Universal Approximation Capability Research Articles

GRINN: a physics-informed neural network for solving hydrodynamic systems in the presence of self-gravity

Modeling self-gravitating gas flows is essential to answering many fundamental questions in astrophysics. This spans many topics including planet-forming disks, star-forming clouds, galaxy formation, and the development of large-scale structures in the Universe. However, the nonlinear interaction between gravity and fluid dynamics offers a formidable challenge to solving the resulting time-dependent partial differential equations (PDEs) in three dimensions (3D). By leveraging the universal approximation capabilities of a neural network within a mesh-free framework, physics informed neural networks (PINNs) offer a new way of addressing this challenge. We introduce the gravity-informed neural network (GRINN), a PINN-based code, to simulate 3D self-gravitating hydrodynamic systems. Here, we specifically study gravitational instability and wave propagation in an isothermal gas. Our results match a linear analytic solution to within 1% in the linear regime and a conventional grid code solution to within 5% as the disturbance grows into the nonlinear regime. We find that the computation time of the GRINN does not scale with the number of dimensions. This is in contrast to the scaling of the grid-based code for the hydrodynamic and self-gravity calculations as the number of dimensions is increased. Our results show that the GRINN computation time is longer than the grid code in one- and two- dimensional calculations but is an order of magnitude lesser than the grid code in 3D with similar accuracy. Physics-informed neural networks like GRINN thus show promise for advancing our ability to model 3D astrophysical flows.

A finite-time path-tracking control algorithm for nonholonomic mobile robots with unknown dynamics and subject to wheel slippage/skid disturbances

Path planning and tracking control are two performance-critical tasks for wheeled mobile robots, particularly when nonholonomic constraints are imposed on robots in dynamically uncertain conditions. Accomplishing certain performance and safety considerations related to path-tracking, such as global stability, transient performance, and smooth finite-time convergence, becomes more difficult for nonholonomic robots. This paper is concerned with proposing a new adaptive robust finite-time tracking control approach for a large class of differential drive autonomous nonholonomic wheeled mobile robots (NWMRs) that are subject to structured uncertainties and extraneous disturbances with fully unknown dynamics. For this purpose, nonlinear kinodynamics of a type of rear-wheel drive NWMRs are developed by incorporating the skid/slippage constituents of the wheel motion. Then, a path-tracking controller is proposed using a continuous finite-time adaptive integral sliding mode control coupled with an integral backstepping approach (FTAISM-IBC). For the adaptive controller design, the entire nonlinear dynamics of the robot, including nonlinear vector functions and control gain functions, together with extraneous disturbances, are estimated by leveraging the universal approximation capabilities of radial basis neural networks (RBFNNs). The finite-time stability proof is presented by utilizing the Lyapunov stability theorem. Furthermore, the adaptive gains are derived to ensure the finite-time stability of the system subject to unknown functions, parametric variations, and unknown but bounded disturbances. Finally, the effectiveness of the proposed controller is evaluated through simulations in terms of several key performance indicators against several reported studies.

Completeness of atomic structure representations

In this paper, we address the challenge of obtaining a comprehensive and symmetric representation of point particle groups, such as atoms in a molecule, which is crucial in physics and theoretical chemistry. The problem has become even more important with the widespread adoption of machine-learning techniques in science, as it underpins the capacity of models to accurately reproduce physical relationships while being consistent with fundamental symmetries and conservation laws. However, some of the descriptors that are commonly used to represent point clouds— notably those based on discretized correlations of the neighbor density that power most of the existing ML models of matter at the atomic scale—are unable to distinguish between special arrangements of particles in three dimensions. This makes it impossible to machine learn their properties. Atom-density correlations are provably complete in the limit in which they simultaneously describe the mutual relationship between all atoms, which is impractical. We present a novel approach to construct descriptors of finite correlations based on the relative arrangement of particle triplets, which can be employed to create symmetry-adapted models with universal approximation capabilities, and have the resolution of the neighbor discretization as the sole convergence parameter. Our strategy is demonstrated on a class of atomic arrangements that are specifically built to defy a broad class of conventional symmetric descriptors, showing its potential for addressing their limitations.

Extended Features Based Random Vector Functional Link Network for Classification Problem

Random vector functional link (RVFL) network has been successfully employed in diverse domains such as computer vision and machine learning, due to its universal approximation capability. Recently, the shallow RVFL architecture has been extended to deep architectures. In deep architectures, multiple hidden layers are stacked for extracting informative features from the original feature space. Therefore, having rich features, deep models are very successful compared to shallow models. In this article, we propose an extended feature RVFL (efRVFL) model that is trained over extended feature space generated analytically from the original feature space. The proposed efRVFL model has three types of features, i.e., original features, supervised randomized (newly generated) features, and unsupervised randomized features, in its feature matrix. The proposed efRVFL model with additional features has capability to capture nonlinear hidden relationships within the dataset. The proposed efRVFL model is an unstable classifier, and thus, its performance can be improved further via ensemble learning. Ensemble models are stable and accurate and have better generalization performance than single models. Therefore, we also propose an ensemble of extended feature RVFL (en-efRVFL) model. Each base model of en-efRVFL is trained over different feature spaces so that more accurate and diverse base models can be generated. The outcome of the base models is integrated via average voting scheme. Empirical evaluation over <inline-formula> <tex-math notation="LaTeX">$46$</tex-math> </inline-formula> UCI classification datasets demonstrates that the proposed efRVFL and en-efRVFL models have better performance than RVFL and other given deep models. Furthermore, the experimental results over <inline-formula> <tex-math notation="LaTeX">$12$</tex-math> </inline-formula> sparse datasets show that the proposed en-efRVFL model has a winning performance among several deep feedforward neural networks (FNNs).

Designing Universally-Approximating Deep Neural Networks: A First-Order Optimization Approach.

Universal approximation capability, also referred to as universality, is an important property of deep neural networks, endowing them with the potency to accurately represent the underlying target function in learning tasks. In practice, the architecture of deep neural networks largely influences the performance of the models. However, most existing methodologies for designing neural architectures, such as the heuristic manual design or neural architecture search, ignore the universal approximation property, thus losing a potential safeguard about the performance. In this paper, we propose a unified framework to design the architectures of deep neural networks with a universality guarantee based on first-order optimization algorithms, where the forward pass is interpreted as the updates of an optimization algorithm. The (explicit or implicit) network is designed by replacing each gradient term in the algorithm with a learnable module similar to a two-layer network or its derivatives Specifically, we explore the realm of width-bounded neural networks, a common practical scenario, showcasing their universality. Moreover, adding operations of normalization, downsampling, and upsampling does not hurt the universality. To the best of our knowledge, this is the first work that width-bounded networks with universal approximation guarantee can be designed in a principled way. Our framework can inspire a variety of neural architectures including some renowned structures such as ResNet and DenseNet, as well as novel innovations. The experimental results on image classification problems demonstrate that the newly inspired networks are competitive and surpass the baselines of ResNet, DenseNet, as well as the advanced ConvNeXt and ViT, testifying to the effectiveness of our framework.

The convergence of a generalized convex neural network

ABSTRACT The convergence of neural networks is a central area of research that is essential in understanding the universal approximation capability and the structural complexity of these systems. In this study, we investigate a generalized convex incremental iteration method. This iteration method is presented in a more general form than previous studies and is capable of scraping a broader range of weight parameters. Additionally, we systematically demonstrate the convergence rate of the convex iteration. Our findings are also easily extensible to deep convolutional neural networks, which helps to explain their effectiveness in various real-world applications. Furthermore, we provide a discrete statistical perspective to address issues of input data non-compactness and unknowability of the objective function in practical settings. To support our conclusions, we propose two algorithms, namely back propagation and random search, the latter of which can prevent the network from getting stuck in a local minimum during training. Finally, we present results from several regression problems, which indicate that our algorithms exhibit superior performance and are consistent with our theoretical predictions. These results provide a more comprehensive understanding of the convergence of neural networks and its significance in the universal approximation capability of these systems.

An array of microresonators as a photonic extreme learning machine

Machine learning technologies have found fertile ground in optics due to their promising features based on speed and parallelism. Feed-forward neural networks are one of the most widely used machine learning algorithms due to their simplicity and universal approximation capability. However, the typical training procedure, where all weights are optimized, can be time and energy consuming. An alternative approach is the Extreme Learning Machine, a feed-forward neural network in which only the output weights are trained, while the internal connections are random. Here we present an experimental implementation of a photonic extreme learning machine (PELM) in an integrated silicon chip. The PELM is based on the processing of the image of the scattered light by an array of 18 gratings coupled to microresonators. Light propagation in the microresonator array is a linear process while light detection by the video camera is a nonlinear process. Training is done offline by analyzing the recorded scattered light image with a linear classifier. We provide a proof-of-concept demonstration of the PELM by solving both binary and analog tasks, and show how the performance depends on the number of microresonators used in the readout procedure.

Nonparametric Multivariate Probability Density Forecast in Smart Grids With Deep Learning

This paper proposes a nonparametric multivariate density forecast model based on deep learning. It not only offers the whole marginal distribution of each random variable in forecasting targets, but also reveals the future correlation between them. Differing from existing multivariate density forecast models, the proposed method requires no a priori hypotheses on the forecasted joint probability distribution of forecasting targets. In addition, based on the universal approximation capability of neural networks, the real joint cumulative distribution functions of forecasting targets are well-approximated by a special positive-weighted deep neural network in the proposed method. Numerical tests from different scenarios were implemented under a comprehensive verification framework for evaluation, including the very short-term forecast of the wind speed, wind power, and the day-ahead forecast of the aggregated electricity load. Testing results corroborate the superiority of the proposed method over current multivariate density forecast models considering the accordance with reality, prediction interval width, and correlations between different random variables.

Deep learning algorithms for solving differential equations: a survey

ABSTRACT Differential equations (DEs) are widely employed in the mathematical modelling of a wide range of scientific and engineering problems. The analytical solution of these DEs is typically unknown for a variety of practical problems of relevance. Several numerical methods have been developed over time to find the solution to such DEs and numerous new approaches are still being proposed daily. In recent years, deep learning has emerged as a promising method for solving high-dimensional DEs. Due to the universal approximation capability of neural networks, there is no doubt that studies in this field will continue to grow in the near future. However, there is a need to understand the best-performing neural network architectures and algorithms that demonstrated their effectiveness and ability over traditional algorithms for solving various types of high-dimensional DEs. In this survey, we provide a review of deep learning algorithms classified as artificial neural networks (ANNs) and deep neural networks (DNNs) for solutions of DEs, that have been published in the last decade (between 2011 and 2022). The key purpose of this study is to explore the research papers published in the area of numerical solutions of DEs in order to get a deeper understanding of the current situation.

Chattering suppression and hydrodynamic disturbance estimation of underwater manipulators using adaptive fuzzy sliding mode control

Robotic manipulators are nowadays widely used in various underwater scenarios, but their motion control remains a challenging task due to hydrodynamic effects. This article proposes a novel adaptive fuzzy sliding mode control (AFSMC) strategy for precise and robust control of underwater manipulators. To simulate the movement of the robotic manipulators in the underwater environment, the Unified Robot Description Format (URDF) file of a custom-designed electric underwater manipulator is imported into the Simscape Multibody and the hydrodynamic disturbance is modeled according to Morison’s equation. Furthermore, the proposed control strategy takes advantage of the universal approximation capability of fuzzy systems to avoid chattering and observe disturbances by adjusting the control gains of classical sliding mode control (CSMC). And adaptive laws are designed to update the parameters of the fuzzy systems. The strong friction caused by the seal is also compensated by actual test data. In the simulation experiments, a special environment of water flow and variable loads is considered. The results demonstrate that the AFSMC strategy can achieve high precision and strong robustness against disturbances for trajectory tracking. More importantly, the chattering caused by CSMC can be eliminated and the hydrodynamic disturbance can be estimated with high precision through the proposed control strategy.

Random vector functional link network: Recent developments, applications, and future directions

Neural networks have been successfully employed in various domains such as classification, regression and clustering, etc. Generally, the back propagation (BP) based iterative approaches are used to train the neural networks, however, it results in the issues of local minima, sensitivity to learning rate and slow convergence. To overcome these issues, randomization based neural networks such as random vector functional link (RVFL) network have been proposed. RVFL model has several characteristics such as fast training speed, direct links, simple architecture, and universal approximation capability, that make it a viable randomized neural network. This article presents the first comprehensive review of the evolution of RVFL model, which can serve as the extensive summary for the beginners as well as practitioners. We discuss the shallow RVFLs, ensemble RVFLs, deep RVFLs and ensemble deep RVFL models. The variations, improvements and applications of RVFL models are discussed in detail. Moreover, we discuss the different hyperparameter optimization techniques followed in the literature to improve the generalization performance of the RVFL model. Finally, we present potential future research directions/opportunities that can inspire the researchers to improve the RVFL’s architecture and learning algorithm further.

Fast Finite-Time Control for Nonaffine Stochastic Nonlinear Systems Against Multiple Actuator Constraints via Output Feedback.

This research addresses the finite-time control problem for nonaffine stochastic nonlinear systems with actuator faults and input saturation. Specifically, a new finite-time control scheme is constructed based on the adaptive backstepping framework, with the usage of a state observer and taking advantage of the universal approximation capability of the fuzzy-logic system (FLS). The novelty of this work is that it considers the output feedback problem of a completely nonaffine stochastic system and incorporates the idea of the dynamic surface control (DSC) design. By using the Lyapunov stability theory, all the signals of the controlled system can be semiglobal finite-time stable in probability (SGFSP) while the system is imposed with multiple actuator constraints. In the meantime, the problem of "complexity explosion" is avoided. Two simulation examples are given to demonstrate the validity of the presented strategy.

Greedy stochastic configuration networks for ill-posed problems

The inequality constraint in stochastic configuration networks (SCNs) is key to their universal approximation capability. SCNs incrementally add new units to neural networks under a supervisory mechanism. However, with an increasing number of hidden nodes, the output weights which are evaluated by the least squares method, are frequently unstable. This instability phenomenon (arbitrarily small perturbations leading to large changes in the solutions) can be classified as an ill-posed problem. Instability is one of the most important manifestations of ill-posed problems, and the Tikhonov regularization method, which is widely used to address ill-posed problems, is undesirable. This study examined the supervisory mechanism of SCNs and algebraic properties of the hidden output matrix, thereby proposing a new greedy stochastic configuration network (GSCN) for ill-posed problems. First, under the premise of ensuring universal approximation capability, GSCN tries to optimize the randomly assigned input weights and biases based on the hunter–prey optimization (HPO) algorithm. The optimized hidden parameters which can reduce residual error the most are selected for the newly added hidden nodes. Second, singular value decomposition (SVD) and orthogonal–triangular (QR) decomposition with column pivoting are introduced to extract a linearly independent subset of the hidden output matrix for ill-posed problems in SCNs. Finally, one function approximation, six benchmark regressions, two classification datasets, and bearing fault diagnosis are employed to evaluate the performance of GSCN. Experimental results indicate that GSCN shows superior performance with respect to fast convergence, generalization, and stability compared with other contrast methods. This code is available at https://github.com/Gzu-public-bigdata/GSCN.

Intelligent Discrete Deep Learning Based Classification Methodology in Chemometrics.

In recent years, deep learning models have attracted much attention for classification purposes in chemometrics. The popularity of deep learning models in this field comes from their unique features like universal approximation capability with the desired accuracy. Deep learning classifiers use several intelligent processing layers to model mixed, complex, and nonlinear patterns in the underlying data sets, which is why the development of deep learning based models has never been stopped in the chemometrics literature. Despite the variety of deep learning classification models used in this field, they all use a continuous distance-based cost function in their learning processes. Although using a continuous cost function for learning deep classifiers is a common approach, it conflicts with the discrete nature of the classification problem. In fact, applying a continuous cost function for inherently discrete classification problems can reduce the performance of the classification. In this research, a novel discrete learning based classification approach is proposed and implemented on a deep feed-forward neural network as one of the most commonly used deep learning models to develop a different learning process for deep classification models. The basis of the proposed learning approach is maximizing a discrete matching function of the actual and fitted values instead of minimizing a continuous distance-based cost function. The proposed classification approach is evaluated on five benchmark data sets in the chemistry field. The empirical results indicated the superiority of the proposed discrete deep learning approach over its classic continuous form. The results of this study demonstrate the important effect of discrete learning processes on the performances of deep learning classification models. Therefore, the proposed methodology can be a powerful alternative to common classification approaches to analyze chemical data in the chemometrics field.

Stochastic configuration networks with chaotic maps and hierarchical learning strategy

Stochastic configuration networks (SCNs) have universal approximation capability and fast modeling properties, which have been successfully employed in large-scale data analytics. Based on SCNs, Stochastic configuration networks with block increments (BSC) use the node block increments mechanism to improve training speed but increase the complexity of the model. This paper presents a parallel configuration method (PCM), develops an extension of the original BSC with chaos theory and proposes stochastic configuration networks with chaotic maps (SCNCM), and establishes a hierarchical learning strategy (HLS) to enhance the compactness and construction speed of the model. Firstly, PCM randomly assigns the input weights w and biases b of hidden layer nodes by using uniform and normal distributions. In PCM, an iterative learning algorithm is intended to generate the scope control set and improve configuration efficiency. Secondly, the paper presents two kinds of stochastic configuration networks with chaotic maps, which are SCNCM-I and SCNCM-II. SCNCM-I adjusts block size by using multiple error values and chaotic maps to improve the training speed. Based on SCNCM-I, SCNCM-II utilizes node removal mechanism to enhance the compactness. Finally, HLS integrates with SCNCM-I, SCNCM-II, and the Harris-hawks optimization algorithm (HHO). The purpose of training is to enhance the training speed and compactness for three algorithms. The experiments are conducted on four benchmark data sets and an industrial application shows its effectiveness.

Online Adaptable Trajectory using artificial Neural Networks for Unknown Environment

Walking in unknown environment is complex process. If robots want to move in complex environment, they have to adapt the changes in trajectory due to uncertain circumstances(obstacles and ditches). In this paper, online trajectories are generated for given boundary conditions using neural networks due to it's universal approximation capabilities. Here, NN is used for coefficients calculation which can adapt a change in constraints during tracking. To investigate the idea that suggested trajectory will have continuous velocity and smooth acceleration, the trajectory has been formulated using neural networks for 4, 6, 7 and 9 constraints and compared. A trajectory is trained only in 2.82 seconds for 6 constraints and in 2 minutes 91 seconds for 7 constraints in matlabs on intel Core i5-4200 CPU, 2 Cores, 4 GB RAM laptop. We have tried to find the best trajectory fit in less computation and time. These NN trajectories can adapt the online changes in constraints due to uncertain environment conditions.

Research on intelligent recognition method for self‐blast state of glass insulator based on mixed data augmentation

AbstractAutomatically and accurately detecting the self‐blast state of glass insulators is of great significance to operation and maintenance of transmission lines. To solve the shortcomings of the existing open‐loop cognitive models to detect the self‐blast state of glass insulators, this study explores a mixed data augmentation‐based intelligent recognition method to detect the self‐blast state of the glass insulator, by imitating the human cognitive mode. Firstly, generative adversarial network is utilised to obtain the high‐quality generative self‐blast samples of the glass insulator, and the non‐generative data augmentation techniques is used to obtain rich sample features. Secondly, considering the characteristics of aerial images such as large scale variations, variable shooting angles and complex backgrounds, feature maps with strong semantics and adaptive multi‐scale fusion are extracted using the feature pyramid network with adaptive hierarchy and the multi‐deformable convolutional network. Then, the extracted feature maps are transmitted to a two‐dimensional stochastic configuration network that can adaptively generate hidden nodes and basis functions so as to develop the self‐blast state classification criteria with universal approximation capability. Thirdly, based on the generalised error and entropy theory, the semantic error entropy evaluation indices of recognition results are defined to evaluate in real time, the credibility of the uncertain recognition results for the self‐blast state of the glass insulator. Then, based on transfer learning and the established self‐optimising feedback mechanism for feature pyramid network, the self‐optimising adjustment and reconstruction of the feature map space with strong semantics and multi‐scale fusion and its classification criteria are realised. Finally, the stacking method is applied to integrate the recognition results of the feature pyramid network with adaptive hierarchy and multi‐channel deformable convolutional networks to improve the robustness of the recognition model. Results of experimental comparison with other machine learning and deep learning methods verify the feasibility and effectiveness of the proposed method.

Evolution-Driven Randomized Graph Convolutional Networks

Randomized neural networks (NNs), such as random vector functional link (RVFL) and extreme learning machine (ELM), have been widely applied in various classification problems owing to their computational efficiency and universal approximation capability. However, such approaches are designed for regular Euclidean data and lack the ability to generalize to complex structured data. Moreover, their randomly generated parameters often lead to a suboptimal decision boundary with a growing requirement of hidden neurons. In this article, we first propose a plain framework, termed randomized graph convolutional networks (RGCNs), to generalize the classic randomized NNs to the non-Euclidean domain. Then, a hybrid framework called evolution-driven RGCN (EvoRGCN) is presented by using adaptive differential evolution with novelty search strategy to seek the globally optimal graph embedding for the plain RGCN. Finally, we recast the classic ELM and RVFL under the proposed frameworks, resulting in four novel semi-supervised models, including the plain models [i.e., graph convolutional extreme learning machines (GCELMs) and graph convolutional RVFL (GCRVFL)] and the optimized models (i.e., O-GCELM and O-GCRVFL). We show that our approaches are the natural generalization of the traditional randomized NNs in the non-Euclidean domain. Furthermore, our approaches not only retain the advantages of the classic approaches but also enable them to handle graph data. We compare our approaches against many existing methods across regular datasets and graph benchmarks, demonstrating that the proposed approaches dramatically outperform the compared methods with better generalization ability and robustness. Particularly, we quantitatively show the performance ranking of different randomized NNs, i.e., O-GCRVFL <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$&gt; $ </tex-math></inline-formula> O-GCELM <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\approx $ </tex-math></inline-formula> GCRVFL <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$&gt; $ </tex-math></inline-formula> GCELM <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\approx $ </tex-math></inline-formula> RVFL <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$&gt; $ </tex-math></inline-formula> ELM.

Data Stream Classification Based on Extreme Learning Machine: A Review

Many daily applications are generating massive amount of data in the form of stream at an ever higher speed, such as medical data, clicking stream, internet record and banking transaction, etc. In contrast to the traditional static data, data streams are of some inherent properties, to name a few, infinite length, concept drift, multiple labels and concept evolution. Among all the data mining tasks, classification is one of the basic topics in data stream mining and has gained more and more attentions among different research communities. Extreme Learning Machine (ELM) has drawn much interests in data classification due to its high efficiency, universal approximation capability, generalization ability, and simplicity, which have greatly inspired the development of many ELM-based algorithms and their applications during the past decades. In this paper, we mainly provide a comprehensive review on ELM theoretical research and its variants in data stream classification, and categorize these algorithms from different perspectives. Firstly, we briefly introduce the basic principles of ELM and its characteristics. Secondly, we give an overview of different ELM variants to address the particular issues of data stream classification. Thirdly, we present an overview of different strategies to optimize the ELM, which have further improved the stability, accuracy and generalization ability of ELM, and briefly introduce some practical applications of ELM in data stream classification. Finally, we conduct several groups of experiments to compare the performance of ELM based models addressing the focused issues. Also, the open issues and prospects of ELM models used for stream classification are discussed, which are worthwhile to be further studied in the future.

Adaptive neural finite‐time output‐feedback tracking control for nonlinear stochastic nonstrict‐feedback systems with input saturation

SummaryIn this paper, an observer‐based adaptive neural output‐feedback control scheme is developed for a class of nonlinear stochastic nonstrict‐feedback systems with input saturation in finite‐time interval. The mean value theorem and the property of the smooth function are applied to cope with the difficulties caused by the existence of input saturation. According to the universal approximation capability of the radial basis function neural network, it will be utilized to compensate the unknown nonlinear functions. Based on the state observer, the finite‐time Lyapunov stability theorem, we propose an adaptive neural output‐feedback control scheme for nonlinear stochastic systems in nonstrict‐feedback form. The developed controller guarantees that the system output signal can track the given reference signal trajectory, and all closed‐loop signals are semi‐globally finite‐time stability in probability. The observer errors and the tracking error can converge to a small neighborhood of the origin. Finally, simulation results demonstrate the effectiveness of the developed control scheme.