In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the sol...In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the soliton selffrequency shift is remarkable and the photonic crystal fibre （PCF） parameters vary with the frequency considerably. The precision of numerical simulation by using ASSFM is higher than that by using split-step Fourier method in the process of laser pulse propagation in PCFs due to the fact that the variation of fibre parameters with the peak frequency in the pulse spectrum can be taken into account fully.展开更多
Full-color imaging is essential in digital pathology for accurate tissue analysis.Utilizing advanced optical modulation and phase retrieval algorithms,Fourier ptychographic microscopy(FPM)offers a powerful solution fo...Full-color imaging is essential in digital pathology for accurate tissue analysis.Utilizing advanced optical modulation and phase retrieval algorithms,Fourier ptychographic microscopy(FPM)offers a powerful solution for high-throughput digital pathology,combining high resolution,large field of view,and extended depth of field(DOF).However,the full-color capabilities of FPM are hindered by coherent color artifacts and reduced computational efficiency,which significantly limits its practical applications.Color-transferbased FPM(CFPM)has emerged as a potential solution,theoretically reducing both acquisition and reconstruction threefold time.Yet,existing methods fall short of achieving the desired reconstruction speed and colorization quality.In this study,we report a generalized dual-color-space constrained model for FPM colorization.This model provides a mathematical framework for model-based FPM colorization,enabling a closed-form solution without the need for redundant iterative calculations.Our approach,termed generalized CFPM(gCFPM),achieves colorization within seconds for megapixel-scale images,delivering superior colorization quality in terms of both colorfulness and sharpness,along with an extended DOF.Both simulations and experiments demonstrate that gCFPM surpasses state-of-the-art methods across all evaluated criteria.Our work offers a robust and comprehensive workflow for high-throughput full-color pathological imaging using FPM platforms,laying a solid foundation for future advancements in methodology and engineering.展开更多
Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)at...Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.展开更多
Fourier Ptychographic Microscopy(FPM)is a high-throughput computational optical imaging technology reported in 2013.It effectively breaks through the trade-off between high-resolution imaging and wide-field imaging.In...Fourier Ptychographic Microscopy(FPM)is a high-throughput computational optical imaging technology reported in 2013.It effectively breaks through the trade-off between high-resolution imaging and wide-field imaging.In recent years,it has been found that FPM is not only a tool to break through the trade-off between field of view and spatial resolution,but also a paradigm to break through those trade-off problems,thus attracting extensive attention.Compared with previous reviews,this review does not introduce its concept,basic principles,optical system and series of applications once again,but focuses on elaborating the three major difficulties faced by FPM technology in the process from“looking good”in the laboratory to“working well”in practical applications:mismatch between numerical model and physical reality,long reconstruction time and high computing power demand,and lack of multi-modal expansion.It introduces how to achieve key technological innovations in FPM through the dual drive of Artificial Intelligence(AI)and physics,including intelligent reconstruction algorithms introducing machine learning concepts,optical-algorithm co-design,fusion of frequency domain extrapolation methods and generative adversarial networks,multi-modal imaging schemes and data fusion enhancement,etc.,gradually solving the difficulties of FPM technology.Conversely,this review deeply considers the unique value of FPM technology in potentially feeding back to the development of“AI+optics”,such as providing AI benchmark tests under physical constraints,inspirations for the balance of computing power and bandwidth in miniaturized intelligent microscopes,and photoelectric hybrid architectures.Finally,it introduces the industrialization path and frontier directions of FPM technology,pointing out that with the promotion of the dual drive of AI and physics,it will generate a large number of industrial application case,and looks forward to the possibilities of future application scenarios and expansions,for instance,body fluid biopsy and point-of-care testing at the grassroots level represent the expansion of the growth market.展开更多
In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both...In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.展开更多
We present the Fourier lightfield multiview stereoscope(FiLM-Scope).This imaging device combines concepts from Fourier lightfield microscopy and multiview stereo imaging to capture high-resolution 3D videos over large...We present the Fourier lightfield multiview stereoscope(FiLM-Scope).This imaging device combines concepts from Fourier lightfield microscopy and multiview stereo imaging to capture high-resolution 3D videos over large fields of view.The FiLM-Scope optical hardware consists of a multicamera array,with 48 individual microcameras,placed behind a high-throughput primary lens.This allows the FiLM-Scope to simultaneously capture 48 unique 12.8 megapixel images of a 28×37 mm field-of-view,from unique angular perspectives over a 21 deg×29 deg range,with down to 22μm lateral resolution.We also describe a self-supervised algorithm to reconstruct 3D height maps from these images.Our approach demonstrates height accuracy down to 11μm.To showcase the utility of our system,we perform tool tracking over the surface of an ex vivo rat skull and visualize the 3D deformation in stretching human skin,with videos captured at up to 100 frames per second.The FiLM-Scope has the potential to improve 3D visualization in a range of microsurgical settings.展开更多
In oceanic and atmospheric science,finer resolutions have become a prevailing trend in all aspects of development.For high-resolution fluid flow simulations,the computational costs of widely used numerical models incr...In oceanic and atmospheric science,finer resolutions have become a prevailing trend in all aspects of development.For high-resolution fluid flow simulations,the computational costs of widely used numerical models increase significantly with the resolution.Artificial intelligence methods have attracted increasing attention because of their high precision and fast computing speeds compared with traditional numerical model methods.The resolution-independent Fourier neural operator(FNO)presents a promising solution to the still challenging problem of high-resolution fluid flow simulations based on low-resolution data.Accordingly,we assess the potential of FNO for high-resolution fluid flow simulations using the vorticity equation as an example.We assess and compare the performance of FNO in multiple high-resolution tests varying the amounts of data and the evolution durations.When assessed with finer resolution data(even up to number of grid points with 1280×1280),the FNO model,trained at low resolution(number of grid points with 64×64)and with limited data,exhibits a stable overall error and good accuracy.Additionally,our work demonstrates that the FNO model takes less time than the traditional numerical method for high-resolution simulations.This suggests that FNO has the prospect of becoming a cost-effective and highly precise model for high-resolution simulations in the future.Moreover,FNO can make longer high-resolution predictions while training with less data by superimposing vorticity fields from previous time steps as input.A suitable initial learning rate can be set according to the frequency principle,and the time intervals of the dataset need to be adjusted according to the spatial resolution of the input when training the FNO model.Our findings can help optimize FNO for future fluid flow simulations.展开更多
Quantum information masking(QIM)is a crucial technique for protecting quantum data from being accessed by local subsystems.In this paper,we introduce a novel method for achieving1-uniform QIM in multipartite systems u...Quantum information masking(QIM)is a crucial technique for protecting quantum data from being accessed by local subsystems.In this paper,we introduce a novel method for achieving1-uniform QIM in multipartite systems utilizing a Fourier matrix.We further extend this approach to construct an orthogonal array with the aid of a Hadamard matrix,which is a specific type of Fourier matrix.This allows us to explore the relationship between 2-uniform QIM and orthogonal arrays.Through this framework,we derive two distinct 2-uniform quantum states,enabling the 2-uniform masking of original information within multipartite systems.Furthermore,we prove that the maximum number of quantum bits required for achieving a2-uniformly masked state is 2^(n)-1,and the minimum is 2^(n-1)+3.Moreover,our scheme effectively demonstrates the rich quantum correlations between multipartite systems and has potential application value in quantum secret sharing.展开更多
The aim of this paper is to price power option with its underlying asset price following exponential normal inverse gaussian(NIG)process.We first find the risk neutral equivalent martingale measure Q by Esscher transf...The aim of this paper is to price power option with its underlying asset price following exponential normal inverse gaussian(NIG)process.We first find the risk neutral equivalent martingale measure Q by Esscher transform.Then,using the Fourier transform and its inverse,we derive the analytical pricing formulas of power options which are expressed in the form of Fourier integral.In addition,the fast Fourier transform(FFT)algorithm is applied to calculate these pricing formulas.Finally,Shangzheng 50ETF options are chosen to test our results.Estimating the parameters in NIG process by maximum likelihood method,we show that the NIG prices are much closer to market prices than the Black-Scholes-Merton(BSM)ones.展开更多
The capacity to diagnose faults in rolling bearings is of significant practical importance to ensure the normal operation of the equipment.Frequency-domain features can effectively enhance the identification of fault ...The capacity to diagnose faults in rolling bearings is of significant practical importance to ensure the normal operation of the equipment.Frequency-domain features can effectively enhance the identification of fault modes.However,existing methods often suffer from insufficient frequency-domain representation in practical applications,which greatly affects diagnostic performance.Therefore,this paper proposes a rolling bearing fault diagnosismethod based on aMulti-Scale FusionNetwork(MSFN)using the Time-Division Fourier Transform(TDFT).The method constructs multi-scale channels to extract time-domain and frequency-domain features of the signal in parallel.A multi-level,multi-scale filter-based approach is designed to extract frequency-domain features in a segmented manner.A cross-attention mechanism is introduced to facilitate the fusion of the extracted time-frequency domain features.The performance of the proposed method is validated using the CWRU and Ottawa datasets.The results show that the average accuracy of MSFN under complex noisy signals is 97.75%and 94.41%.The average accuracy under variable load conditions is 98.68%.This demonstrates its significant application potential compared to existing methods.展开更多
With the growing demand for high-precision flow field simulations in computational science and engineering,the super-resolution reconstruction of physical fields has attracted considerable research interest.However,tr...With the growing demand for high-precision flow field simulations in computational science and engineering,the super-resolution reconstruction of physical fields has attracted considerable research interest.However,tradi-tional numerical methods often entail high computational costs,involve complex data processing,and struggle to capture fine-scale high-frequency details.To address these challenges,we propose an innovative super-resolution reconstruction framework that integrates a Fourier neural operator(FNO)with an enhanced diffusion model.The framework employs an adaptively weighted FNO to process low-resolution flow field inputs,effectively capturing global dependencies and high-frequency features.Furthermore,a residual-guided diffusion model is introduced to further improve reconstruction performance.This model uses a Markov chain for noise injection in phys-ical fields and integrates a reverse denoising procedure,efficiently solved by an adaptive time-step ordinary differential equation solver,thereby ensuring both stability and computational efficiency.Experimental results demonstrate that the proposed framework significantly outperforms existing methods in terms of accuracy and efficiency,offering a promising solution for fine-grained data reconstruction in scientific simulations.展开更多
文摘In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the soliton selffrequency shift is remarkable and the photonic crystal fibre （PCF） parameters vary with the frequency considerably. The precision of numerical simulation by using ASSFM is higher than that by using split-step Fourier method in the process of laser pulse propagation in PCFs due to the fact that the variation of fibre parameters with the peak frequency in the pulse spectrum can be taken into account fully.
基金supported by the National Natural Science Foundation of China(Grant Nos.12104500 and 82430062)the Key Research and Development Projects of Shaanxi Province(Grant No.2023-YBSF-263),the Shenzhen Engineering Research Centre(Grant No.XMHT20230115004)the Shenzhen Science and Technology Innovation Commission(Grant No.KCXFZ20201221173207022).
文摘Full-color imaging is essential in digital pathology for accurate tissue analysis.Utilizing advanced optical modulation and phase retrieval algorithms,Fourier ptychographic microscopy(FPM)offers a powerful solution for high-throughput digital pathology,combining high resolution,large field of view,and extended depth of field(DOF).However,the full-color capabilities of FPM are hindered by coherent color artifacts and reduced computational efficiency,which significantly limits its practical applications.Color-transferbased FPM(CFPM)has emerged as a potential solution,theoretically reducing both acquisition and reconstruction threefold time.Yet,existing methods fall short of achieving the desired reconstruction speed and colorization quality.In this study,we report a generalized dual-color-space constrained model for FPM colorization.This model provides a mathematical framework for model-based FPM colorization,enabling a closed-form solution without the need for redundant iterative calculations.Our approach,termed generalized CFPM(gCFPM),achieves colorization within seconds for megapixel-scale images,delivering superior colorization quality in terms of both colorfulness and sharpness,along with an extended DOF.Both simulations and experiments demonstrate that gCFPM surpasses state-of-the-art methods across all evaluated criteria.Our work offers a robust and comprehensive workflow for high-throughput full-color pathological imaging using FPM platforms,laying a solid foundation for future advancements in methodology and engineering.
文摘Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.
基金National Natural Science Foundation of China(No.12574332)the Space Optoelectronic Measurement and Perception Lab.,Beijing Institute of Control Engineering(No.LabSOMP-2023-10)Major Science and Technology Innovation Program of Xianyang City(No.L2024-ZDKJ-ZDCGZH-0021)。
文摘Fourier Ptychographic Microscopy(FPM)is a high-throughput computational optical imaging technology reported in 2013.It effectively breaks through the trade-off between high-resolution imaging and wide-field imaging.In recent years,it has been found that FPM is not only a tool to break through the trade-off between field of view and spatial resolution,but also a paradigm to break through those trade-off problems,thus attracting extensive attention.Compared with previous reviews,this review does not introduce its concept,basic principles,optical system and series of applications once again,but focuses on elaborating the three major difficulties faced by FPM technology in the process from“looking good”in the laboratory to“working well”in practical applications:mismatch between numerical model and physical reality,long reconstruction time and high computing power demand,and lack of multi-modal expansion.It introduces how to achieve key technological innovations in FPM through the dual drive of Artificial Intelligence(AI)and physics,including intelligent reconstruction algorithms introducing machine learning concepts,optical-algorithm co-design,fusion of frequency domain extrapolation methods and generative adversarial networks,multi-modal imaging schemes and data fusion enhancement,etc.,gradually solving the difficulties of FPM technology.Conversely,this review deeply considers the unique value of FPM technology in potentially feeding back to the development of“AI+optics”,such as providing AI benchmark tests under physical constraints,inspirations for the balance of computing power and bandwidth in miniaturized intelligent microscopes,and photoelectric hybrid architectures.Finally,it introduces the industrialization path and frontier directions of FPM technology,pointing out that with the promotion of the dual drive of AI and physics,it will generate a large number of industrial application case,and looks forward to the possibilities of future application scenarios and expansions,for instance,body fluid biopsy and point-of-care testing at the grassroots level represent the expansion of the growth market.
基金Supported by the National Natural Science Foundation of China (Grant No. 12301521)the Natural Science Foundation of Shanxi Province (Grant No. 20210302124081)。
文摘In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.
基金supported by the National Cancer Institute(NCI)of the National Institutes of Health(Grant No.R44CA250877)the Office of Research Infrastructure Programs(ORIP),Office of the Director,National Institutes of Health,and the National Institute of Environmental Health Sciences(NIEHS)of the National Institutes of Health(Grant No.R44OD024879)+2 种基金the National Institute of Biomedical Imaging and Bioengineering(NIBIB)of the National Institutes of Health(Grant No.R43EB030979)the National Science Foundation(Grant Nos.2036439 and 2238845)the Duke Coulter Translational Part-nership Award,the Fitzpatrick Institute at the Duke University.
文摘We present the Fourier lightfield multiview stereoscope(FiLM-Scope).This imaging device combines concepts from Fourier lightfield microscopy and multiview stereo imaging to capture high-resolution 3D videos over large fields of view.The FiLM-Scope optical hardware consists of a multicamera array,with 48 individual microcameras,placed behind a high-throughput primary lens.This allows the FiLM-Scope to simultaneously capture 48 unique 12.8 megapixel images of a 28×37 mm field-of-view,from unique angular perspectives over a 21 deg×29 deg range,with down to 22μm lateral resolution.We also describe a self-supervised algorithm to reconstruct 3D height maps from these images.Our approach demonstrates height accuracy down to 11μm.To showcase the utility of our system,we perform tool tracking over the surface of an ex vivo rat skull and visualize the 3D deformation in stretching human skin,with videos captured at up to 100 frames per second.The FiLM-Scope has the potential to improve 3D visualization in a range of microsurgical settings.
基金The National Natural Science Foundation of China under contract No.42425606the Basic Scientific Fund for the National Public Research Institute of China(Shu-Xingbei Young Talent Program)under contract No.2023S01+1 种基金the Ocean Decade International Cooperation Center Scientific and Technological Cooperation Project under contract No.GHKJ2024005China-Korea Joint Ocean Research Center Project under contract Nos PI-20240101(China)and 20220407(Korea).
文摘In oceanic and atmospheric science,finer resolutions have become a prevailing trend in all aspects of development.For high-resolution fluid flow simulations,the computational costs of widely used numerical models increase significantly with the resolution.Artificial intelligence methods have attracted increasing attention because of their high precision and fast computing speeds compared with traditional numerical model methods.The resolution-independent Fourier neural operator(FNO)presents a promising solution to the still challenging problem of high-resolution fluid flow simulations based on low-resolution data.Accordingly,we assess the potential of FNO for high-resolution fluid flow simulations using the vorticity equation as an example.We assess and compare the performance of FNO in multiple high-resolution tests varying the amounts of data and the evolution durations.When assessed with finer resolution data(even up to number of grid points with 1280×1280),the FNO model,trained at low resolution(number of grid points with 64×64)and with limited data,exhibits a stable overall error and good accuracy.Additionally,our work demonstrates that the FNO model takes less time than the traditional numerical method for high-resolution simulations.This suggests that FNO has the prospect of becoming a cost-effective and highly precise model for high-resolution simulations in the future.Moreover,FNO can make longer high-resolution predictions while training with less data by superimposing vorticity fields from previous time steps as input.A suitable initial learning rate can be set according to the frequency principle,and the time intervals of the dataset need to be adjusted according to the spatial resolution of the input when training the FNO model.Our findings can help optimize FNO for future fluid flow simulations.
基金supported by the National Natural Science Foundation of China under Grant No.12301590Natural Science Foundation of Hebei Province under Grant No.A2022210002。
文摘Quantum information masking(QIM)is a crucial technique for protecting quantum data from being accessed by local subsystems.In this paper,we introduce a novel method for achieving1-uniform QIM in multipartite systems utilizing a Fourier matrix.We further extend this approach to construct an orthogonal array with the aid of a Hadamard matrix,which is a specific type of Fourier matrix.This allows us to explore the relationship between 2-uniform QIM and orthogonal arrays.Through this framework,we derive two distinct 2-uniform quantum states,enabling the 2-uniform masking of original information within multipartite systems.Furthermore,we prove that the maximum number of quantum bits required for achieving a2-uniformly masked state is 2^(n)-1,and the minimum is 2^(n-1)+3.Moreover,our scheme effectively demonstrates the rich quantum correlations between multipartite systems and has potential application value in quantum secret sharing.
基金Supported by National Natural Science Foundation of China(11571089,11501164)Natural Science Founda-tion of Hebei Province(A2019205299)+1 种基金the Foundation of Hebei Education Department(ZD2018065,ZD2019053)Hebei Normal University(L2019Z01).
文摘The aim of this paper is to price power option with its underlying asset price following exponential normal inverse gaussian(NIG)process.We first find the risk neutral equivalent martingale measure Q by Esscher transform.Then,using the Fourier transform and its inverse,we derive the analytical pricing formulas of power options which are expressed in the form of Fourier integral.In addition,the fast Fourier transform(FFT)algorithm is applied to calculate these pricing formulas.Finally,Shangzheng 50ETF options are chosen to test our results.Estimating the parameters in NIG process by maximum likelihood method,we show that the NIG prices are much closer to market prices than the Black-Scholes-Merton(BSM)ones.
基金fully supported by the Frontier Exploration Projects of Longmen Laboratory(No.LMQYTSKT034)Key Research and Development and Promotion of Special(Science and Technology)Project of Henan Province,China(No.252102210158)。
文摘The capacity to diagnose faults in rolling bearings is of significant practical importance to ensure the normal operation of the equipment.Frequency-domain features can effectively enhance the identification of fault modes.However,existing methods often suffer from insufficient frequency-domain representation in practical applications,which greatly affects diagnostic performance.Therefore,this paper proposes a rolling bearing fault diagnosismethod based on aMulti-Scale FusionNetwork(MSFN)using the Time-Division Fourier Transform(TDFT).The method constructs multi-scale channels to extract time-domain and frequency-domain features of the signal in parallel.A multi-level,multi-scale filter-based approach is designed to extract frequency-domain features in a segmented manner.A cross-attention mechanism is introduced to facilitate the fusion of the extracted time-frequency domain features.The performance of the proposed method is validated using the CWRU and Ottawa datasets.The results show that the average accuracy of MSFN under complex noisy signals is 97.75%and 94.41%.The average accuracy under variable load conditions is 98.68%.This demonstrates its significant application potential compared to existing methods.
基金supported by the National Natural Science Foundation of China(Grant Nos.42005003 and 41475094)National Key R&D Program of China(Grant No.2018YFC1506704).
文摘With the growing demand for high-precision flow field simulations in computational science and engineering,the super-resolution reconstruction of physical fields has attracted considerable research interest.However,tradi-tional numerical methods often entail high computational costs,involve complex data processing,and struggle to capture fine-scale high-frequency details.To address these challenges,we propose an innovative super-resolution reconstruction framework that integrates a Fourier neural operator(FNO)with an enhanced diffusion model.The framework employs an adaptively weighted FNO to process low-resolution flow field inputs,effectively capturing global dependencies and high-frequency features.Furthermore,a residual-guided diffusion model is introduced to further improve reconstruction performance.This model uses a Markov chain for noise injection in phys-ical fields and integrates a reverse denoising procedure,efficiently solved by an adaptive time-step ordinary differential equation solver,thereby ensuring both stability and computational efficiency.Experimental results demonstrate that the proposed framework significantly outperforms existing methods in terms of accuracy and efficiency,offering a promising solution for fine-grained data reconstruction in scientific simulations.
