Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via PINN deep learning

In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential in the deep learning.     

Using deep neural networks to diagnose engine pre-ignition

Engine downsizing and boosting have been recognized as effective strategies for improving engine efficiency. However, operating the engines at high load promotes abnormal combustion events, such as pre-ignition and potential superknock. Currently the most effective method for detecting pre-ignition is by using in-cylinder pressure sensors that have high precision and sensitivity, but also high cost. Due to rapid advances in automotive technology such as autonomous driving, computer-aided designs and future connectivity, we propose to use a complimentary data-driven strategy for diagnosing abnormal combustion events. To this end, a data-driven diagnostics approach for pre-ignition detection with deep neural networks is proposed. The success of convolutional neural networks (CNNs) in object detection and recurrent neural networks (RNNs) in sequence forecasting inspired us to develop these models for pre-ignition detection. For a cost-effective strategy, we use data from less expensive sensors, such as lambda and low-resolution exhaust back pressure (EBP), instead of high resolution in-cylinder pressure measurements. The first deep learning model is combined with a commonly used dimensionality reduction tool–Principal Component Analysis (PCA). The second model eliminates this step and directly processes time-series data. Results indicate that the first model with reduced input dimensions, and correspondingly smaller size of the network, shows better performance in detecting pre-ignition cycles with an F1 score of 79%. Overall, the proposed deep learning approach is a promising alternative for abnormal combustion diagnostics using data from low resolution sensors.     

An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation

The Hirota bilinear method has been studied in a lot of local equations, but there are few of works to solve nonlocal equations by Hirota bilinear method. In this letter, we show that the nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation admits multiple complex soliton solutions. A variety of exact solutions including the single bright soliton solutions and two bright soliton solutions are derived via constructing an improved Hirota bilinear method for nonlocal complex MKdV equation. From the gauge equivalence, we can see the difference between the solution of nonlocal integrable complex MKdV equation and the solution of local complex MKdV equation.     

Data-driven parity-time-symmetric vector rogue wave solutions of multi-component nonlinear Schrödinger equation

Rogue waves are a class of nonlinear waves with extreme amplitudes, which usually appear suddenly and disappear without any trace. Recently, the parity-time ($\mathcal {PT}$)-symmetric vector rogue waves (RWs) of multi-component nonlinear Schrödinger equation ($n$-NLSE) are usually derived by the methods of integrable systems. In this paper, we utilize the multi-stage physics-informed neural networks (MS-PINNs) algorithm to derive the data-driven $\mathcal {PT}$ symmetric vector RWs solution of coupled NLS system in elliptic and X-shapes domains with nonzero boundary condition. The results of the experiment show that the multi-stage physics-informed neural networks are quite feasible and effective for multi-component nonlinear physical systems in the above domains and boundary conditions.     

Soliton,breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

The nonlinear Schro¨dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schro¨dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schro¨dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.     

Data-driven modeling of a four-dimensional stochastic projectile system

The dynamical modeling of projectile systems with sufficient accuracy is of great difficulty due to high-dimensional space and various perturbations. With the rapid development of data science and scientific tools of measurement recently, there are numerous data-driven methods devoted to discovering governing laws from data. In this work, a data-driven method is employed to perform the modeling of the projectile based on the Kramers-Moyal formulas. More specifically, the four-dimensional projectile system is assumed as an Itô stochastic differential equation. Then the least square method and sparse learning are applied to identify the drift coefficient and diffusion matrix from sample path data, which agree well with the real system. The effectiveness of the data-driven method demonstrates that it will become a powerful tool in extracting governing equations and predicting complex dynamical behaviors of the projectile.     

Dynamics of a D’Alembert wave and a soliton molecule for an extended BLMP equation

The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial differential systems.In this paper,we construct a(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli(eBLMP)equation which fails to pass the Painleve property.The D’Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable.The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation.The dynamics of the three-soliton molecule,the three-kink soliton molecule,the soliton molecule bound by an asymmetry soliton and a one-soliton,and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.     

Incorporating prior knowledge via volumetric deep residual network to optimize the reconstruction of sparsely sampled MRI

For sparse sampling that accelerates magnetic resonance (MR) image acquisition, non-linear reconstruction algorithms have been developed, which incorporated patient specific a prior information. More generic a prior information could be acquired via deep learning and utilized for image reconstruction. In this study, we developed a volumetric hierarchical deep residual convolutional neural network, referred to as T-Net, to provide a data-driven end-to-end mapping from sparsely sampled MR images to fully sampled MR images, where cartilage MR images were acquired using an Ultra-short TE sequence and retrospectively undersampled using pseudo-random Cartesian and radial acquisition schemes. The network had a hierarchical architecture that promoted the sparsity of feature maps and increased the receptive field, which were valuable for signal synthesis and artifact suppression. Relatively dense local connections and global shortcuts were established to facilitate residual learning and compensate for details lost in hierarchical processing. Additionally, volumetric processing was adopted to fully exploit spatial continuity in three-dimensional space. Data consistency was further enforced. The network was trained with 336 three-dimensional images (each consisting of 32 slices) and tested by 24 images. The incorporation of a priori information acquired via deep learning facilitated high acceleration factors (as high as 8) while maintaining high image fidelity (quantitatively evaluated using the structural similarity index measurement). The proposed T-Net had an improved performance as compared to several state-of-the-art networks.     

On the investigation of nonlinear waves of the Hirota and the Maxwell–Bloch equation in nonlinear optics

In this paper, considering the Hirota and Maxwell-Bloch (H-MB) equations which is governed by femtosecond pulse propagation through two-level doped fibre system, we construct the Darboux transformation of this system through linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions (both bright and dark breathers) of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.     

Using physics-informed enhanced super-resolution generative adversarial networks for subfilter modeling in turbulent reactive flows

Turbulence is still one of the main challenges in accurate prediction of reactive flows. Therefore, the development of new turbulence closures that can be applied to combustion problems is essential. Over the last few years, data-driven modeling has become popular in many fields as large, often extensively labeled datasets are now available and training of large neural networks has become possible on graphics processing units (GPUs) that speed up the learning process tremendously. However, the successful application of deep neural networks in fluid dynamics, such as in subfilter modeling in the context of large-eddy simulations (LESs), is still challenging. Reasons for this are the large number of degrees of freedom in natural flows, high requirements of accuracy and error robustness, and open questions, for example, regarding the generalization capability of trained neural networks in such high-dimensional, physics-constrained scenarios. This work presents a novel subfilter modeling approach based on a generative adversarial network (GAN), which is trained with unsupervised deep learning (DL) using adversarial and physics-informed losses. A two-step training method is employed to improve the generalization capability, especially extrapolation, of the network. The novel approach gives good results in a priori and a posteriori tests with decaying turbulence including turbulent mixing, and the importance of the physics-informed continuity loss term is demonstrated. The applicability of the network in complex combustion scenarios is furthermore discussed by employing it in reactive and inert LESs of the Spray A case defined by the Engine Combustion Network (ECN).     

A deep learning method for solving high-order nonlinear soliton equations

We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higher-order nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.     

High-order rogue waves for the Hirota equation

The Hirota equation is better than the nonlinear Schrödinger equation when approximating deep ocean waves. In this paper, high-order rational solutions for the Hirota equation are constructed based on the parameterized Darboux transformation. Several types of this kind of solutions are classified by their structures.     

Multi-task learning approach for modulation and wireless signal classification for 5G and beyond: Edge deployment via model compression

Future communication networks must address the scarce spectrum to accommodate extensive growth of heterogeneous wireless devices. Efforts are underway to address spectrum coexistence, enhance spectrum awareness, and bolster authentication schemes. Wireless signal recognition is becoming increasingly more significant for spectrum monitoring, spectrum management, secure communications, among others. Consequently, comprehensive spectrum awareness on the edge has the potential to serve as a key enabler for the emerging beyond 5G (fifth generation) networks. State-of-the-art studies in this domain have (i) only focused on a single task – modulation or signal (protocol) classification – which in many cases is insufficient information for a system to act on, (ii) consider either radar or communication waveforms (homogeneous waveform category), and (iii) does not address edge deployment during neural network design phase. In this work, for the first time in the wireless communication domain, we exploit the potential of deep neural networks based multi-task learning (MTL) framework to simultaneously learn modulation and signal classification tasks while considering heterogeneous wireless signals such as radar and communication waveforms in the electromagnetic spectrum. The proposed MTL architecture benefits from the mutual relation between the two tasks in improving the classification accuracy as well as the learning efficiency with a lightweight neural network model. We additionally include experimental evaluations of the model with over-the-air collected samples and demonstrate first-hand insight on model compression along with deep learning pipeline for deployment on resource-constrained edge devices. We demonstrate significant computational, memory, and accuracy improvement of the proposed model over two reference architectures. In addition to modeling a lightweight MTL model suitable for resource-constrained embedded radio platforms, we provide a comprehensive heterogeneous wireless signals dataset for public use.     

A physics-constrained deep residual network for solving the sine-Gordon equation

Despite some empirical successes for solving nonlinear evolution equations using deep learning,there are several unresolved issues.First,it could not uncover the dynamical behaviors of some equations where highly nonlinear source terms are included very well.Second,the gradient exploding and vanishing problems often occur for the traditional feedforward neural networks.In this paper,we propose a new architecture that combines the deep residual neural network with some underlying physical laws.Using the sine-Gordon equation as an example,we show that the numerical result is in good agreement with the exact soliton solution.In addition,a lot of numerical experiments show that the model is robust under small perturbations to a certain extent.     

深度学习在超声检测缺陷识别中的应用与发展*

深度学习（Deep Learning）是目前最强大的机器学习算法之一,其中卷积神经网络（Convolutional Neural Network, CNN）模型具有自动学习特征的能力,在图像处理领域较其他深度学习模型有较大的性能优势。本文先简述了深度学习的发展史,然后综述了深度学习在超声检测缺陷识别中的应用与发展,从早期浅层神经网络到现在深度学习的应用现状,并借鉴医学影像识别和射线图像识别领域的方法,分析了卷积神经网络对超声图像缺陷识别的适用性。最后,探讨归纳了目前在超声检测图像识别中使用CNN存在的一些问题,及其主要应对策略的研究方向。     

Dynamical analysis of diversity lump solutions to the (2+1)-dimensional dissipative Ablowitz–Kaup–Newell–Segure equation

The lump solution is one of the exact solutions of the nonlinear evolution equation. In this paper, we study the lump solution and lump-type solutions of (2+1)-dimensional dissipative Ablowitz–Kaup–Newell–Segure (AKNS) equation by the Hirota bilinear method and test function method. With the help of Maple, we draw three-dimensional plots of the lump solution and lump-type solutions, and by observing the plots, we analyze the dynamic behavior of the (2+1)-dimensional dissipative AKNS equation. We find that the interaction solutions come in a variety of interesting forms.     

Co-Training for Visual Object Recognition Based on Self-Supervised Models Using a Cross-Entropy Regularization

Automatic recognition of visual objects using a deep learning approach has been successfully applied to multiple areas. However, deep learning techniques require a large amount of labeled data, which is usually expensive to obtain. An alternative is to use semi-supervised models, such as co-training, where multiple complementary views are combined using a small amount of labeled data. A simple way to associate views to visual objects is through the application of a degree of rotation or a type of filter. In this work, we propose a co-training model for visual object recognition using deep neural networks by adding layers of self-supervised neural networks as intermediate inputs to the views, where the views are diversified through the cross-entropy regularization of their outputs. Since the model merges the concepts of co-training and self-supervised learning by considering the differentiation of outputs, we called it Differential Self-Supervised Co-Training (DSSCo-Training). This paper presents some experiments using the DSSCo-Training model to well-known image datasets such as MNIST, CIFAR-100, and SVHN. The results indicate that the proposed model is competitive with the state-of-art models and shows an average relative improvement of 5% in accuracy for several datasets, despite its greater simplicity with respect to more recent approaches.     

Why Does Deep and Cheap Learning Work So Well?

We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of practical interest can frequently be approximated through “cheap learning” with exponentially fewer parameters than generic ones. We explore how properties frequently encountered in physics such as symmetry, locality, compositionality, and polynomial log-probability translate into exceptionally simple neural networks. We further argue that when the statistical process generating the data is of a certain hierarchical form prevalent in physics and machine learning, a deep neural network can be more efficient than a shallow one. We formalize these claims using information theory and discuss the relation to the renormalization group. We prove various “no-flattening theorems” showing when efficient linear deep networks cannot be accurately approximated by shallow ones without efficiency loss; for example, we show that n variables cannot be multiplied using fewer than \(2^n\) neurons in a single hidden layer.     

Line positions,intensities, and Einstein A coefficients for 3-0 band of 12C16O: A spectroscopy learning method

Based on the model- and data-driven strategy, a spectroscopy learning method that can extract the novel and hidden information from the line list databases has been applied to the R branch emission spectra of 3-0 band of the ground electronic state of ¹²C¹⁶O. The labeled line lists such as line intensities and Einstein A coefficients quoted in HITRAN2020 are collected to enhance the dataset. The quantified spectroscopy-learned spectroscopic constants is beneficial for improving the extrapolative accuracy beyond the measurements. Explicit comparisons are made for line positions, line intensities, Einstein A coefficients, which demonstrate that the model- and data-driven spectroscopy learning approach is a promising and an easy-to-implement strategy.     

A deep learning method for solving third-order nonlinear evolution equations

It has still been difficult to solve nonlinear evolution equations analytically. In this paper, we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly. Specifically, the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters. In particular, numerical experiments on several third-order nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation, modified KdV equation, KdV–Burgers equation and Sharma–Tasso–Olver equation, demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.