quantum model achieved a perfect score of 1.000, indicating that the model correctly identified all positive instances without any false positives or false negatives.

[0086] This example implementation of anomaly detection demonstrated QML's effectiveness in handling imbalanced datasets prevalent in a wide range of industries, such as aerospace, defense, and automotive. The HQCNN surpassed traditional methods by addressing data challenges and achieving computational efficiency, and its versatility positions it as a valuable tool for various applications.

[0087] FIG. 5 Illustrates an operational environment 500, according to one or more of the implementations herein. As illustrated in FIG. 5, the operational environment 500 may include actors, including a user device 510, a network 520, an system 530 having at least a computing resource, for example a processor 532, and an electronic storage device 534, and a quantum processor 536.

[0088] The user device 510 may include any variety of devices a user may use to interface with the system 530 via the network 520, including, for example, a server, a desktop computer, a laptop computer, a handheld computer, a tablet computing platform, a Netbook, a Smartphone, a gaming console, and / or other computing platforms.

[0089] The network 520 may include any variety of devices configured to enable a device to communicate with other devices, such as via a wired connection and / or a wireless connection, for example, via the internet and / or other networks using, forexample, TCP / IP or cellular hardware enabling wired or wireless (e.g., cellular, 2G, 3G, 4G, 4G LTE, 5G, or wireless local area network) communication. For example, the network 520 may include, for example, a receiver, a transmitter, a transceiver, a modem, a network interface card, and / or an antenna.

[0090] The system 530 may include any variety of devices configurable to perform the implementations and methods disclosed herein and interface with the user device 510 via the network 520, including, for example, a server, a desktop computer, a laptop computer, a handheld computer, a tablet computing platform, a Netbook, a Smartphone, a gaming console, and / or other computing platforms.

[0091] The system 530 may include the processor 532. The processor 532 may include, for example, one or more processor(s) configured to execute machine-readable instructions for implementing all or some of the implementations herein. The processor 532 may be configured to access the electronic storage device 534 to retrieve and / or write electronic data from and to the electronic storage device 534. The processor 532 may be further configured to interface directly or remotely with the quantum processor 536.

[0092] The system 530 may include the electronic storage device 534. The electronic storage device 534 may be configured to electronically store data (e.g., host) corresponding to one or more databases or other forms of data storage for use in implementations herein. The electronic storage device 534 may be accessible by the processor 532.

[0093] The system 530 may include the quantum processor 536. The quantum processor 536 may include any variety of quantum computing devices that may interface with the processor 532. The quantum processor 536 may comprise one or more quantum bits (qubits). The quantum bit may be configured for superposition. The quantum bit may be configured for entailment. The quantum processor 536 may comprise a quantum bit interface module. The quantum bit interface module may be configured to receive a quantum input from the processor. The quantum bit interface module may be configured to manipulate the quantum bit based on the quantum input (e.g., via electrical signals, electromagnetic signals, etc.). The quantum bit interface module may be configured to measure the quantum bit. The quantum bit interface module may be configured to determine a quantum output by measuring the quantum bit. The quantum bit interface module may be configured to output the measurement of the quantum bit to the processor 532. The quantum processor 536 may comprise a plurality of quantum bits including the quantum bit.The quantum processor 536 may comprise a quantum logic gate comprising the quantum bit. The quantum computing system may further comprise a graphics processing unit. The quantum computing system may further comprise a tensor processing unit. The quantum computing system may further comprise a field- programmable gate array. The processor 532, the quantum processor 536, and at least one of a graphics processing unit, a tensor processing unit, or a field- programmable gate array may be configured in a high-performance computing heterogeneous architecture. The quantum processor 536 be an emulated quantum processor. The quantum processor 536 may be remote from the processor. Communicating the quantum input using the processor to the quantum bit interface module may be via a network (e.g., the internet). The quantum computing system may include a refrigeration system configured to maintain the quantum bit at a temperature below an ambient temperature of the quantum computing system. The quantum computing system may include a vacuum system configured to maintain the quantum bit in low vacuum, medium vacuum, high vacuum, ultra-high vacuum, or extreme-high vacuum.

[0094] FIG. 6 is a diagram of example components of a device 600, according to one or more implementations herein. The device 600 may correspond to one or more device, network, resource, or service of FIG. 5. In some implementations, one or more device, network, resource, or service of FIG. 5 may include one or more of the devices 600 and / or one or more components of the device 600, for example, according to a client / server architecture, a peer-to-peer architecture, and / or other architectures, which may include a plurality of hardware, software, and / or firmware components operating together to provide the functionality attributed herein to the device 600. In some implementations, the device 600 may include a distributed computing architecture (e.g., one or more individual computing platforms operating in concert to accomplish a computing task). For example, the device 600 may be implemented by a cloud of computing platforms operating together as the device 600. By way of non-limiting example, a given device 600 may include one or more of a server, a desktop computer, a laptop computer, a handheld computer, a tablet computing platform, a Netbook, a Smartphone, a gaming console, and / or other computing platforms.

[0095] As used herein, the term “component” is intended to be broadly construed as hardware, firmware, or a combination of hardware and software. It will be apparent that systems and / or methods described herein may be implemented in different forms of hardware, firmware, and / or a combination of hardware and software. Theactual specialized control hardware or software code used to implement these systems and / or methods is not limiting of the implementations. Thus, the operation and behavior of the systems and / or methods are described herein without reference to specific software code — it being understood that software and hardware can be used to implement the systems and / or methods based on the description herein.

[0096] The device 600 may include a bus 610, a processor 620, a memory 630, an electronic storage component 640, an input component 650, an output component 660, a communication component 670, and a quantum processor 680.

[0097] The bus 610 includes a component that enables wired and / or wireless communication among the components of device 600. The bus 610 may enable various components of a computer system to communicate with each other, allowing for the transfer of data from one part to another.

[0098] The processor 620 may include a central processing unit, a graphics processing unit, a microprocessor, a controller, a microcontroller, a digital signal processor, a field-programmable gate array (FPGA), an application-specific integrated circuit, and / or another type of processing component. The processor 620 may be implemented in hardware, firmware, or a combination of hardware and software. In some implementations, the processor 620 may include one or more processors capable of being programmed to perform a function. Such processors may or may not be all integral to the same physical device and may in some embodiments be distributed among several devices.

[0099] The processor 620 may be configured to execute one or more of the modules disclosed herein, and / or other modules by software; hardware; firmware; some combination of software, hardware, and / or firmware; and / or other mechanisms for configuring processing capabilities on the processor 620. As used herein, the term “module” may refer to any component or set of components that perform the functionality attributed to the module. This may include one or more physical processors during execution of processor readable instructions, the processor readable instructions, circuitry, hardware, storage media, or any other components. Various modules or portions thereof may be implemented in any of various ways, including procedure-based techniques, component-based techniques, and / or object-oriented techniques, among others. For example, the program instructions may be implemented using system libraries, language libraries, model-viewcontroller (MVC) principles, application programming interfaces (APIs), large language models (LLMs), system-specific programming languages and principles,cross-platform programming languages and principles, pre-compiled programming languages, markup programming languages, stylesheet languages, “bytecode” programming languages, object-oriented programming principles or languages, other programming principles or languages, C, C++, C#, Java, JavaScript, Python, PHP, HTML, CSS, TypeScript, R, Elm, Unity, VB.Net, Visual Basic, Swift, Objective-C, Perl, Ruby, Go, SQL, Haskell, Scala, Arduino, assembly language, Microsoft Foundation Classes (MFC), Streaming SIMD Extension (SSE), or other technologies or methodologies, as desired.

[0100] It should be appreciated that although some modules disclosed herein may be illustrated for example as being implemented within a single processing unit, in embodiments in which the processor 620 includes multiple processing units, one or more of modules disclosed herein may be implemented remotely from the other modules. The description of the functionality provided by the different modules disclosed herein is for illustrative purposes, and is not intended to be limiting, as any of modules described herein may provide more or less functionality than is described. For example, one or more of modules disclosed herein may be eliminated, and some or all of its functionality may be provided by other ones of modules disclosed herein. As another example, the processor 620 may be configured to execute one or more additional modules that may perform some or all of the functionality attributed herein to one of modules disclosed herein.

[0101] The memory 630 may include a random-access memory, a read only memory, and / or another type of memory (e.g., a flash memory, a magnetic memory, and / or an optical memory).

[0102] The electronic storage component 640 may store information and / or software related to the operation of the device 600. For example, the electronic storage component 640 may include a solid-state disk drive, a hard disk drive, a magnetic disk drive, an optical disk drive, a compact disc, a digital versatile disc, and / or another type of non-transitory computer-readable medium. Implementations of the electronic storage component 640 may include one or more of optically readable storage media (e.g., optical disks, etc.), magnetically readable storage media (e.g., magnetic tape, magnetic hard drive, floppy drive, etc.), electrical charge-based storage media (e.g., EEPROM, RAM, etc.), solid-state storage media (e.g., flash drive, etc.), and / or other electronically readable storage media. Implementations of the electronic storage component 640 may include one or both of system storage provided integrally ( / .e., substantially non-removable) to the device 600 and / or removable storage that is removably connectable to the device 600 via, forexample, a port (e.g., a serial port, a USB port, an IEEE 1394 port, a THUNDERBOLT™ port, etc.) or a drive (e.g., disk drive, flash drive, or solid-state drive etc.). The electronic storage component 640 may also or alternatively include one or more virtual storage resources (e.g., cloud storage, a virtual private network, and / or other virtual storage resources). An electronic storage may store software algorithms, information determined by one or more processors, information received from one or more computing platforms, information received from one or more remote platforms, databases (e.g., structured query language (SQL) databases (e.g., MYSQL®, MARIADB®, MONGODB®), NO-SQL databases, among others), data files, compiled data, analyzed data, charts, tables, videos, images, presentations, and 3D content in the respective format and / or other information enabling a computing platform to function as described herein.

[0103] The input component 650 may enable the device 600 to receive input, such as user input and / or sensed inputs. For example, the input component 650 may include a touch screen, a keyboard, a keypad, a mouse, a button, a microphone, a switch, a sensor (internal and / or external), a global positioning system component, an accelerometer, a gyroscope, and / or an actuator.

[0104] The output component 660 may enable the device 600 to provide output, such as via a display, a speaker, and / or one or more light-emitting diodes.

[0105] The communication component 670 may enable the device 600 to communicate with other devices, such as via a wired connection and / or a wireless connection, for example, via the internet and / or other networks using, for example, TCP / IP or cellular hardware enabling wired or wireless (e.g., cellular, 2G, 3G, 4G, 4G LTE, 5G, wireless local area network, near field communication (NFC), BLUETOOTH®) communication. For example, the communication component 670 may include a receiver, a transmitter, a transceiver, a modem, a network interface card, and / or an antenna.

[0106] As used herein, “internet” may include an interconnected network of systems and a suite of protocols for the end-to-end transfer of data therebetween. A model describing may be the Transport Control Protocol and Internet Protocol (TCP / IP), which may also be referred to as the internet protocol suite. TCP / IP provides a model of four layers of abstraction: an application layer, a transport layer, an internet layer, and a link layer. The link layer may include hosts accessible without traversing a router, and thus may be determined by the configuration of the network (e.g., a hardware network implementation, a local area network, a virtual privatenetwork, or a networking tunnel). The link layer may be used to move packets of data between the internet layer interfaces of different hosts on the same link. The link layer may interface with hardware for end-to-end transmission of data. The internet layer may include the exchange of datagrams across network boundaries (e.g., from a source network to a destination network), which may be referred to as routing, and is performed using host addressing and identification over an internet protocol (IP) addressing system (e.g., IPv4, IPv6). A datagram may include a self- contained, independent, basic unit of data, including a header (e.g., including a source address, a destination address, and a type) and a payload (e.g., the data to be transported), to be transferred across a packet-switched network. The transport layer may utilize the user datagram protocol (UDP) to provide for basic data channels (e.g., via network ports) usable by applications for data exchange by establishing end-to-end, host-to-host connectivity independent of any underlying network or structure of user data. The application layer may include various user and support protocols used by applications users may use to create and exchange data, utilize services, or provide services over network connections established by the lower layers, including, for example, routing protocols, the hypertext transfer protocol (HTTP), the file transfer protocol (FTP), the simple mail transfer protocol (SMTP), and the dynamic host configuration protocol (DHCP). Such data creation and exchange in the application layer may utilize, for example, a client-server model or a peer-to-peer networking model. Data from the application layer may be encapsulated into UDP datagrams or TCP streams for interfacing with the transport layer, which may then effectuate data transfer via the lower layers.

[0107] The communication component 670 may further implement an internet-of-things (“loT”) configuration, which may include a network of physical objects — devices, vehicles, buildings, and other items — embedded with electronics, software, sensors, and network connectivity that enables these objects to collect and exchange data via the Internet. Each loT product / device may be an endpoint device having its own Internet address (e.g., IPv4, IPv6 address). The loT allows objects to be sensed and controlled remotely across an existing network infrastructure (e.g., the Internet), creating opportunities for more direct integration of the physical world into computer-based systems.

[0108] The quantum processor 680 may be similar to the quantum processor 536 or as otherwise described herein and may be in operative communication with the processor 620, the bus 610, or another component of the device 600 directly or indirectly.

[0109] The device 600 may perform one or more processes described herein. For example, a non-transitory computer-readable medium (e.g., the memory 630 and / or the storage component 640) may store a set of instructions (e.g., one or more instructions, code, software code, and / or program code) for execution by the processor 620. The processor 620 or the quantum processor 680, alone or in combination, may execute the set of instructions to perform one or more processes described herein. In some implementations, execution of the set of instructions, by one or more processors 620 or one or more quantum processors 680, alone or in combination, causes the one or more processors 620 or the one or more quantum processors 680, alone or in combination, and / or the device 600 to perform one or more processes described herein. In some implementations, hardwired circuitry may be used instead of or in combination with the instructions to perform one or more processes described herein. Thus, implementations described herein are not limited to any specific combination of hardware circuitry and software.

[0110] The number and arrangement of components shown in FIG. 6 are provided as an example. The device 600 may include additional components, fewer components, different components, or differently arranged components than those shown in FIG. 6. Additionally, or alternatively, a set of components (e.g., one or more components) of the device 600 may perform one or more functions described as being performed by another set of components of the device 600.

[0111] In addition to the example configuration described herein in FIG. 6, various steps, functions, and / or operations of the device 600 and the methods disclosed herein may be carried out by one or more of, for example, electronic circuits, logic gates, multiplexers, programmable logic devices, ASICs, analog or digital controls / switches, microcontrollers, or computing systems. Program instructions implementing methods such as those described herein may be transmitted over or stored on carrier medium. The carrier medium may include a storage medium such as a read-only memory, a random-access memory, a magnetic or optical disk, a non-volatile memory, a solid-state memory, a magnetic tape, and the like. A carrier medium may include a transmission medium such as a wire, cable, or wireless transmission link.

[0112] FIG. 7 is a diagram of example components of a device 770, according to one or more implementations herein. The device 770 may correspond to the communication component 670 or another device or component illustrated in FIG. 5 or FIG. 6. In some implementations, the communication component 670 or another device or component illustrated in FIG. 5 or FIG. 6 may include one or more of thedevices 770 and / or one or more components of the device 770. As shown in FIG. 7, the device 770 may include one or more input components 772 (herein referred to collectively as the input components 772 or individually as the input component 772), a switching component 774, one or more output components 776 (herein referred to collectively as the output components 776 or individually as the output component 776), and a controller 778.

[0113] The input component 772 may be one or more points of attachment for one or more input physical links 771 (herein referred to collectively as the input physical links 771 or individually as the input physical link 771 ) and include one or more points of entry for incoming traffic, such as packets. The input component 772 may process incoming traffic, such as by performing data link layer encapsulation or decapsulation. In some implementations, the input component 772 may transmit and / or receive packets. In some implementations, the input component 772 may include an input line card that includes one or more packet processing components (e.g., in the form of integrated circuits), such as one or more interface cards (IFCs), packet forwarding components, line card controller components, input ports, processors, memories, and / or input queues. In some implementations, the device 770 may include one or more the input components 772.

[0114] The switching component 774 may interconnect the input components 772 with the output components 776. In some implementations, the switching component 774 may be implemented via one or more crossbars, via busses, and / or with shared memories. The shared memories may act as temporary buffers to store packets from the input components 772 before the packets are eventually scheduled for delivery to the output components 776. In some implementations, the switching component 774 may enable the input components 772, the output components 776, and / or the controller 778 to communicate with one another.

[0115] The output component 776 may store packets and may schedule packets for transmission on the output physical link(s) 779 (herein referred to collectively as the output physical links 779 or individually as the output physical link 779). The output component 776 may support data link layer encapsulation or decapsulation, and / or a variety of higher-level protocols. In some implementations, the output component 776 may transmit packets and / or receive packets. In some implementations, the output component 776 may include an output line card that includes one or more packet processing components (e.g., in the form of integrated circuits), such as one or more IFCs, packet forwarding components, line card controller components, output ports, processors, memories, and / or output queues. In someimplementations, the device 770 may include one or more output components 776. In some implementations, the input component 772 and the output component 776 may be implemented by the same set of components (e.g., an input / output component may be a combination of the input component 772 and the output component 776).

[0116] The controller 778 includes a processor in the form of, for example, a CPU, a GPU, an APU, a microprocessor, a microcontroller, a DSP, an FPGA, an ASIC, and / or another type of processor. The processor is implemented in hardware, firmware, or a combination of hardware and software. In some implementations, the controller 778 may include one or more processors that can be programmed to perform a function.

[0117] In some implementations, the controller 778 may include a RAM, a ROM, and / or another type of dynamic or static storage device (e.g., a flash memory, a magnetic memory, an optical memory, etc.) that stores information and / or instructions for use by the controller 778.

[0118] In some implementations, the controller 778 may communicate with other devices, networks, and / or systems connected to the device 770 to exchange information regarding network topology. The controller 778 may create routing tables based on the network topology information, may create forwarding tables based on the routing tables, and may forward the forwarding tables to the input components 772 and / or the output components 776. The input components 772 and / or the output components 776 may use the forwarding tables to perform route lookups for incoming and / or outgoing packets.

[0119] The controller 778 may perform one or more processes described herein. The controller 778 may perform these processes in response to executing software instructions stored by a non-transitory computer-readable medium. A computer- readable medium is defined herein as a non-transitory (e.g., the medium itself ( / .e., tangible, not a signal) as opposed to a limitation on data storage persistency (e.g., RAM vs. ROM)) memory device. A memory device includes memory space within a single physical storage device or memory space spread across multiple physical storage devices.

[0120] Software instructions may be read into a memory and / or storage component associated with the controller 778 from another computer-readable medium or from another device via a communication interface. When executed, software instructions stored in a memory and / or storage component associated with thecontroller 778 may cause the controller 778 to perform one or more processes described herein. Additionally, or alternatively, hardwired circuitry may be used in place of or in combination with software instructions to perform one or more processes described herein. Thus, implementations described herein are not limited to any specific combination of hardware circuitry and software.

[0121] The number and arrangement of components shown in FIG. 7 are provided as an example. In practice, the device 770 may include additional components, fewer components, different components, or differently arranged components than those shown in FIG. 7. Additionally, or alternatively, a set of components (e.g., one or more components) of the device 770 may perform one or more functions described as being performed by another set of components of the device 770.

[0122] The following figures illustrate example methods and operations thereof. In some implementations, a method illustrated herein may include additional operations, fewer operations, differently arranged operations, or different operations than the operations depicted in the following figures. Moreover, or in the alternative, two or more of the operations depicted in the following figures may be performed at least partially in parallel.

[0123] In implementations of the methods illustrated in the following figures, various operations may be performed by one or more hardware processors configured by machine-readable instructions (e.g., instructions stored electronically on an electronic storage medium), which may include a module in accordance with one or more embodiments. Such a hardware processor may include one or more processing devices (e.g., one or more digital processors, analog processors, digital circuits designed to process information, analog circuits designed to process information, state machines, and / or other mechanisms for electronically processing information). The one or more processing devices may include one or more devices configured through hardware, firmware, and / or software, which may be specifically designed for execution of one or more of the operations of methods illustrated herein.

[0124] FIG. 8 is a flowchart illustrating an example method 800, according to one or more implementations herein. In some implementations, one or more operations illustrated in FIG. 8 may be performed by one or more of the devices or components depicted in FIGs. 1A, 2, 5, 6, 7, 9, or 10, in concert, in the alternative, or in combinations thereof. In some implementations, one or more operations may be performed by another device, system, or group of devices or systems separatefrom or including these. Additionally, or alternatively other devices, components, or systems, may be employed to perform the operations.

[0125] An operation 802 may include receiving, at a processor of a computing platform from an electronic storage device in electronic communication with the processor, a classical neural network, and may be performed alone or in combination with one or more other operations depicted in FIG. 8. The classical neural network may include, for example, a pre-trained classical neural network. The classical neural network may include a convolutional neural network or other neural network.

[0126] An operation 804 may include generating, using the processor, a hybrid neural network, and may be performed alone or in combination with one or more other operations depicted in FIG. 8. The hybrid neural network may comprise a quantum neural network configured for processing input data using a quantum processor and yield intermediate data and a classical neural network configured for processing the intermediate data to yield output data. The quantum neural network may comprise a quantum convolutional neural network. The resulting hybrid neural network may include a physics-informed neural network. The generating of the hybrid neural network may be by quantum transfer learning based on the pre-trained classical neural network.

[0127] An operation 806 may include storing, using the processor, the hybrid neural network to the electronic storage device, and may be performed alone or in combination with one or more other operations depicted in FIG. 8. After storage, a data set (e.g., a training data set or a production data set) may be applied either immediately or later as input data to the hybrid neural network.

[0128] Implementations may implement machine learning, a type of artificial intelligence (Al) that provides computers with an ability to learn how to process data without being explicitly programmed. Machine learning focuses on the development of computer programs that can teach themselves to grow and change when exposed to new data. Machine learning explores the study and construction of algorithms that can learn from and make predictions based on data. Such algorithms may overcome following strictly static program instructions by making data-driven predictions or decisions, through building a model from sample inputs.

[0129] Machine learning may refer to a variety of Al software algorithms, which may be used to perform supervised learning, unsupervised learning, reinforcement learning, deep learning, or any combination thereof. A variety of different machine learning algorithms may be employed in implementations. Examples of machine learningalgorithms may include, inter alia, artificial neural network algorithms, Gaussian process regression algorithms, fuzzy logic-based algorithms, or decision tree algorithms.

[0130] In some implementations, more than one machine learning algorithm may be employed. For example, automated classification may be implemented using one type of machine learning algorithm, and adaptive real-time process control may be implemented using a different type of machine learning algorithm. In some implementations, hybrid machine learning algorithms including features and properties drawn from two, three, four, five, or more different types of machine learning algorithms may be employed in implementations.

[0131] Supervised learning algorithms may use labeled training data to infer a relationship between one or more identifiable aspects of a given entity and a classification of the entity according to a specified set of criteria or to infer a relationship between input process control parameters and desired outcomes. The training data may include paired training examples. For example, each training data example may include aspects identified for a given entity and the resultant classification of the given entity. As a further example, each training data example may include process control parameters used in a process and a known outcome of the process.

[0132] Unsupervised learning algorithms may be used to draw inferences from training data including entity data not paired with labeled entity classification data, or input process control parameter data not paired with labeled process outcomes. An example unsupervised learning algorithm is cluster analysis, which may be used for exploratory data analysis to find hidden patterns or groupings in process data.

[0133] Semi-supervised learning algorithms may use both labeled and unlabeled object classification or process data for training. Semi-supervised learning algorithms may typically use a small amount of labeled data with a large amount of unlabeled data.

[0134] Reinforcement learning algorithms may be used, for example, to optimize a process (e.g., steps or actions of the process) to maximize a process reward function or minimize a process loss function. In machine learning environments, reinforcement learning algorithms may be formulated as Markov decision processes. Reward functions or loss functions, which may also be referred to as cost functions or error functions, may map values of one or more process variables and / or outcomes to a real number that represents a reward or cost, respectively, associated with a given process outcome or event. Examples of process parameters and process outcomes include, inter alia, process throughput, process yield, production quality, orproduction cost. In some cases, the definition of the reward or loss function to be maximized or minimized, respectively, may depend on the choice of machine learning algorithm used to run the process control method, or vice versa. For example, if an objective is to maximize a total reward / value function, a reinforcement learning algorithm may be chosen. If the objective is to minimize a mean squared error loss function, a decision tree regression algorithm or linear regression algorithm may be chosen. In general, the machine learning algorithm used to run the process control method will seek to optimize the reward function or minimize the loss function by identifying the current state of the process; comparing the current state to the reference state, which may be a target intermediate or final state; and adjusting one or more process control parameters to minimize a difference between the two states. This adjustment may include reference to past learning provided by a training data set. Reinforcement learning algorithms differ from supervised learning algorithms in that correct training data input / output pairs are not presented, nor are sub-optimal actions explicitly corrected. Implementations of these algorithms tend to focus on real-time performance by finding a balance between exploration of possible outcomes based on updated input data and exploitation of past training.

[0135] Deep learning, which may also be known as deep structured learning, hierarchical learning, or deep machine learning, may be based on a set of algorithms that attempt to model high level abstractions in data. Deep learning algorithms may be inspired by the structure and function of the human brain and are part of a broader family of machine learning methods based on learning representations of data. Rooted in neural network technology, deep learning may involve a probabilistic graph model having many neuron layers, commonly known as a deep architecture. Deep learning technology may process information such as, inter alia, image, text, or sound information in a hierarchical manner. An observation (e.g., a feature to be extracted for reference) can be represented in many ways including, for example, a vector of intensity values, a set of edges, regions of shape, or in another abstract manner. Some representations may simplify the learning task (e.g., face recognition or facial expression recognition). Deep learning can provide efficient algorithms for unsupervised or semi-supervised feature learning and hierarchical feature extraction, implementations employing deep learning can further benefit from the advantage of deep learning concepts in solving a normally intractable representation inversion problem.

[0136] A deep learning module may be configured as a neural network. The deep learning module may further be a deep neural network with a set of weights that model the world based on training using training data. Neural networks can be understood to implement a computational approach — based on a relatively large collection of neural units — to loosely model the way a human brain solves problems with large clusters of biological neurons connected by axons. Each neural unit may be connected to one or more others, and links can be enforcing or inhibitory in their effect on the activation state of connected neural units. These systems may be selflearning and trained rather than explicitly programmed. Neural network systems excel in areas where a solution or feature detection is difficult to express in a traditional computer program.

[0137] An example of a deep learning algorithm may be an artificial neural network (ANN). Large ANNs including many layers may be used, for example, to map entity data to entity classification decisions or to map input process control parameters to desired process outcomes. ANNs will be discussed in further detail below.

[0138] Neural networks typically include multiple layers, and the signal path may traverse from front to back. The goal of neural networks may be to solve problems in a similar manner to the human brain, although several neural networks may be much more abstract. In a simple example of a neural network, there may be two layers ( / .e., sets) of neurons: an input layer that receives an input signal and an output layer that sends an output signal. When the input layer receives an input, it may pass a modified version of the input to the next layer. In a deep network, there may be many layers between the input layer and output layer, allowing the algorithm to use multiple processing layers, which may include multiple linear and non-linear transformations. Modern neural networks typically work with a few thousand to a few million neural units and millions of connections. Neural networks may have various suitable architectures and / or configurations known in the art.

[0139] There are many variants of neural networks with deep architecture depending on the probability specification and network architecture, including, inter alia, deep belief networks (DBN), restricted Boltzmann machines (RBM), random forests, and autoencoders. Implementations of neural networks may vary depending on the size of input data, the number of features to be analyzed, and the nature of the problem. Other layers may be included in the deep learning module besides the neural networks disclosed herein.

[0140] Another type of deep neural network may be a convolutional neural network (CNN), which can be used for analysis of an entity or process. CNNs are commonly composed of layers of different types: convolution, pooling, upscaling, and fully connected layers. In some cases, an activation function such as a rectified linear unit (ReLU) function may be used in some of the layers. In a CNN architecture, there can be one or more layers for each type of operation performed. A CNN architecture may include any number of layers in total, and any number of layers for the different types of operations performed. The simplest CNN architecture starts with an input layer followed by a sequence of convolutional layers and pooling layers (e.g., layers otherwise configured for reducing the dimensionality of the feature map generated by the one or more convolutional layers while retaining the most important features, for example, max pooling layers) and ends with fully connected layers (e.g., a layer in which each of the nodes is connected to each of the nodes in the previous layer). Each convolution layer may include a plurality of parameters used for performing the convolution operations. Each convolution layer may also include one or more filters, which in turn may include one or more weighting factors or other adjustable parameters. In some instances, the parameters may include biases (e.g., parameters that permit an activation function to be shifted). In some cases, the convolutional layers may be followed by an ReLU activation function layer. Other activation functions can also be used, for example, inter alia, saturating hyperbolic tangent, identity, binary step, logistic, arctan, softsign, parametric rectified linear unit, exponential linear unit, softPlus, bent identity, softExponential, Sinusoid, Sine, Gaussian, or sigmoid functions. The convolutional, pooling and ReLU layers may function as learnable feature extractors, while the fully connected layers may function as machine learning classifiers. As with other artificial neural networks, the convolutional layers and fully connected layers of CNN architectures may include various computational parameters, for example, weights, bias values, and threshold values, which may be trained in a training phase.

[0141] Another type of deep neural network may be a visual geometry group (VGG) network. For example, VGG networks may be created by increasing the number of convolutional layers while fixing other parameters of the architecture. Adding convolutional layers to increase depth may be made possible by using substantially small convolutional filters in all of the layers. VGG networks may also include convolutional layers followed by fully connected layers.

[0142] Another type of deep neural network may be a deep residual network. Like some other networks described herein, a deep residual network may include convolutional layers followed by fully connected layers, which may be, in combination, configured and trained for feature property extraction. A deep residual network’s layers may be configured to learn residual functions with reference to layer inputs, instead of learning unreferenced functions. Instead of relying on a direct fit of few stacked layers to a desired underlying mapping, a deep residual network’s layers may be explicitly allowed to fit a residual mapping, which may be realized by feedforward neural networks having shortcut connections ( / .e., connections that skip one or more layers). A deep residual network may be created by inserting shortcut connections into a plain neural network structure including convolutional layers, thereby modifying the plain neural network into a residual learning network.

[0143] In some implementations, the machine learning module may include a support vector machine (SVM), an artificial neural network (ANN), a decision tree-based expert learning system, an autoencoder, a clustering machine learning algorithm, or a nearest neighbor (e.g., kNN) machine learning algorithm, or combinations thereof, some of which will be described in further detail below.

[0144] Support vector machines (SVMs) may be supervised learning algorithms used for classification and regression analysis of entity classification data or process control. Given a set of training data examples (e.g., entity or process data), each marked as belonging to a category, an SVM training algorithm may build a model that assigns new examples (e.g., data from a new entity or process) to a given category.

[0145] FIG. 9 illustrates an artificial neural network (ANN) 900, according to an implementation. ANN 900 may be used for, inter alia, classification or process control optimization according to various implementations.

[0146] ANN 900 may include any type of neural network module, such as, inter alia, a feedforward neural network, radial basis function network, recurrent neural network, or convolutional neural network.

[0147] In implementations implementing ANN 900 for entity classification, ANN 900 may be employed to map entity data to entity classification data. In implementations implementing ANN 900 for process optimization, ANN 900 may be employed to determine an optimal set or sequence of process control parameter settings for adaptive control of a process in real-time based on a stream of process monitoring data and / or entity classification data provided by, for example, observation or fromone or more sensors. ANN 900 may include an untrained ANN, a trained ANN, pretrained ANN, a continuously updated ANN (e.g., an ANN utilizing training data that is continuously updated with real time classification data or process control and monitoring data from a single local system, from a plurality of local systems, or from a plurality of geographically distributed systems).

[0148] ANN 900 may include interconnected nodes (e.g., xi-x:, Xi'-Xj, and yi-yk) organized into n layers of nodes, where xi-x, represents a group of / nodes in an input layer 902 (e.g., layer 1), x / -x represents a group of j nodes in one or more hidden layers 903 (e.g., layer(s) 2 through n - 1 ), and yi-yk represents a group of k nodes in a final layer 904 (e.g., layer n). Input layer 902 may be configured to receive input data 901 (e.g., sensor data, image data, sound data, observed data, automatically retrieved data, manually input data, etc.). Final layer 904 may be configured to provide result data 905.

[0149] There may be one or more hidden layers 903, and the number; of nodes in the one or more hidden layers 903 may vary from implementation to implementation. Thus, ANN 900 may include any total number of layers (e.g., the one or more hidden layers 903). One or more of the hidden layers 903 may function as trainable feature extractors, which may allow mapping of input data 901 to preferred result data 905.

[0150] FIG. 10 illustrates a node 1000, according to an implementation. Each layer of a neural network may include one or more nodes similar to node 1000, for example, nodes xy-x,, x / -x , and yi-yk depicted in FIG. 9. Each node may be analogous to a biological neuron.

[0151] Node 1000 may receive node inputs 1001 (e.g., ai-an) either directly from the ANN’S input data (e.g., input data 901) or from the output of one or more nodes in a different layer or the same layer. With node inputs 1001 , the node 1000 may perform an operation 1003, which while depicted in FIG. 10 as a summation operation, would be readily understood to include various other operations known in the art.

[0152] In some cases, node inputs 1001 may be associated with one or more weights1002 (e.g., wi-wn), which may represent weighting factors. For example, operation1003 may sum the products of each of node inputs 1001 and associated weights 1002 (e.g., a,w,).

[0153] The result of operation 1003 may be offset with one or more biases 1004 (e.g., bias b), which may be a value or a function.

[0154] Output 1006 of node 1000 may be gated using an activation (or threshold) function 1005 (e.g., function f), which may be a linear or a nonlinear function. Activation function 1005 may be, for example, a ReLU activation function or other function such as a saturating hyperbolic tangent, identity, binary step, logistic, arctan, softsign, parametric rectified linear unit, exponential linear unit, softPlus, bent identity, softExponential, Sinusoid, Sine, Gaussian, or sigmoid function, or any combination thereof.

[0155] Weights 1002, biases 1004, or threshold values of activation function 1005, or other computational parameters of the neural network, can be “taught” or “learned” in a training phase using one or more sets of training data. For example, the parameters may be trained using input data from a training data set and a gradient descent or backward propagation method so that the output value(s) (e.g., a set of predicted adjustments to classification or process control parameter settings) computed by the ANN may be consistent with the examples included in the training data set. The parameters may be obtained, for example, from a back propagation neural network training process, which may or may not be performed using the same hardware as that used for automated classification or adaptive, real-time deposition process control.

[0156] Decision tree-based expert systems may be supervised learning algorithms designed to solve entity classification problems or process control problems by applying a series of conditional (e.g., if-then) rules. Expert systems may include two subsystems: an inference engine and a knowledge base. The knowledge base may include a set of facts (e.g., a training data set including entity data for a series of entities, and the associated entity classification data provided by, for example, a skilled operator, technician, or inspector) and derived rules (e.g., derived entity classification rules). The inference engine may then apply the rules to input data for a current entity classification problem or process control problem to determine a classification of the entity or a next set of process control adjustments.

[0157] Autoencoders (also sometimes referred to as an auto-associator or Diabolo network), may be an ANN used for unsupervised and efficient mapping of input data (e.g., entity data or process data), to an output value (e.g., an entity classification or optimized process control parameters). Autoencoders may be used for the purpose of dimensionality reduction, that is, a process of reducing the number of random variables under consideration by deducing a set of principal component variables. Dimensionality reduction may be performed, for example, for the purpose of feature selection (e.g., selecting a subset of the original variables) orfeature extraction (e.g., transforming of data in a high-dimensional space to a space of fewer dimensions).

[0158] FIG. 11 illustrates a method 1100 of training a machine learning model of a machine learning module, according to an implementation. Use of method 1100 may provide for use of training data to train a machine learning model for concurrent or later use.

[0159] At 1101 , a machine learning model including one or more machine learning algorithms may be provided.

[0160] At 1102, training data may be provided. T raining data may include one or more of process simulation data, process characterization data, in-process or post-process inspection data (including inspection data provided by a skilled operator and / or inspection data provided by any of a variety of automated inspection tools), or any combination thereof, for past processes that are the same as or different from that of the current process. One or more sets of training data may be used to train the machine learning algorithm used for object defect detection and classification. In some cases, the type of data included in the training data set may vary depending on the specific type of machine learning algorithm employed.

[0161] At 1103, the machine learning model may be trained using the training data. For example, training the model may include inputting the training data to the machine learning model and modifying one or more parameters of the model until the output of the model is the same as (or substantially the same as) external validation data. Model training may generate one or more trained models. One or more trained models may be selected for further validation or deployment, which may be performed using validation data. The results produced by each trained model for the validation data input to the training model may be compared to the validation data to determine which of the models is the best model. For example, the trained model that produces results most closely matching the validation data may be selected as the best model. Test data may then be used to evaluate the selected model. The selected model may also be sent to model deployment in which the best model may be sent to the processor for use in a post-training mode.

[0162] FIG. 12 illustrates a method 1200 of analyzing input data using a machine learning module, according to an implementation. Use of the machine learning module described by method 1200 may enable, for example, automatic classification of an entity or optimized process control.

[0163] At 1201 , a trained machine learning model may be provided to the machine learning module. The trained machine learning model may have been trained, or under continuous or periodic training by one or more other systems or methods. The machine learning model may be pre-generated and trained, enabling functionality of the module as described herein, which can then be used to perform one or more post-training functions of the machine learning module.

[0164] For example, the provided trained machine learning model may be similar to ANN 900, include nodes similar to node 1000, and may have been trained (or be under continuous or periodic training) using a method similar to method 1100.

[0165] At 1202, input data may be provided to the machine learning module for input into the machine learning model. The input data may result from or be derived from a variety of different sources, similar to input data 901 .

[0166] The provision of input data at 1202 may further include removing noise from the data prior to providing it to the machine learning algorithm. Examples of data processing algorithms suitable for use in removing noise from the input data may include, inter alia, signal averaging algorithms, smoothing filter algorithms, Kalman filter algorithms, nonlinear filter algorithms, total variation minimization algorithms, or any combination thereof.

[0167] The provision of input data at 1202 may further include subtraction of a reference data set from the input data to increase contrast between aspects of interest of an entity or process and those not of interest, thereby facilitating classification or process control optimization. For example, a reference data set may include input data for a real or contrived ideal example of the entity or process. If an image sensor or machine vision system is used for entity observation, the reference data set may include an image or set of images (e.g., representing different views) of an ideal entity.

[0168] At 1203, the machine learning module may process the input data using the trained machine learning model to yield results from the machine learning module. Such results may include, for example, an entity classification or one or more optimized process control parameters.

[0169] The following clauses may provide additional context for the present disclosure but should be taken in no way as limiting.

[0170] Clause 1. A method, comprising: receiving, at a processor of a computing platform from an electronic storage device in electronic communication with the processor, aclassical neural network; generating, using the processor, a hybrid neural network comprising: a quantum neural network configured for processing input data using a quantum processor and yield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and storing, using the processor, the hybrid neural network to the electronic storage device.

[0171] Clause 2. The method of clause 1 , further comprising applying, by the processor, a data set as the input data to the hybrid neural network.

[0172] Clause 3. The method of one or more of clauses 1-2, wherein the classical neural network is a pre-trained classical neural network.

[0173] Clause 4. The method of clause 3, wherein generating the hybrid neural network is by quantum transfer learning based on the pre-trained classical neural network.

[0174] Clause 5. The method of one or more of clauses 1-4, wherein the quantum neural network comprises a quantum convolutional neural network.

[0175] Clause 6. The method of one or more of clauses 1-5, wherein the classical neural network comprises a convolutional neural network.

[0176] Clause 7. The method of one or more of clauses 1-7, wherein the hybrid neural network comprises a physics-informed neural network.

[0177] Clause 8. A system, comprising: a processor of a computing platform; and an electronic storage device in electronic communication with the processor; wherein the processor is configured to: receive a classical neural network; generate a hybrid neural network comprising: a quantum neural network configured for processing input data using a quantum processor and yield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and store the hybrid neural network to the electronic storage device.

[0178] Clause 9. The system of clause 8, wherein the processor is further configured to apply a data set as the input data to the hybrid neural network.

[0179] Clause 10. The system of one or more of clauses 8-9, wherein the classical neural network is a pre-trained classical neural network.

[0180] Clause 11 . The system of clause 10, wherein generating the hybrid neural network is by quantum transfer learning based on the pre-trained classical neural network.

[0181] Clause 12. The system of one or more of clauses 8-11 , wherein the quantum neural network comprises a quantum convolutional neural network.

[0182] Clause 13. The system of one or more of clauses 8-12, wherein the classical neural network comprises a convolutional neural network.

[0183] Clause 14. The system of one or more of clauses 8-14, wherein the hybrid neural network comprises a physics-informed neural network.

[0184] Clause 15. A tangible, non-transitory, computer-readable medium having instructions thereupon which when implemented by a processor cause the processor to perform a method comprising: receiving, from an electronic storage device in electronic communication with the processor, a classical neural network; generating a hybrid neural network comprising: a quantum neural network configured for processing input data using a quantum processor and yield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and storing the hybrid neural network to the electronic storage device.

[0185] Clause 16. The tangible, non-transitory, computer-readable medium of clause 15, wherein the method further comprises applying a data set as the input data to the hybrid neural network.

[0186] Clause 17. The tangible, non-transitory, computer-readable medium of one or more of clauses 15-16, wherein the classical neural network is a pre-trained classical neural network.

[0187] Clause 18. The tangible, non-transitory, computer-readable medium of clause 17, wherein generating the hybrid neural network is by quantum transfer learning based on the pre-trained classical neural network.

[0188] Clause 19. The tangible, non-transitory, computer-readable medium of one or more of clauses 15-18, wherein the quantum neural network comprises a quantum convolutional neural network.

[0189] Clause 20. The tangible, non-transitory, computer-readable medium of one or more of clauses 15-19, wherein the classical neural network comprises a convolutional neural network.

[0190] Clause 21 . The tangible, non-transitory, computer-readable medium of one or more of clauses 15-20, wherein the hybrid neural network comprises a physics-informed neural network.

[0191] The invention is limited only by the appended claims. Variations, characteristics, advantages, implementations, constructions, arrangements, terminology, materials, dimensions, embodiments, illustrations, depictions, and examples composing theabove description and accompanying drawings show some possible implementations of the invention without limiting the invention. It is not necessary that every implementation of the invention achieve or possess every advantage, purpose, or characteristic identified herein, and as such, one skilled in the art may effect various additions, changes, modifications, or omissions without departing from the scope or spirit of the invention or its legal equivalents.

[0192] All ranges are inclusive of the stated limits, the orders of magnitude thereof, and all values and ranges substantially therebetween unless otherwise defined. Unless otherwise stated, every use of “and” forms an inclusive list comprising at least the conjoined elements, and every use of “or” forms an inclusive list comprising at least one element of conjoined elements. Unless otherwise stated, singular usage (e.g., 'a', 'an', or 'the') includes plurals of the same.

[0193] The order of recitations in a claim do not imply a temporal or ordered relationship unless unavoidable by the plain language of that claim. No claim may be interpreted to invoke 35 U.S.C. § 112(f) unless that claim recites “means for” or “step for.”

Claims

CLAIMSWe claim:

1. A method, comprising: receiving, at a processor of a computing platform from an electronic storage device in electronic communication with the processor, a classical neural network; generating, using the processor, a hybrid neural network comprising: a quantum neural network configured for processing input data using a quantum processor and yield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and storing, using the processor, the hybrid neural network to the electronic storage device.

2. The method of claim 1 , further comprising applying, by the processor, a data set as the input data to the hybrid neural network.

3. The method of claim 1 , wherein the classical neural network is a pre-trained classical neural network.

4. The method of claim 3, wherein generating the hybrid neural network is by quantum transfer learning based on the pre-trained classical neural network.

5. The method of claim 1 , wherein the quantum neural network comprises a quantum convolutional neural network.

6. The method of claim 1 , wherein the classical neural network comprises a convolutional neural network.

7. The method of claim 1 , wherein the hybrid neural network comprises a physics- informed neural network.

8. A system, comprising: a processor of a computing platform; and an electronic storage device in electronic communication with the processor; wherein the processor is configured to: receive a classical neural network; generate a hybrid neural network comprising:a quantum neural network configured for processing input data using a quantum processor and yield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and store the hybrid neural network to the electronic storage device.

9. The system of claim 8, wherein the processor is further configured to apply a data set as the input data to the hybrid neural network.

10. The system of claim 8, wherein the classical neural network is a pre-trained classical neural network.

11. The system of claim 10, wherein generating the hybrid neural network is by quantum transfer learning based on the pre-trained classical neural network.

12. The system of claim 8, wherein the quantum neural network comprises a quantum convolutional neural network.

13. The system of claim 8, wherein the classical neural network comprises a convolutional neural network.

14. The system of claim 8, wherein the hybrid neural network comprises a physics- informed neural network.

15. A tangible, non-transitory, computer-readable medium having instructions thereupon which when implemented by a processor cause the processor to perform a method comprising: receiving, from an electronic storage device in electronic communication with the processor, a classical neural network; generating a hybrid neural network comprising: a quantum neural network configured for processing input data using a quantum processor and yield intermediate data; and a classical neural network configured for processing the intermediate data to yield output data; and storing the hybrid neural network to the electronic storage device.

16. The tangible, non-transitory, computer-readable medium of claim 15, wherein the method further comprises applying a data set as the input data to the hybrid neural network.

17. The tangible, non-transitory, computer-readable medium of claim 15, wherein the classical neural network is a pre-trained classical neural network.

18. The tangible, non-transitory, computer-readable medium of claim 17, wherein generating the hybrid neural network is by quantum transfer learning based on the pre-trained classical neural network.

19. The tangible, non-transitory, computer-readable medium of claim 15, wherein the quantum neural network comprises a quantum convolutional neural network.

20. The tangible, non-transitory, computer-readable medium of claim 15, wherein the classical neural network comprises a convolutional neural network.

21. The tangible, non-transitory, computer-readable medium of claim 15, wherein the hybrid neural network comprises a physics-informed neural network.

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