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Soliton, breather, and rogue wave solutions for solving the nonlinear Schrodinger equation using a deep learning method with physical constraints 被引量:6
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作者 Jun-Cai Pu Jun Li Yong Chen 《Chinese Physics B》 SCIE EI CAS CSCD 2021年第6期77-87,共11页
The nonlinear Schrodinger 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 particu... The nonlinear Schrodinger 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 Schrodinger 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 Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method. 展开更多
关键词 deep learning method neural network soliton solutions breather solution rogue wave solutions
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A deep learning method for solving high-order nonlinear soliton equations 被引量:1
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作者 Shikun Cui Zhen Wang +2 位作者 Jiaqi Han Xinyu Cui Qicheng Meng 《Communications in Theoretical Physics》 SCIE CAS CSCD 2022年第7期57-69,共13页
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 higherorder nonlinear soliton equa... 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 higherorder 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. 展开更多
关键词 deep learning method physics-informed neural networks high-order nonlinear soliton equations interaction between solitons the numerical driven solution
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LexDeep:Hybrid Lexicon and Deep Learning Sentiment Analysis Using Twitter for Unemployment-Related Discussions During COVID-19
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作者 Azlinah Mohamed Zuhaira Muhammad Zain +5 位作者 Hadil Shaiba Nazik Alturki Ghadah Aldehim Sapiah Sakri Saiful Farik Mat Yatin Jasni Mohamad Zain 《Computers, Materials & Continua》 SCIE EI 2023年第4期1577-1601,共25页
The COVID-19 pandemic has spread globally,resulting in financialinstability in many countries and reductions in the per capita grossdomestic product.Sentiment analysis is a cost-effective method for acquiringsentiment... The COVID-19 pandemic has spread globally,resulting in financialinstability in many countries and reductions in the per capita grossdomestic product.Sentiment analysis is a cost-effective method for acquiringsentiments based on household income loss,as expressed on social media.However,limited research has been conducted in this domain using theLexDeep approach.This study aimed to explore social trend analytics usingLexDeep,which is a hybrid sentiment analysis technique,on Twitter to capturethe risk of household income loss during the COVID-19 pandemic.First,tweet data were collected using Twint with relevant keywords before(9 March2019 to 17 March 2020)and during(18 March 2020 to 21 August 2021)thepandemic.Subsequently,the tweets were annotated using VADER(lexiconbased)and fed into deep learning classifiers,and experiments were conductedusing several embeddings,namely simple embedding,Global Vectors,andWord2Vec,to classify the sentiments expressed in the tweets.The performanceof each LexDeep model was evaluated and compared with that of a supportvector machine(SVM).Finally,the unemployment rates before and duringCOVID-19 were analysed to gain insights into the differences in unemploymentpercentages through social media input and analysis.The resultsdemonstrated that all LexDeep models with simple embedding outperformedthe SVM.This confirmed the superiority of the proposed LexDeep modelover a classical machine learning classifier in performing sentiment analysistasks for domain-specific sentiments.In terms of the risk of income loss,the unemployment issue is highly politicised on both the regional and globalscales;thus,if a country cannot combat this issue,the global economy will alsobe affected.Future research should develop a utility maximisation algorithmfor household welfare evaluation,given the percentage risk of income lossowing to COVID-19. 展开更多
关键词 Sentiment analysis sentiment lexicon machine learning imbalanced data deep learning method unemployment rate
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Survey on Deep Learning Approaches for Detection of Email Security Threat
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作者 Mozamel M.Saeed Zaher Al Aghbari 《Computers, Materials & Continua》 SCIE EI 2023年第10期325-348,共24页
Emailing is among the cheapest and most easily accessible platforms,and covers every idea of the present century like banking,personal login database,academic information,invitation,marketing,advertisement,social engi... Emailing is among the cheapest and most easily accessible platforms,and covers every idea of the present century like banking,personal login database,academic information,invitation,marketing,advertisement,social engineering,model creation on cyber-based technologies,etc.The uncontrolled development and easy access to the internet are the reasons for the increased insecurity in email communication.Therefore,this review paper aims to investigate deep learning approaches for detecting the threats associated with e-mail security.This study compiles the literature related to the deep learning methodologies,which are applicable for providing safety in the field of cyber security of email in different organizations.Relevant data were extracted from different research depositories.The paper discusses various solutions for handling these threats.Different challenges and issues are also investigated for e-mail security threats including social engineering,malware,spam,and phishing in the existing solutions to identify the core current problem and set the road for future studies.The review analysis showed that communication media is the common platform for attackers to conduct fraudulent activities via spoofed e-mails and fake websites and this research has combined the merit and demerits of the deep learning approaches adaption in email security threat by the usage of models and technologies.The study highlighted the contrasts of deep learning approaches in detecting email security threats.This review study has set criteria to include studies that deal with at least one of the six machine models in cyber security. 展开更多
关键词 Attackers deep learning methods e-mail security threats machine learning PHISHING
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Spatial-temporal simulation and prediction of root zone soil moisture based on Hydrus-1D and CNN-LSTM-attention models in Yutian Oasis,southern Xinjiang,China
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作者 Xiaobo LÜ Ilyas NURMEMET +4 位作者 Sentian XIAO Jing ZHAO Xinru YU Yilizhati AILI Shiqin LI 《Pedosphere》 2025年第5期846-857,共12页
Root zone soil moisture(RZSM)plays a critical role in land-atmosphere hydrological cycles and serves as the primary water source for vegetation growth.However,the correlations between RZSM and its associated variables... Root zone soil moisture(RZSM)plays a critical role in land-atmosphere hydrological cycles and serves as the primary water source for vegetation growth.However,the correlations between RZSM and its associated variables,including surface soil moisture(SSM),often exhibit nonlinearities that are challenging to identify and quantify using conventional statistical techniques.Therefore,this study presents a hybrid convolutional neural network(CNN)-long short-term memory neural network(LSTM)-attention(CLA)model for predicting RZSM.Owing to the scarcity of soil moisture(SM)observation data,the physical model Hydrus-1D was employed to simulate a comprehensive dataset of spatial-temporal SM.Meteorological data and moderate resolution imaging spectroradiometer vegetation characterization parameters were used as predictor variables for the training and validation of the CLA model.The results of the CLA model for SM prediction in the root zone were significantly enhanced compared with those of the traditional LSTM and CNN-LSTM models.This was particularly notable at the depth of 80–100 cm,where the fitness(R^(2))reached nearly 0.9298.Moreover,the root mean square error of the CLA model was reduced by 49%and 57%compared with those of the LSTM and CNN-LSTM models,respectively.This study demonstrates that the integration of physical modeling and deep learning methods provides a more comprehensive and accurate understanding of spatial-temporal SM variations in the root zone. 展开更多
关键词 arid region convolutional neural network deep learning method hybrid prediction model leaf area index long short-term memory neural network normalized difference vegetation index physical model surface soil moisture
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Deep Learning Applied to Computational Mechanics:A Comprehensive Review,State of the Art,and the Classics 被引量:1
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作者 Loc Vu-Quoc Alexander Humer 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第11期1069-1343,共275页
Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularl... Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularly deep learning(DL),applied and relevant to computational mechanics(solid,fluids,finite-element technology)are reviewed in detail.Both hybrid and pure machine learning(ML)methods are discussed.Hybrid methods combine traditional PDE discretizations with ML methods either(1)to help model complex nonlinear constitutive relations,(2)to nonlinearly reduce the model order for efficient simulation(turbulence),or(3)to accelerate the simulation by predicting certain components in the traditional integration methods.Here,methods(1)and(2)relied on Long-Short-Term Memory(LSTM)architecture,with method(3)relying on convolutional neural networks.Pure ML methods to solve(nonlinear)PDEs are represented by Physics-Informed Neural network(PINN)methods,which could be combined with attention mechanism to address discontinuous solutions.Both LSTM and attention architectures,together with modern and generalized classic optimizers to include stochasticity for DL networks,are extensively reviewed.Kernel machines,including Gaussian processes,are provided to sufficient depth for more advanced works such as shallow networks with infinite width.Not only addressing experts,readers are assumed familiar with computational mechanics,but not with DL,whose concepts and applications are built up from the basics,aiming at bringing first-time learners quickly to the forefront of research.History and limitations of AI are recounted and discussed,with particular attention at pointing out misstatements or misconceptions of the classics,even in well-known references.Positioning and pointing control of a large-deformable beam is given as an example. 展开更多
关键词 deep learning breakthroughs network architectures backpropagation stochastic optimization methods from classic to modern recurrent neural networks long short-term memory gated recurrent unit attention transformer kernel machines Gaussian processes libraries Physics-Informed Neural Networks state-of-the-art history limitations challenges Applications to computational mechanics Finite-element matrix integration improved Gauss quadrature Multiscale geomechanics fluid-filled porous media Fluid mechanics turbulence proper orthogonal decomposition Nonlinear-manifold model-order reduction autoencoder hyper-reduction using gappy data control of large deformable beam
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Highly Accurate and Efficient Deep Learning Paradigm for Full-Atom Protein Loop Modeling with KarmaLoop
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作者 Tianyue Wang Xujun Zhang +13 位作者 Odin Zhang Guangyong Chen Peichen Pan Ercheng Wang Jike Wang Jialu Wu Donghao Zhou Langcheng Wang Ruofan Jin Shicheng Chen Chao Shen Yu Kang Chang-Yu Hsieh Tingjun Hou 《Research》 2025年第1期823-839,共17页
Protein loop modeling is a challenging yet highly nontrivial task in protein structure prediction.Despite recent progress,existing methods including knowledge-based,ab initio,hybrid,and deep learning(DL)methods fall s... Protein loop modeling is a challenging yet highly nontrivial task in protein structure prediction.Despite recent progress,existing methods including knowledge-based,ab initio,hybrid,and deep learning(DL)methods fall substantially short of either atomic accuracy or computational efficiency.To overcome these limitations,we present KarmaLoop,a novel paradigm that distinguishes itself as the first DL method centered on full-atom(encompassing both backbone and side-chain heavy atoms)protein loop modeling.Our results demonstrate that KarmaLoop considerably outperforms conventional and DL-based methods of loop modeling in terms of both accuracy and efficiency,with the average RMSDs of 1.77 and 1.95Åfor the CASP13+14 and CASP15 benchmark datasets,respectively,and manifests at least 2 orders of magnitude speedup in general compared with other methods.Consequently,our comprehensive evaluations indicate that KarmaLoop provides a state-of-the-art DL solution for protein loop modeling,with the potential to hasten the advancement of protein engineering,antibody-antigen recognition,and drug design. 展开更多
关键词 deep learning dl methods accuracy karmaloop deep learning protein loop modeling protein structure predictiondespite full atom modeling
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Efficient People Detection with Infrared Images
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作者 Maria da Conceição Proença 《Journal of Computer and Communications》 2024年第4期31-39,共9页
This work focuses on the problem of monitoring the coastline, which in Portugal’s case means monitoring 3007 kilometers, including 1793 maritime borders with the Atlantic Ocean to the south and west. The human burden... This work focuses on the problem of monitoring the coastline, which in Portugal’s case means monitoring 3007 kilometers, including 1793 maritime borders with the Atlantic Ocean to the south and west. The human burden on the coast becomes a problem, both because erosion makes the cliffs unstable and because pollution increases, making the fragile dune ecosystem difficult to preserve. It is becoming necessary to increase the control of access to beaches, even if it is not a popular measure for internal and external tourism. The methodology described can also be used to monitor maritime borders. The use of images acquired in the infrared range guarantees active surveillance both day and night, the main objective being to mimic the infrared cameras already installed in some critical areas along the coastline. Using a series of infrared photographs taken at low angles with a modified camera and appropriate filter, a recent deep learning algorithm with the right training can simultaneously detect and count whole people at close range and people almost completely submerged in the water, including partially visible targets, achieving a performance with F1 score of 0.945, with 97% of targets correctly identified. This implementation is possible with ordinary laptop computers and could contribute to more frequent and more extensive coverage in beach/border surveillance, using infrared cameras at regular intervals. It can be partially automated to send alerts to the authorities and/or the nearest lifeguards, thus increasing monitoring without relying on human resources. 展开更多
关键词 Beach Overload People Counting Border Control People Detection deep learning methods Remote Surveillance
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Text Classification Techniques:A Holistic Review,Observational Analysis,and Experimental Investigation
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作者 Kamal Taha Paul D.Yoo +1 位作者 Chan Yeun Aya Taha 《Big Data Mining and Analytics》 2025年第3期624-660,共37页
This review article provides a thorough assessment of modern and innovative algorithms for text classification through both observational and experimental evaluations.We propose a new classification system,grounded in... This review article provides a thorough assessment of modern and innovative algorithms for text classification through both observational and experimental evaluations.We propose a new classification system,grounded in methodology,to categorize text classification algorithms into an organized structure from general categories down to particular fine-grained techniques.Drawing on more than 100 academic papers from prominent publishers,our extensive review spans a wide range of algorithms,encompassing traditional,deep learning,and emerging approaches.Through observational studies and comparative experiments among various algorithms,techniques,and methodological categories,we offer detailed insights into the area of text classification.The goal of this survey is to assist scholars in choosing the right methods for specific projects while encouraging further advancements in this area.This detailed examination not only contributes to the scholarly conversation on text classification but also seeks to direct future progress by identifying promising avenues for innovation and enhancement.The primary contributions of this article include the sophisticated methodological classification,a thorough review and examination of state-of-the-art algorithms,along with observational and experimental assessments,and a visionary outlook on the future development of text classification methods. 展开更多
关键词 text classification classical methods deep learning methods experimental evaluation
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State of charge estimation of lithium-ion battery based on state of temperature estimation using weight clustered-convolutional neural network-long short-term memory
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作者 Chaoran Li Sichen Zhu +5 位作者 Liuli Zhang Xinjian Liu Menghan Li Haiqin Zhou Qiang Zhang Zhonghao Rao 《Green Energy and Intelligent Transportation》 2025年第1期31-43,共13页
State of charge(SOC)plays a vital role in the safe,efficient,and stable operation of lithium-ion batteries.Since the difference between the surface temperature and core temperature of batteries under severe conditions... State of charge(SOC)plays a vital role in the safe,efficient,and stable operation of lithium-ion batteries.Since the difference between the surface temperature and core temperature of batteries under severe conditions can reach 5–10 C,using the surface temperature as input feature of SOC estimation is unreasonable.Due to the high requirement for storage space,SOC estimation methods based on deep learning methods are limited to implement in embedded devices.In this paper,to achieve reasonable and high accuracy SOC estimation and provide support for battery thermal management,SOC estimation based on state of temperature(SOT)is implemented.And weight clustered-convolutional neural network-long short-term memory(WC-CNN-LSTM)is proposed to achieve high accuracy SOT and SOC estimation with small model sizes.A self-established dataset is used to verify the effectiveness of the proposed method and model.The WC-CNN-LSTM model with the number of clusters of 400 could achieve comparative accuracy with the baseline model with a 52.98%smaller model size and 25.08%more time consumption for model training on SOT estimation.And it could also achieve consistent and even better accuracy on SOC estimation with the baseline model with a small model size. 展开更多
关键词 State of charge State of temperature Lithium-ion battery deep learning method
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A CGAN Framework for Predicting Crack Patterns and Stress-Strain Behavior in Concrete Random Media
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作者 Xing Lin Junning Wu Shixue Liang 《Computer Modeling in Engineering & Sciences》 2025年第10期215-239,共25页
Random media like concrete and ceramics exhibit stochastic crack propagation due to their heterogeneous microstructures.This study establishes a Conditional Generative Adversarial Network(CGAN)combined with randomfiel... Random media like concrete and ceramics exhibit stochastic crack propagation due to their heterogeneous microstructures.This study establishes a Conditional Generative Adversarial Network(CGAN)combined with randomfieldmodeling for the efficient prediction of stochastic crack patterns and stress-strain responses.Atotal dataset of 500 samples,including crack propagation images and corresponding stress-strain curves,is generated via random Finite Element Method(FEM)simulations.This dataset is then partitioned into 400 training and 100 testing samples.Themodel demonstrates robust performance with Intersection overUnion(IoU)scores of 0.8438 and 0.8155 on training and testing datasets,and R2 values of 0.9584 and 0.9462 in stress-strain curve predictions.By using these results,the CGAN is subsequently implemented as a surrogate model for large-scale Monte Carlo(MC)Simulations to capture the key statistical characteristics such as crack density and spatial distribution.Compared to conventional FEM-based methods,this approach reduces the computational cost to about 1/250 while maintaining high prediction accuracy.The methodology establishes a viable pathway for probabilistic fracture analysis in quasi-brittle materials,balancing computational efficiency with physical fidelity in capturing material stochasticity. 展开更多
关键词 Random media concrete stochastic crack patterns random field deep learning method surrogate model
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Number of Solitons Emerged in the Initial Profile of Shallow Water Using Convolutional Neural Networks 被引量:1
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作者 WANG Zhen CUI Shikun 《Journal of Systems Science & Complexity》 SCIE EI CSCD 2024年第2期463-479,共17页
The soliton resolution conjecture proposes that the initial value problem can evolve into a dispersion part and a soliton part.However,the problem of determining the number of solitons that form in a given initial pro... The soliton resolution conjecture proposes that the initial value problem can evolve into a dispersion part and a soliton part.However,the problem of determining the number of solitons that form in a given initial profile remains unsolved,except for a few specific cases.In this paper,the authors use the deep learning method to predict the number of solitons in a given initial value of the Korteweg-de Vries(KdV)equation.By leveraging the analytical relationship between Asech^(2)(x)initial values and the number of solitons,the authors train a Convolutional Neural Network(CNN)that can accurately identify the soliton count from spatio-temporal data.The trained neural network is capable of predicting the number of solitons with other given initial values without any additional assistance.Through extensive calculations,the authors demonstrate the effectiveness and high performance of the proposed method. 展开更多
关键词 Convolutional neural network deep learning method KdV equation SOLITON
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MTSC-Net:A Semi-Supervised Counting Network for Estimating the Number of Slash pine New Shoots
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作者 Zhaoxu Zhang Yanjie Li +3 位作者 Yue Cao Yu Wang Xuchao Guo Xia Hao 《Plant Phenomics》 CSCD 2024年第4期953-966,共14页
The new shoot density of slash pine serves as a vital indicator for assessing its growth and photosynthetic capacity,while the number of new shoots offers an intuitive reflection of this density.With deep learning met... The new shoot density of slash pine serves as a vital indicator for assessing its growth and photosynthetic capacity,while the number of new shoots offers an intuitive reflection of this density.With deep learning methods becoming increasingly popular,automated counting of new shoots has greatly improved in recent years but is still limited by tedious and expensive data collection and labeling.To resolve these issues,this paper proposes a semi-supervised counting network(MTSC-Net)for estimating the number of slash pine new shoots.First,based on the mean-teacher framework,we introduce the improved VGG19 to extract multiscale new shoot features.Second,to connect local new shoot feature information with global channel features,attention feature fusion module is introduced to achieve effective feature fusion.Finally,the new shoot density map and density probability distribution are processed in a fine-grained manner through multiscale dilated convolution of the regression head and classification head.In addition,a masked image modeling strategy is introduced to encourage the contextual understanding of global new shoot features and improve the counting performance.The experimental results show that MTSC-Net outperforms other semi-supervised counting models with labeled percentages ranging from 5%to 50%.When the labeled percentage is 5%,the mean absolute error and root mean square error are 17.71 and 25.49,respectively.These findings demonstrate that our work can be used as an efficient semi-supervised counting method to provide automated support for tree breeding and genetic utilization. 展开更多
关键词 mean teacher framework deep learning methods new shoots slash pine multiscale dilated convolution new shoot density semi supervised counting attention feature fusion
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Intelligent algorithms: new avenues for designing nanophotonic devices [Invited] 被引量:5
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作者 Lifeng Ma Jing Li +4 位作者 Zhouhui Liu Yuxuan Zhang Nianen Zhang Shuqiao Zheng Cuicui Lu 《Chinese Optics Letters》 SCIE EI CAS CSCD 2021年第1期31-63,共33页
The research on nanophotonic devices has made great progress during the past decades. It is the unremitting pursuit of researchers that realize various device functions to meet practical applications. However, most of... The research on nanophotonic devices has made great progress during the past decades. It is the unremitting pursuit of researchers that realize various device functions to meet practical applications. However, most of the traditional methods rely on human experience and physical inspiration for structural design and parameter optimization, which usually require a lot of resources, and the performance of the designed device is limited. Intelligent algorithms, which are composed of rich optimized algorithms, show a vigorous development trend in the field of nanophotonic devices in recent years. The design of nanophotonic devices by intelligent algorithms can break the restrictions of traditional methods and predict novel configurations, which is universal and efficient for different materials, different structures, different modes, different wavelengths, etc. In this review, intelligent algorithms for designing nanophotonic devices are introduced from their concepts to their applications, including deep learning methods, the gradient-based inverse design method, swarm intelligence algorithms, individual inspired algorithms, and some other algorithms. The design principle based on intelligent algorithms and the design of typical new nanophotonic devices are reviewed. Intelligent algorithms can play an important role in designing complex functions and improving the performances of nanophotonic devices, which provide new avenues for the realization of photonic chips. 展开更多
关键词 intelligent algorithms nanophotonic devices deep learning methods swarm intelligence genetic algorithm
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A Variational Neural Network Approach for Glacier Modelling with Nonlinear Rheology
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作者 Tiangang Cui Zhongjian Wang Zhiwen Zhang 《Communications in Computational Physics》 SCIE 2023年第9期934-954,共21页
We propose a mesh-free method to solve the full Stokes equation for modeling the glacier dynamics with nonlinear rheology.Inspired by the Deep-Ritz method proposed in[13],we first formulate the solution to the non-New... We propose a mesh-free method to solve the full Stokes equation for modeling the glacier dynamics with nonlinear rheology.Inspired by the Deep-Ritz method proposed in[13],we first formulate the solution to the non-Newtonian Stokes equation as the minimizer of a variational problem with boundary constraints.Then,we approximate its solution space by a deep neural network.The loss function for training the neural network is a relaxed version of the variational form,in which penalty terms are used to present soft constraints due to mixed boundary conditions.Instead of introducing mesh grids or basis functions to evaluate the loss function,our method only requires uniform sampling from the physical domain and boundaries.Furthermore,we introduce a re-normalization technique in the neural network to address the significant variation in the scaling of real-world problems.Finally,we illustrate the performance of our method by several numerical experiments,including a 2D model with the analytical solution,the Arolla glacier model with realistic scaling and a 3D model with periodic boundary conditions.Numerical results show that our proposed method is efficient in solving the non-Newtonian mechanics arising from glacier modeling with nonlinear rheology. 展开更多
关键词 deep learning method variational problems mesh-free method non-Newtonian mechanics nonlinear rheology glacier modelling
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