Adaptive Learning Rate Research Articles

A multi-objective indirect neural adaptive processes control design for minimization of energy consumption: An experimental validation on a transesterification reactor

Nowadays, energy management environment is a very important issue that technologies have focused on in order to save costs and minimize energy waste. This objective can be achieved by means of an energy resource management approach through an appropriate optimization technique. However, energy savings can conflict with other objective functions, to solve the problem a multi-objective optimization that considers the minimization of the control energy can be adopted. In this paper, we propose to use a multi-objective indirect neural adaptive control concept for nonlinear systems with unknown dynamics. The control scheme consists of an adaptive instantaneous neural emulator and neural controller built on fully connected real-time recurrent learning networks (RTRL). The multi-objective particle swarm optimization (MOPSO) algorithm is used as a mechanism to find NE and NC adaptive learning rates that optimize the proposed objective functions: control energy minimization and closed-loop preservation. Comparative studies and experimental validation on a semi-batch reactor, used to generate cleaner biofuels, are performed to confirm the effectiveness of the proposed strategy.

Adaptive learning rate in dynamical binary environments: the signature of adaptive information processing

Adaptive learning rate in dynamical binary environments: the signature of adaptive information processing

Analytical Neural Network System for the Helicopter Turboshaft Engines Operating Modes Classification

A novel neural network-based analytical system has been developed for classifying helicopter turboshaft engine operating modes during flight operations. This system utilizes a neural network architecture comprising an input layer with three neurons, two fully connected hidden layers, and an output layer with two neurons. The proposed approach demonstrates an exceptional recognition accuracy of 0.997 (99.7%) across steady-state, unsteady, and transient operating modes of helicopter turboshaft engines. A new method for training this neural network has been introduced, employing forward propagation, loss calculation, backpropagation, and weight updates, enhanced by an adaptive learning rate and the cross-entropy function as the loss criterion. The method also incorporates a novel modified Smooth ReLU activation function for hidden layer neurons. This innovation led to a near-perfect accuracy in network training and reduced the loss to 0.025 (2.5%), highlighting the high quality and reliability of the neural network in classifying engine operating modes during flight. Furthermore, it has been empirically shown that the application of this neural network significantly reduces type I errors by a factor of 2.09 to 2.14 and type II errors by 2.05 to 2.21 times compared to traditional classifiers based on ART-1 and BAM networks. This advancement marks a substantial improvement in classification accuracy and error minimization for helicopter turboshaft engine operating modes.

Deep Reinforcement Learning with Local Attention for Single Agile Optical Satellite Scheduling Problem

This paper investigates the single agile optical satellite scheduling problem, which has received increasing attention due to the rapid growth in earth observation requirements. Owing to the complicated constraints and considerable solution space of this problem, the conventional exact methods and heuristic methods, which are sensitive to the problem scale, demand high computational expenses. Thus, an efficient approach is demanded to solve this problem, and this paper proposes a deep reinforcement learning algorithm with a local attention mechanism. A mathematical model is first established to describe this problem, which considers a series of complex constraints and takes the profit ratio of completed tasks as the optimization objective. Then, a neural network framework with an encoder–decoder structure is adopted to generate high-quality solutions, and a local attention mechanism is designed to improve the generation of solutions. In addition, an adaptive learning rate strategy is proposed to guide the actor–critic training algorithm to dynamically adjust the learning rate in the training process to enhance the training effectiveness of the proposed network. Finally, extensive experiments verify that the proposed algorithm outperforms the comparison algorithms in terms of solution quality, generalization performance, and computation efficiency.

A Unified Analysis of AdaGrad With Weighted Aggregation and Momentum Acceleration.

Integrating adaptive learning rate and momentum techniques into stochastic gradient descent (SGD) leads to a large class of efficiently accelerated adaptive stochastic algorithms, such as AdaGrad, RMSProp, Adam, AccAdaGrad, and so on. In spite of their effectiveness in practice, there is still a large gap in their theories of convergences, especially in the difficult nonconvex stochastic setting. To fill this gap, we propose weighted AdaGrad with unified momentum and dubbed AdaUSM, which has the main characteristics that: 1) it incorporates a unified momentum scheme that covers both the heavy ball (HB) momentum and the Nesterov accelerated gradient (NAG) momentum and 2) it adopts a novel weighted adaptive learning rate that can unify the learning rates of AdaGrad, AccAdaGrad, Adam, and RMSProp. Moreover, when we take polynomially growing weights in AdaUSM, we obtain its O(log(T)/√T) convergence rate in the nonconvex stochastic setting. We also show that the adaptive learning rates of Adam and RMSProp correspond to taking exponentially growing weights in AdaUSM, thereby providing a new perspective for understanding Adam and RMSProp. Finally, comparative experiments of AdaUSM against SGD with momentum, AdaGrad, AdaEMA, Adam, and AMSGrad on various deep learning models and datasets are also carried out.

A novel method for a technology enhanced learning recommender system considering changing user interest based on neural collaborative filtering

This study introduces an advanced recommender system for technology enhanced learning (TEL) that synergizes neural collaborative filtering, sentiment analysis, and an adaptive learning rate to address the limitations of traditional TEL systems. Recognizing the critical gap in existing approaches—primarily their neglect of user emotional feedback and static learning paths—our model innovatively incorporates sentiment analysis to capture and respond to nuanced emotional feedback from users. Utilizing Bidirectional Encoder Representations from Transformers for sentiment analysis, our system not only understands but also respects user privacy by processing feedback without revealing sensitive information. The adaptive learning rate, inspired by AdaGrad, allows our model to adjust its learning trajectory based on the sentiment scores associated with user feedback, ensuring a dynamic response to both positive and negative sentiments. This dual approach enhances the system’s adaptability to changing user preferences and improves its contentment understanding. Our methodology involves a comprehensive analysis of both the content of learning materials and the behaviors and preferences of learners, facilitating a more personalized learning experience. By dynamically adjusting recommendations based on real-time user data and behavioral analysis, our system leverages the collective insights of similar users and relevant content. We validated our approach against three datasets—MovieLens, Amazon, and a proprietary TEL dataset—and saw significant improvements in recommendation precision, F-score, and mean absolute error. The results indicate the potential of integrating sentiment analysis and adaptive learning rates into TEL recommender systems, marking a step forward in developing more responsive and user-centric educational technologies. This study paves the way for future advancements in TEL systems, emphasizing the importance of emotional intelligence and adaptability in enhancing the learning experience.

Speed and accuracy investigations of neural network algorithms for ionospheric modelling at an equatorial region

Neural networks are very efficient tools for modeling, including ionospheric modeling. The training algorithm is important for achieving the optimum performance of the trained network. This research is therefore meant to evaluate and compare the performances of ten neural network training algorithms based on their prediction accuracies, and the duration/times taken by each of the algorithms to establish the optimum result. The neural networks were trained using electron density measurements by Radio Occultation (RO) technique from the Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) satellites. Data for the period 2006 through 2021 was used. The networks were trained using about 2.9 million data points collected from the Kano region, Nigeria (5-degree rectangular region around geographic: 12.00° N, 8.59° E) after performing data quality control. The training algorithms considered in the work include: Bayesian Regularization (BR); Broyden-Fletcher-Goldfarb-Shanno Quasi-Newton (BFG); Conjugate Gradient with Powell/Beale (CGB); Fletcher-Reeves Conjugate Gradient (CGF); Gradient descent with momentum and adaptive learning rate (GDX); Levenberg Marquardt (LM); One Step Secant (OSS); Polak-Ribiére Conjugate Gradient (CGP); Resilient Backpropagation (RP); and Scaled Conjugate Gradient (SCG). The results showed that the BR and the LM algorithms gave the best performances in minimizing the errors of prediction (the mean RMSEs are respectively 112 and 114 ×103 electrons/cm3), but the RP algorithm, which came third in terms of accuracy, was significantly faster than both the LM and BR algorithms. The worst-performing algorithm in terms of accuracy was the GDX algorithm, although it was the fastest algorithm. The BFG algorithm was the worst-performing algorithm in terms of a combination of speed and accuracy. The developed neural network model was validated using ionosonde electron density measurements obtained from Ilorin, Nigeria (geographic: 8.5° N, 4.5° E; geomagnetic: 1.8° S). A comparison of the neural network, the NeQuick, and the IRI model predictions relative to the ionosonde measurements indicate that the neural network model was the best-performing model; the NN model predictions minimized the mean absolute errors (MAEs) in ∼44% of 399 ionosonde profiles investigated, the IRI model did so in ∼32%, and the NeQuick did so in ∼24%. The MAEs of the NeQuick however exhibited the best (least) variance. In overall, the NN model gave the least (best) mean of the MAEs (∼73 × 103 cm−3), compared to ∼82 × 103 cm−3 given by both the NeQuick and the IRI models, further supporting the idea that neural networks are excellent for present-day ionospheric modeling.

Brain Tumor Segmentation from Optimal MRI Slices Using a Lightweight U-Net

The timely detection and accurate localization of brain tumors is crucial in preserving people’s quality of life. Thankfully, intelligent computational systems have proven invaluable in addressing these challenges. In particular, the UNET model can extract essential pixel-level features to automatically identify the tumor’s location. However, known deep learning-based works usually directly feed the 3D volume into the model, which causes excessive computational complexity. This paper presents an approach to boost the UNET network, reducing computational workload while maintaining superior efficiency in locating brain tumors. This concept could benefit portable or embedded recognition systems with limited resources for operating in real time. This enhancement involves an automatic slice selection from the MRI T2 modality volumetric images containing the most relevant tumor information and implementing an adaptive learning rate to avoid local minima. Compared with the original model (7.7 M parameters), the proposed UNET model uses only 2 M parameters and was tested on the BraTS 2017, 2020, and 2021 datasets. Notably, the BraTS2021 dataset provided outstanding binary metric results: 0.7807 for the Intersection Over the Union (IoU), 0.860 for the Dice Similarity Coefficient (DSC), 0.656 for the Sensitivity, and 0.9964 for the Specificity compared to vanilla UNET.

CMA-ES with Learning Rate Adaptation

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful methods for solving continuous black-box optimization problems. A practically useful aspect of CMA-ES is that it can be used without hyperparameter tuning. However, the hyperparameter settings still have a considerable impact on performance, especially for difficult tasks, such as solving multimodal or noisy problems. This study comprehensively explores the impact of learning rate on CMA-ES performance and demonstrates the necessity of a small learning rate by considering ordinary differential equations. Thereafter, it discusses the setting of an ideal learning rate. Based on these discussions, we develop a novel learning rate adaptation mechanism for CMA-ES that maintains a constant signal-to-noise ratio. Additionally, we investigate the behavior of CMA-ES with the proposed learning rate adaptation mechanism through numerical experiments and compare the results with those obtained for CMA-ES with a fixed learning rate and with population size adaptation. The results show that CMA-ES with the proposed learning rate adaptation works well for multimodal and/or noisy problems without extremely expensive learning rate tuning.

DISTRIBUTED HIGH-PERFORMANCE COMPUTING METHODS FOR ACCELERATING DEEP LEARNING TRAINING

This paper comprehensively analyzes distributed high-performance computing methods for accelerating deep learning training. We explore the evolution of distributed computing architectures, including data parallelism, model parallelism, and pipeline parallelism, and their hybrid implementations. The study delves into optimization techniques crucial for large-scale training, such as distributed optimization algorithms, gradient compression, and adaptive learning rate methods. We investigate communication-efficient algorithms, including Ring All Reduce variants and decentralized training approaches, which address the scalability challenges in distributed systems. The research examines hardware acceleration and specialized systems, focusing on GPU clusters, custom AI accelerators, high-performance interconnects, and distributed storage systems optimized for deep learning workloads. Finally, we discuss this field's challenges and future directions, including scalability-efficiency trade-offs, fault tolerance, energy efficiency in large-scale training, and emerging trends like federated learning and neuromorphic computing. Our findings highlight the synergy between advanced algorithms, specialized hardware, and optimized system designs in pushing the boundaries of large-scale deep learning, paving the way for future breakthroughs in artificial intelligence.

Multi-Dimensional Adaptive Learning Rate Gradient Descent Optimization Algorithm for Network Training in Magneto-Optical Defect Detection

Multi-Dimensional Adaptive Learning Rate Gradient Descent Optimization Algorithm for Network Training in Magneto-Optical Defect Detection

Lossy Image Compression with Stochastic Quantization

Introduction. Lossy image compression algorithms play a crucial role in various domains, including graphics, and image processing. As image information density increases, so do the resources required for processing and transmission. One of the most prominent approaches to address this challenge is color quantization, proposed by Orchard et al. (1991). This technique optimally maps each pixel of an image to a color from a limited palette, maintaining image resolution while significantly reducing information content. Color quantization can be interpreted as a clustering problem (Krishna et al. (1997), Wan (2019)), where image pixels are represented in a three-dimensional space, with each axis corresponding to the intensity of an RGB channel. The purpose of the paper. Scaling of traditional algorithms like K-Means can be challenging for large data, such as modern images with millions of colors. This paper reframes color quantization as a three-dimensional stochastic transportation problem between the set of image pixels and an optimal color palette, where the number of colors is a predefined hyperparameter. We employ Stochastic Quantization (SQ) with a seeding technique proposed by Arthur et al. (2007) to enhance the scalability of color quantization. This method introduces a probabilistic element to the quantization process, potentially improving efficiency and adaptability to diverse image characteristics. Results. To demonstrate the efficiency of our approach, we present experimental results using images from the ImageNet dataset. These experiments illustrate the performance of our Stochastic Quantization method in terms of compression quality, computational efficiency, and scalability compared to traditional color quantization techniques. Conclusions. This study introduces a scalable algorithm for solving the color quantization problem without memory constraints, demonstrating its efficiency on a subset of images from the ImageNet dataset. The convergence speed of the algorithm can be further enhanced by modifying the update rule with alternative methods to Stochastic Gradient Descent (SGD) that incorporate adaptive learning rates. Moreover, the stochastic nature of the proposed solution enables the utilization of parallelization techniques to simultaneously update the positions of multiple quants, potentially leading to significant performance improvements. This aspect of parallelization and its impact on algorithm efficiency presents a topic for future research. The proposed method not only addresses the limitations of existing color quantization techniques but also opens up new possibilities for optimizing image compression algorithms in resource-constrained environments. Keywords: non-convex optimization, stochastic optimization, stochastic quantization, color quantization, lossy compression.

Individuals with anxiety and depression use atypical decision strategies in an uncertain world.

Previous studies on reinforcement learning have identified three prominent phenomena: (1) individuals with anxiety or depression exhibit a reduced learning rate compared to healthy subjects; (2) learning rates may increase or decrease in environments with rapidly changing (i.e. volatile) or stable feedback conditions, a phenomenon termed learning rate adaptation; and (3) reduced learning rate adaptation is associated with several psychiatric disorders. In other words, multiple learning rate parameters are needed to account for behavioral differences across participant populations and volatility contexts in this flexible learning rate (FLR) model. Here, we propose an alternative explanation, suggesting that behavioral variation across participant populations and volatile contexts arises from the use of mixed decision strategies. To test this hypothesis, we constructed a mixture-of-strategies (MOS) model and used it to analyze the behaviors of 54 healthy controls and 32 patients with anxiety and depression in volatile reversal learning tasks. Compared to the FLR model, the MOS model can reproduce the three classic phenomena by using a single set of strategy preference parameters without introducing any learning rate differences. In addition, the MOS model can successfully account for several novel behavioral patterns that cannot be explained by the FLR model. Preferences for different strategies also predict individual variations in symptom severity. These findings underscore the importance of considering mixed strategy use in human learning and decision-making and suggest atypical strategy preference as a potential mechanism for learning deficits in psychiatric disorders.

Individuals with anxiety and depression use atypical decision strategies in an uncertain world

Previous studies on reinforcement learning have identified three prominent phenomena: (1) individuals with anxiety or depression exhibit a reduced learning rate compared to healthy subjects; (2) learning rates may increase or decrease in environments with rapidly changing (i.e. volatile) or stable feedback conditions, a phenomenon termed learning rate adaptation; and (3) reduced learning rate adaptation is associated with several psychiatric disorders. In other words, multiple learning rate parameters are needed to account for behavioral differences across participant populations and volatility contexts in this flexible learning rate (FLR) model. Here, we propose an alternative explanation, suggesting that behavioral variation across participant populations and volatile contexts arises from the use of mixed decision strategies. To test this hypothesis, we constructed a mixture-of-strategies (MOS) model and used it to analyze the behaviors of 54 healthy controls and 32 patients with anxiety and depression in volatile reversal learning tasks. Compared to the FLR model, the MOS model can reproduce the three classic phenomena by using a single set of strategy preference parameters without introducing any learning rate differences. In addition, the MOS model can successfully account for several novel behavioral patterns that cannot be explained by the FLR model. Preferences for different strategies also predict individual variations in symptom severity. These findings underscore the importance of considering mixed strategy use in human learning and decision-making and suggest atypical strategy preference as a potential mechanism for learning deficits in psychiatric disorders.

Sigmoidal learning rate optimizer for deep neural network training using a two-phase adaptation approach

The optimization of machine learning models is an open research question since an optimal procedure to change the learning rate throughout training is still unknown. Manually defining a learning rate schedule involves troublesome, time-consuming try-and-error procedures to determine hyperparameters, such as learning rate decay epochs and learning rate decay rates, in order to navigate the intricate landscape of loss functions. Although adaptive learning rate optimizers automatize this process, recent studies suggest they may produce overfitting and reduce performance compared to fine-tuned learning rate schedules. Considering that the new machine learning loss function approaches present landscapes with much more saddle points than local minima, we proposed the Training Aware Sigmoidal Optimizer (TASO). This automated two-phase learning rate adaptation mechanism significantly reduces the need for manual hyperparameter tuning. The first phase uses a high learning rate to quickly traverse the numerous saddle points in the error surface, while the second phase uses a low learning rate to gradually approach the center of the local minimum previously found. We compared the proposed approach with commonly used adaptive learning rate schedules such as Adam, RMSProp, and Adagrad. The validation experiments were performed in image and text datasets and showed that TASO outperformed all competing methods in optimal (i.e., performing hyperparameter validation) and suboptimal (i.e., using default hyperparameters) scenarios. In our benchmark tests, TASO demonstrated promising performance, achieving an 8.32% increase in accuracy and a significant 46.62% decrease in training loss on average across various datasets and models. This remarkable performance positions TASO ahead of well-established adaptive optimizers, suggesting higher effectiveness and consistency in performance.

Efficient adaptive learning rate for convolutional neural network based on quadratic interpolation egret swarm optimization algorithm

Convolutional neural network (CNN) has recently become popular for addressing multi-domain image classification. However, most existing methods frequently suffer from poor performance, especially in performance and convergence for various datasets. Herein, we have proposed an algorithm for multi-domain image classification by introducing a novel adaptive learning rate rule to the conventional CNN. Specifically, we adopt the CNN to extract rich feature representations. Given that the hyperparameters of the learning rate have a positive effect on the prediction error, the Egret Swarm Optimization Algorithm (ESOA) is introduced to update the learning rate, which can jump out of local extrema during exploration. Therefore, combined with quadratic interpolation, the objective function can be approximated by a polynomial, thereby improving its prediction accuracy. To verify the robustness of the proposed algorithm, we conducted comprehensive experiments in five domain public datasets to fulfil the task of image classification. Meanwhile, the highest accuracy rate of 97.15 % was obtained on the test set. The performances of our method on 24 benchmark functions (CEC2017 and CEC2022) are compared with Particle Swarm Optimization (PSO), Genetic Algorithm(GA), Whale Optimization Algorithm(WOA), Catch Fish Optimization Algorithm(CFOA), GOOSE Algorithm(GO) and ESOA. In two benchmark sets, the performance metric values of our algorithm rank no. 1, especially in all unimodal functions in contrast with other baseline algorithms.

Classification of Pneumonia via a Hybrid ZFNet-Quantum Neural Network Using a Chest X-ray Dataset.

Deep neural networks (DNNs) have made significant contributions to diagnosing pneumonia from chest X-ray imaging. However, certain aspects of diagnosis and planning can be further enhanced through the implementation of a quantum deep neural network (QDNN). Therefore, we introduced a technique that integrates neural networks with quantum algorithms named the ZFNet-quantum neural network for detecting pneumonia using 5863 X-ray scans with binary cases. The hybrid model efficiently pre-processes complex and high-dimensional data by extracting significant features from the ZFNet model. These significant features are given to the quantum circuit algorithm and further embedded into a quantum device. The parameterized quantum circuit algorithm using qubits, superposition theorem, and entanglement phenomena generates 4 features from 4098 features extracted from images via a deep transfer learning model. Moreover, to validate the outcome measures of the proposed technique, we used various PennyLane quantum devices to detect pneumonia and normal control images. By using the Adam optimizer, which exploits an adaptive learning rate that is fixed to 10-6 and six layers of a quantum circuit composed of quantum gates, the proposed model achieves an accuracy of 96.5%, corresponding to 25 epochs. The integrated ZFNet-quantum learning network outperforms the deep transfer learning network in terms of testing accuracy, as the accuracy gained by the convolutional neural network (CNN) is 94%. Therefore, we use a hybrid classical-quantum model to detect pneumonia in which a variational quantum algorithm enhances the outcomes of a ZFNet transfer learning method. This approach is an efficient and automated method for detecting pneumonia and could significantly enhance outcome measures related to the speed and accuracy of the network in the clinical and healthcare sectors.

IMAML-IDCG: Optimization-based meta-learning with ImageNet feature reusing for few-shot invasive ductal carcinoma grading

Model-Agnostic Meta-Learning (MAML) is a widely used few-shot learning (FSL) technique that reduces reliance on large, labeled datasets in deep learning for medical imaging analysis. However, MAML requires backpropagating through all feature layers for task adaptation, leading to suboptimal computational efficiency. We propose IMAML-IDCG (ImageNet Model-Agnostic Meta-Learning in Invasive Ductal Carcinoma Grading), which enhances computational efficiency for few-shot grading of Invasive Ductal Carcinoma (IDC) through three key techniques: (1) ImageNet feature reusing, (2) ImageNet partial freezing strategy, and (3) adaptive inner learning rate. IMAML-IDCG is initialized with ImageNet pre-trained weights. During the inner optimization loop, only the model’s classifier head layer is optimized, leveraging prior ImageNet knowledge (ImageNet feature reusing) and employing an adaptive learning rate for improved task adaptation. In the outer optimization loop, IMAML-IDCG selectively fine-tunes the last few model layers to enhance efficiency and reduce overfitting (ImageNet partial freezing strategy). We evaluated IMAML-IDCG using the BreaKHis dataset (7,909 images) as the base dataset, and the BCHI (282 images) and PathoIDCG (3,744 images) datasets as the novel datasets. Our empirical results demonstrate that IMAML-IDCG outperforms MAML and other FSL methods in few-shot IDC grading tasks across various cross-magnification domain settings. Notably, IMAML-IDCG achieves a 14.64% improvement over MAML on the BCHI dataset and a 6.04% improvement on the PathoIDCG 40X dataset when meta-trained with the BreaKHis 40X dataset in the 3-way 5-shot scenario.

Automatic Learning Rate Adaption for Memristive Deep Learning Systems.

As a possible device to further enhance the performance of the hybrid complementary metal oxide semiconductor (CMOS) technology in the hardware, the memristor has attracted widespread attention in implementing efficient and compact deep learning (DL) systems. In this study, an automatic learning rate tuning method for memristive DL systems is presented. Memristive devices are utilized to adjust the adaptive learning rate in deep neural networks (DNNs). The speed of the learning rate adaptation process is fast at first and then becomes slow, which consist of the memristance or conductance adjustment process of the memristors. As a result, no manual tuning of learning rates is required in the adaptive back propagation (BP) algorithm. While cycle-to-cycle and device-to-device variations could be a significant issue in memristive DL systems, the proposed method appears robust to noisy gradients, various architectures, and different datasets. Moreover, fuzzy control methods for adaptive learning are presented for pattern recognition, such that the over-fitting issue can be well addressed. To our best knowledge, this is the first memristive DL system using an adaptive learning rate for image recognition. Another highlight of the presented memristive adaptive DL system is that quantized neural network architecture is utilized, and there is therefore a significant increase in the training efficiency, without the loss of testing accuracy.

Structure modification based PID neural network decoupling control for nonlinear multivariable systems

In this research, a structure modification based PID neural network (PIDNN) decoupling strategy is proposed to solve the difficulty in controller caused by the strong coupling in nonlinear multivariable systems. Incomplete differential neurons are first introduced into the hidden layer of the PIDNN, which achieving a decreased amount of control overshoot and better control stability. Then, an improved integration strategy is incorporated in the hidden layer structure, resulting in expedited convergence speed in the PIDNN control. Furthermore, intelligent optimization algorithms are employed to improve the convergence rate of the proposed PIDNN. The stability of the proposed controller is analyzed, and an adaptive learning rate scheme is derived. Finally, to confirm the effectiveness of the controller, three examples and the analysis of experimental results are provided.