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
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
Fourier neural operator(FNO)model is developed for large eddy simulation(LES)of three-dimensional(3D)turbulence.Velocity fields of isotropic turbulence generated by direct numerical simulation(DNS)are used for trainin...Fourier neural operator(FNO)model is developed for large eddy simulation(LES)of three-dimensional(3D)turbulence.Velocity fields of isotropic turbulence generated by direct numerical simulation(DNS)are used for training the FNO model to predict the filtered velocity field at a given time.The input of the FNO model is the filtered velocity fields at the previous several time-nodes with large time lag.In the a posteriori study of LES,the FNO model performs better than the dynamic Smagorinsky model(DSM)and the dynamic mixed model(DMM)in the prediction of the velocity spectrum,probability density functions(PDFs)of vorticity and velocity increments,and the instantaneous flow structures.Moreover,the proposed model can significantly reduce the computational cost,and can be well generalized to LES of turbulence at higher Taylor-Reynolds numbers.展开更多
In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equat...In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equation,Hirota equation,and NLS equation with the generalized PT-symmetric Scarf-II potentials.Specifically,we analyze the state transitions of different types of solitons(e.g.bright solitons,breathers,peakons,rogons,and periodic waves)appearing in these complex nonlinear wave equations.By checking the absolute errors between the predicted solutions and exact solutions,we can find that the FNO with the Ge Lu activation function can perform well in all cases even though these solution parameters have strong influences on the wave structures.Moreover,we find that the approximation errors via the physics-informed neural networks(PINNs)are similar in magnitude to those of the FNO.However,the FNO can learn the entire family of solutions under a given distribution every time,while the PINNs can only learn some specific solution each time.The results obtained in this paper will be useful for exploring physical mechanisms of soliton excitations in nonlinear wave equations and applying the FNO in other nonlinear wave equations.展开更多
Let T_(a,φ)be a Fourier integral operator defined by the oscillatory integral T_(a,φ)u(x)=1/(2π)^(n)∫_(R^(n))^e^(iφ(x,ξ))a(x,ξ)(u)(ξ)dξ,where a∈S_(e,δ)^(m)andφ∈Φ^(2),satisfying the strong non-degenerate ...Let T_(a,φ)be a Fourier integral operator defined by the oscillatory integral T_(a,φ)u(x)=1/(2π)^(n)∫_(R^(n))^e^(iφ(x,ξ))a(x,ξ)(u)(ξ)dξ,where a∈S_(e,δ)^(m)andφ∈Φ^(2),satisfying the strong non-degenerate condition.It is shown that if0<(e)≤1,0≤δ<1 and m≤e^(2)-n/2,thenT_(α,φ)is a bounded operator from L^(∞()R^(n))to BMO(R^(n)).展开更多
Let T_(ϕ,a)be a Fourier integral operator with amplitude a and phase functions ϕ.In this paper,we study the boundedness of Fourier integral operator of rough amplitude a∈L^(∞)S_(ρ)^(m)and rough phase functionsϕ∈L^...Let T_(ϕ,a)be a Fourier integral operator with amplitude a and phase functions ϕ.In this paper,we study the boundedness of Fourier integral operator of rough amplitude a∈L^(∞)S_(ρ)^(m)and rough phase functionsϕ∈L^(m)ϕ^(2)with some measure condition.We prove the global L^(1)boundedness for T_(ϕ,a),when 1/<ρ≤1 and m<ρ-n+1/2.Our theorem improves some known results.展开更多
An efficient data-driven approach for predicting steady airfoil flows is proposed based on the Fourier neural operator(FNO),which is a new framework of neural networks.Theoretical reasons and experimental results are ...An efficient data-driven approach for predicting steady airfoil flows is proposed based on the Fourier neural operator(FNO),which is a new framework of neural networks.Theoretical reasons and experimental results are provided to support the necessity and effectiveness of the improvements made to the FNO,which involve using an additional branch neural operator to approximate the contribution of boundary conditions to steady solutions.The proposed approach runs several orders of magnitude faster than the traditional numerical methods.The predictions for flows around airfoils and ellipses demonstrate the superior accuracy and impressive speed of this novel approach.Furthermore,the property of zero-shot super-resolution enables the proposed approach to overcome the limitations of predicting airfoil flows with Cartesian grids,thereby improving the accuracy in the near-wall region.There is no doubt that the unprecedented speed and accuracy in forecasting steady airfoil flows have massive benefits for airfoil design and optimization.展开更多
We propose a novel metasurface based on a combined pattern of outer C-shaped ring and inner rectangular ring.By Fourier convolution operation to generating different predesigned sequences of metasurfaces,we realize va...We propose a novel metasurface based on a combined pattern of outer C-shaped ring and inner rectangular ring.By Fourier convolution operation to generating different predesigned sequences of metasurfaces,we realize various functionalities to flexible manipulate terahertz waves including vortex terahertz beam splitting,anomalous vortex terahertz wave deflection,vortex terahertz wave splitting and deflection simultaneously.The incident terahertz wave can be flexibly controlled in a single metasurface.The designed metasurface has an extensive application prospect in the field of future terahertz communication and sensing.展开更多
Indoor airflow distribution significantly influences temperature regulation,humidity control,and pollutant dispersion,directly impacting thermal comfort and occupant health.Accurate and efficient prediction of airflow...Indoor airflow distribution significantly influences temperature regulation,humidity control,and pollutant dispersion,directly impacting thermal comfort and occupant health.Accurate and efficient prediction of airflow fields is essential for optimizing ventilation systems and enabling real-time control.However,existing computational approaches for dynamic ventilation are computationally intensive and have limited generalization capabilities.This study leverages the Fourier neural operator(FNO),a method rooted in operator learning and Fourier transform principles,to develop a three-dimensional(3D)airflow simulation model capable of predicting velocity and its components.The model was trained using 200 s of sinusoidal ventilation data(amplitude:0.4)and evaluated under diverse air supply patterns,including sinusoidal(amplitude:0.8),intermittent,and stepwise periodic ventilation.Additionally,the model’s performance was assessed with low-resolution training data and further tested for recursive prediction accuracy.Results reveal that the FNO method achieves high accuracy,with a mean square error of 9.906×10^(-5)for sinusoidal amplitude 0.8 and 4.004×10^(-5)over 400 time steps for sinusoidal,intermittent,and stepwise conditions.Further evaluations,including tests on low-resolution training data and recursive prediction,were conducted to examine the model’s resolution invariance and assess its performance in iterative forecasting.These findings demonstrate the FNO method’s potential for robust,efficient prediction of 3D unsteady airflow fields,providing a pathway for real-time ventilation system optimization.展开更多
Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm（f1, f2）（x） =（∫0^∞︱∫（Rn）^2）e^2πix·（ξ1＋ξ2））m（tξ1, tξ2）f1（ξ1）f2（ξ2）d...Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm（f1, f2）（x） =（∫0^∞︱∫（Rn）^2）e^2πix·（ξ1＋ξ2））m（tξ1, tξ2）f1（ξ1）f2（ξ2）dξ1dξ2︱^2dt/t）^1/2.Let s be an integer with s ∈ [n ＋ 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏i^2=1ω^i^p/p） and each ωi is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p（νω） if p0 〈 p1, p2 〈 ∞ with 1/p = 1/p1 ＋ 1/p2. Moreover,if p0 〉 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p,∞（νω）. The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.展开更多
Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W^s(R^(2n))< ∞ for some s ∈ (n, 2n]. In this paper, it is proved that th...Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W^s(R^(2n))< ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R^n) functions is a compact operator from L^(p1)(R^n, w_1) × L^(p2)(R^n, w_2) to L^p(R^n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R^(2n)).展开更多
In this note,we consider a class of Fourier integral operators with rough amplitudes and rough phases.When the index of symbols in some range,we prove that they are bounded on L^(1) and construct an example to show th...In this note,we consider a class of Fourier integral operators with rough amplitudes and rough phases.When the index of symbols in some range,we prove that they are bounded on L^(1) and construct an example to show that this result is sharp in some cases.This result is a generalization of the corresponding theorems of Kenig-Staubach and Dos Santos Ferreira-Staubach.展开更多
We prove the following properties:(1)Let a∈Λ_(1,0,k,k’)^(m0)(R^(n)×R^(n))with m0=-1|1/p-1/2|(n-1),n≥2(1 n/p,k’>0;2≤p≤∞,k>n/2,k’>0 respectively).Suppose the phase function S is positively homogen...We prove the following properties:(1)Let a∈Λ_(1,0,k,k’)^(m0)(R^(n)×R^(n))with m0=-1|1/p-1/2|(n-1),n≥2(1 n/p,k’>0;2≤p≤∞,k>n/2,k’>0 respectively).Suppose the phase function S is positively homogeneous inξ-variables,non-degenerate and satisfies certain conditions.Then the Fourier integral operator T is L^(p)-bounded.Applying the method of(1),we can obtain the L^(p)-boundedness of the Fourier integral operator if(2)the symbol a∈Λ_(1,δ,k,k’)^(m0),0≤δ≤1,with m0,k,k’and S given as in(1).Also,the method of(1)gives:(3)a∈Λ_(1,δ,k,k’),0≤δ<1 and k,k’given as in(1),then the L^(p)-boundedness of the pseudo-differential operators holds,1<p<∞.展开更多
We first prove the L~2-boundedness of a Fourier integral operator where it’s symbol a ∈S_(1/2,1/2)~0(R~n× R~n) and the phase function S is non-degenerate,satisfies certain conditions and may not be positively h...We first prove the L~2-boundedness of a Fourier integral operator where it’s symbol a ∈S_(1/2,1/2)~0(R~n× R~n) and the phase function S is non-degenerate,satisfies certain conditions and may not be positively homogeneous in ξ-variables.Then we use the above property,Paley’s inequality,covering lemma of Calderon and Zygmund etc.,and obtain the L~p-boundedness of Fourier integral operators if(1) the symbol a ∈ Λ_(k)^(m_(0)) and Supp a = E×R~n,with E a compact set of R~n(m_(0) =-|1/p-1/2|n,1<p≤2,k>n/2;2<p<∞,k>n/p),(2) the symbol a ∈ Λ_(0,k,k’)^(m_(0))(m_(0) =-|1/p-1/2|n,1<p ≤2,k>n/2,k’>n/p;2<p<∞,k>n/p,k’>n/2) with the phase function S(x,ξ) = xξ + h(x,ξ),x,ξ ∈ R~n non-degenerate,satisfying certain conditions and ?ξ h ∈ S_(1,0)~0(R~n× R~n),or(3) the symbol a ∈ Λ_(0,k,k’)^(m_(0)),the requirements for m_(0),k,k’ are the same as in(2),and ?_(ξ)h is not in S_(1,0)~0(R~n× R~n) but the phase function S is non-degenerate,satisfies certain conditions and is positively homogeneous in ξ-variables.展开更多
In this paper,we investigate the L^(2) boundedness of the Fourier integral operator Tφ,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions.In particular,if the symbol a...In this paper,we investigate the L^(2) boundedness of the Fourier integral operator Tφ,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions.In particular,if the symbol a∈L∞Smρ,the phase functionφsatisfies some measure conditions and ∇kξφ(·,ξ)L∞≤C|ξ|−k for all k≥2,ξ≠0,and some∈>0,we obtain that Tφ,a is bounded on L^(2) if m<n2 min{ρ−1,−2}.This result is a generalization of a result of Kenig and Staubach on pseudo-differential operators and it improves a result of Dos Santos Ferreira and Staubach on Fourier integral operators.Moreover,the Fourier integral operator with rough symbols and inhomogeneous phase functions we study in this paper can be used to obtain the almost everywhere convergence of the fractional Schr odinger operator.展开更多
Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions,which may not fall under the scope of the theory of standard Calder¬on–Zygmund classes.For anyb^(→)=(b_(1),...Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions,which may not fall under the scope of the theory of standard Calder¬on–Zygmund classes.For anyb^(→)=(b_(1),b_(2))∈(CMO(R^(n)))^(2),let[T,b_(j)]e_(j)(j=1,2),[T,→b]_(α)be the commutators in the j-th entry and the iterated commutators of T,respectively.In this paper,for all p_(0)>1,p0/2<p<∞,and p0≤p1,p2<∞with 1/p=1/p1+1/p2,we prove that[T,b_(j)]_(ej) and[T,b^(→)]αare weighted compact operators from L^(p1)(w1)×L^(p2)(w2)to L^(p)(νw^(→)),wherew^(→)=(w1,w2)∈A_(p^(→)/p0) andνw^(→)=w_(1)^(p/p1) w_()2)^(p/p2).As applications,we obtain the weighted compactness of commutators in the j-th entry and the iterated commutators of several kinds of bilinear Littlewood–Paley square operators with some mild kernel regularity,including bilinear g function,bilinear gλ^(∗)function and bilinear Lusin’s area integral.In addition,we also get the weighted compactness of commutators in the j-th entry and the iterated commutators of bilinear Fourier multiplier operators,and bilinear square Fourier multiplier operators associated with bilinear g function,bilinear gλ^(∗) function and bilinear Lusin’s area integral,respectively.展开更多
Digital twins of lithium-ion batteries are increasingly used to enable predictive monitoring,control,and design at system scale.Increasing their capabilities involves improving their physical fidelity while maintainin...Digital twins of lithium-ion batteries are increasingly used to enable predictive monitoring,control,and design at system scale.Increasing their capabilities involves improving their physical fidelity while maintaining sub-millisecond computational speed.In this work,we introduce machine learning surrogates that learn physical dynamics.Specifically,we benchmark three operator-learning surrogates for the Single Particle Model(SPM):Deep Operator Networks(DeepONets),Fourier Neural Operators(FNOs)and a newly proposed parameter-embedded Fourier Neural Operator(PE-FNO),which conditions each spectral layer on particle radius and solid-phase diffusivity.We extend the comparison to classical machine-learning baselines by including U-Nets.Models are trained on simulated trajectories spanning four current families(constant,triangular,pulse-train,and Gaussian-random-field)and a full range of State-of-Charge(SOC)(0%to 100%).DeepONet accurately replicates constant-current behaviour but struggles with more dynamic loads.The basic FNO maintains mesh invariance and keeps concentration errors below 1%,with voltage mean-absolute errors under 1.7mV across all load types.Introducing parameter embedding marginally increases error but enables generalisation to varying radii and diffusivities.PE-FNO executes approximately 200 times faster than a 16-thread SPM solver.Consequently,PE-FNO’s capabilities in inverse tasks are explored in a parameter estimation task with Bayesian optimisation,recovering anode and cathode diffusivities with 1.14%and 8.4%mean absolute percentage error,respectively,and 0.5918 percentage points higher error in comparison with classical methods.These results pave the way for neural operators to meet the accuracy,speed and parametric flexibility demands of real-time battery management,design-of-experiments and large-scale inference.PE-FNO outperforms conventional neural surrogates,offering a practical path towards high-speed and high-fidelity electrochemical digital twins.展开更多
This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve t...This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve the singularity of such kind of operators： a delicate method to decompose the operator and balance the L^2 norm estimates; and a method for resolution of singularity of the convolution type. The operator is decomposed into four major pieces instead of infinite dyadic pieces, which reveals that Cotlar＇s Lemma is not essential for the L^2 estimate of the operator. In the end the conclusion is further improved from the degenerate C^∞ phase function to the degenerate C^4 phase function.展开更多
In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as ...In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as well as the global existence of small data weak solution u when the nonlinearity power p is larger than some critical power p_(crit)(μ).Our proof is based on a class of new weighted Strichartz estimates with the weight t^(θ)|(1-μ)^(2)t^(2/|1-μ|)-x^(2)|^(γ)(θ>0andγ>0 are appropriate constants)for the solution of linear generalized Tricomi equation∂_(t)^(2)φ-t^(m)∂_(x)^(2)φ=0 with m being any fixed positive number.展开更多
Recent progress in topology optimization(TO)has seen a growing integration of machine learning to accelerate computation.Among these,online learning stands out as a promising strategy for large-scale TO tasks,as it el...Recent progress in topology optimization(TO)has seen a growing integration of machine learning to accelerate computation.Among these,online learning stands out as a promising strategy for large-scale TO tasks,as it eliminates the need for pre-collected training datasets by updating surrogate models dynamically using intermediate optimization data.Stress-constrained lightweight design is an important class of problem with broad engineering relevance.Most existing frameworks use pixel or voxel-based representations and employ the finite element method(FEM)for analysis.The limited continuity across finite elements often compromises the accuracy of stress evaluation.To overcome this limitation,isogeometric analysis is employed as it enables smooth representation of structures and thus more accurate stress computation.However,the complexity of the stress-constrained design problem together with the isogeometric representation results in a large computational cost.This work proposes a multi-grid,single-mesh online learning framework for isogeometric topology optimization(ITO),leveraging the Fourier Neural Operator(FNO)as a surrogate model.Operating entirely within the isogeometric analysis setting,the framework provides smooth geometry representation and precise stress computation,without requiring traditional mesh generation.A localized training approach is employed to enhance scalability,while a multi-grid decomposition scheme incorporates global structural context into local predictions to boost FNO accuracy.By learning the mapping from spatial features to sensitivity fields,the framework enables efficient single-resolution optimization,avoiding the computational burden of two-resolution simulations.The proposed method is validated through 2D stress-constrained design examples,and the effect of key parameters is studied.展开更多
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.
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(Nos.91952104,92052301,12172161,and 12161141017)National Numerical Windtunnel Project(No.NNW2019ZT1-A04)+4 种基金Shenzhen Science and Technology Program(No.KQTD20180411143441009)Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory(Guangzhou)(No.GML2019ZD0103)CAAI-Huawei Mind Spore open Fundand by Department of Science and Technology of Guangdong Province(No.2019B21203001)supported by Center for Computational Science and Engineering of Southern University of Science and Technology。
文摘Fourier neural operator(FNO)model is developed for large eddy simulation(LES)of three-dimensional(3D)turbulence.Velocity fields of isotropic turbulence generated by direct numerical simulation(DNS)are used for training the FNO model to predict the filtered velocity field at a given time.The input of the FNO model is the filtered velocity fields at the previous several time-nodes with large time lag.In the a posteriori study of LES,the FNO model performs better than the dynamic Smagorinsky model(DSM)and the dynamic mixed model(DMM)in the prediction of the velocity spectrum,probability density functions(PDFs)of vorticity and velocity increments,and the instantaneous flow structures.Moreover,the proposed model can significantly reduce the computational cost,and can be well generalized to LES of turbulence at higher Taylor-Reynolds numbers.
基金the NSFC under Grant Nos.11925108 and 11731014the NSFC under Grant No.11975306
文摘In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equation,Hirota equation,and NLS equation with the generalized PT-symmetric Scarf-II potentials.Specifically,we analyze the state transitions of different types of solitons(e.g.bright solitons,breathers,peakons,rogons,and periodic waves)appearing in these complex nonlinear wave equations.By checking the absolute errors between the predicted solutions and exact solutions,we can find that the FNO with the Ge Lu activation function can perform well in all cases even though these solution parameters have strong influences on the wave structures.Moreover,we find that the approximation errors via the physics-informed neural networks(PINNs)are similar in magnitude to those of the FNO.However,the FNO can learn the entire family of solutions under a given distribution every time,while the PINNs can only learn some specific solution each time.The results obtained in this paper will be useful for exploring physical mechanisms of soliton excitations in nonlinear wave equations and applying the FNO in other nonlinear wave equations.
文摘Let T_(a,φ)be a Fourier integral operator defined by the oscillatory integral T_(a,φ)u(x)=1/(2π)^(n)∫_(R^(n))^e^(iφ(x,ξ))a(x,ξ)(u)(ξ)dξ,where a∈S_(e,δ)^(m)andφ∈Φ^(2),satisfying the strong non-degenerate condition.It is shown that if0<(e)≤1,0≤δ<1 and m≤e^(2)-n/2,thenT_(α,φ)is a bounded operator from L^(∞()R^(n))to BMO(R^(n)).
基金Supported by the National Natural Science Foundation of China(11801518)the Natural Science Foundation of Zhejiang Province(LQ18A010005)the Science Foundation of Zhejiang Education Department(Y201738640)。
文摘Let T_(ϕ,a)be a Fourier integral operator with amplitude a and phase functions ϕ.In this paper,we study the boundedness of Fourier integral operator of rough amplitude a∈L^(∞)S_(ρ)^(m)and rough phase functionsϕ∈L^(m)ϕ^(2)with some measure condition.We prove the global L^(1)boundedness for T_(ϕ,a),when 1/<ρ≤1 and m<ρ-n+1/2.Our theorem improves some known results.
文摘An efficient data-driven approach for predicting steady airfoil flows is proposed based on the Fourier neural operator(FNO),which is a new framework of neural networks.Theoretical reasons and experimental results are provided to support the necessity and effectiveness of the improvements made to the FNO,which involve using an additional branch neural operator to approximate the contribution of boundary conditions to steady solutions.The proposed approach runs several orders of magnitude faster than the traditional numerical methods.The predictions for flows around airfoils and ellipses demonstrate the superior accuracy and impressive speed of this novel approach.Furthermore,the property of zero-shot super-resolution enables the proposed approach to overcome the limitations of predicting airfoil flows with Cartesian grids,thereby improving the accuracy in the near-wall region.There is no doubt that the unprecedented speed and accuracy in forecasting steady airfoil flows have massive benefits for airfoil design and optimization.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61871355 and 61831012)the Talent Project of Zhejiang Provincial Department of Science and Technology(Grant No.2018R52043)the Research Funds for Universities of Zhejiang Province,China(Grant Nos.2020YW20 and 2021YW86)。
文摘We propose a novel metasurface based on a combined pattern of outer C-shaped ring and inner rectangular ring.By Fourier convolution operation to generating different predesigned sequences of metasurfaces,we realize various functionalities to flexible manipulate terahertz waves including vortex terahertz beam splitting,anomalous vortex terahertz wave deflection,vortex terahertz wave splitting and deflection simultaneously.The incident terahertz wave can be flexibly controlled in a single metasurface.The designed metasurface has an extensive application prospect in the field of future terahertz communication and sensing.
基金supported by the National Natural Science Foundation of China[grant number 52078009]the Joint Research Project of the Wind Engineering Research Center,Tokyo Polytechnic University(MEXT(Japan)Promotion of the Distinctive Joint Research Center Program)[Joint Research Assignment number JURC 24242008].
文摘Indoor airflow distribution significantly influences temperature regulation,humidity control,and pollutant dispersion,directly impacting thermal comfort and occupant health.Accurate and efficient prediction of airflow fields is essential for optimizing ventilation systems and enabling real-time control.However,existing computational approaches for dynamic ventilation are computationally intensive and have limited generalization capabilities.This study leverages the Fourier neural operator(FNO),a method rooted in operator learning and Fourier transform principles,to develop a three-dimensional(3D)airflow simulation model capable of predicting velocity and its components.The model was trained using 200 s of sinusoidal ventilation data(amplitude:0.4)and evaluated under diverse air supply patterns,including sinusoidal(amplitude:0.8),intermittent,and stepwise periodic ventilation.Additionally,the model’s performance was assessed with low-resolution training data and further tested for recursive prediction accuracy.Results reveal that the FNO method achieves high accuracy,with a mean square error of 9.906×10^(-5)for sinusoidal amplitude 0.8 and 4.004×10^(-5)over 400 time steps for sinusoidal,intermittent,and stepwise conditions.Further evaluations,including tests on low-resolution training data and recursive prediction,were conducted to examine the model’s resolution invariance and assess its performance in iterative forecasting.These findings demonstrate the FNO method’s potential for robust,efficient prediction of 3D unsteady airflow fields,providing a pathway for real-time ventilation system optimization.
基金supported by National Natural Science Foundation of China (Grant Nos. 11401175, 11501169 and 11471041)the Fundamental Research Funds for the Central Universities (Grant No. 2014KJJCA10)+2 种基金Program for New Century Excellent Talents in University (Grant No. NCET-13-0065)Grantin-Aid for Scientific Research (C) (Grant No. 15K04942)Japan Society for the Promotion of Science
文摘Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm（f1, f2）（x） =（∫0^∞︱∫（Rn）^2）e^2πix·（ξ1＋ξ2））m（tξ1, tξ2）f1（ξ1）f2（ξ2）dξ1dξ2︱^2dt/t）^1/2.Let s be an integer with s ∈ [n ＋ 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏i^2=1ω^i^p/p） and each ωi is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p（νω） if p0 〈 p1, p2 〈 ∞ with 1/p = 1/p1 ＋ 1/p2. Moreover,if p0 〉 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p,∞（νω）. The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.
基金supported by the National Natural Science Foundation of China(No.11371370)
文摘Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W^s(R^(2n))< ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R^n) functions is a compact operator from L^(p1)(R^n, w_1) × L^(p2)(R^n, w_2) to L^p(R^n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R^(2n)).
基金supported by the National Natural Science Foundation of China(No.11871436)the National Key Research and Development Program of China(No.2022YFA1005700)。
文摘In this note,we consider a class of Fourier integral operators with rough amplitudes and rough phases.When the index of symbols in some range,we prove that they are bounded on L^(1) and construct an example to show that this result is sharp in some cases.This result is a generalization of the corresponding theorems of Kenig-Staubach and Dos Santos Ferreira-Staubach.
文摘We prove the following properties:(1)Let a∈Λ_(1,0,k,k’)^(m0)(R^(n)×R^(n))with m0=-1|1/p-1/2|(n-1),n≥2(1 n/p,k’>0;2≤p≤∞,k>n/2,k’>0 respectively).Suppose the phase function S is positively homogeneous inξ-variables,non-degenerate and satisfies certain conditions.Then the Fourier integral operator T is L^(p)-bounded.Applying the method of(1),we can obtain the L^(p)-boundedness of the Fourier integral operator if(2)the symbol a∈Λ_(1,δ,k,k’)^(m0),0≤δ≤1,with m0,k,k’and S given as in(1).Also,the method of(1)gives:(3)a∈Λ_(1,δ,k,k’),0≤δ<1 and k,k’given as in(1),then the L^(p)-boundedness of the pseudo-differential operators holds,1<p<∞.
文摘We first prove the L~2-boundedness of a Fourier integral operator where it’s symbol a ∈S_(1/2,1/2)~0(R~n× R~n) and the phase function S is non-degenerate,satisfies certain conditions and may not be positively homogeneous in ξ-variables.Then we use the above property,Paley’s inequality,covering lemma of Calderon and Zygmund etc.,and obtain the L~p-boundedness of Fourier integral operators if(1) the symbol a ∈ Λ_(k)^(m_(0)) and Supp a = E×R~n,with E a compact set of R~n(m_(0) =-|1/p-1/2|n,1<p≤2,k>n/2;2<p<∞,k>n/p),(2) the symbol a ∈ Λ_(0,k,k’)^(m_(0))(m_(0) =-|1/p-1/2|n,1<p ≤2,k>n/2,k’>n/p;2<p<∞,k>n/p,k’>n/2) with the phase function S(x,ξ) = xξ + h(x,ξ),x,ξ ∈ R~n non-degenerate,satisfying certain conditions and ?ξ h ∈ S_(1,0)~0(R~n× R~n),or(3) the symbol a ∈ Λ_(0,k,k’)^(m_(0)),the requirements for m_(0),k,k’ are the same as in(2),and ?_(ξ)h is not in S_(1,0)~0(R~n× R~n) but the phase function S is non-degenerate,satisfies certain conditions and is positively homogeneous in ξ-variables.
基金Supported by National Natural Science Foundation of China(Grant No.12071437)National key R&D program of China(Grant No.2022YFA1005703)。
文摘In this paper,we investigate the L^(2) boundedness of the Fourier integral operator Tφ,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions.In particular,if the symbol a∈L∞Smρ,the phase functionφsatisfies some measure conditions and ∇kξφ(·,ξ)L∞≤C|ξ|−k for all k≥2,ξ≠0,and some∈>0,we obtain that Tφ,a is bounded on L^(2) if m<n2 min{ρ−1,−2}.This result is a generalization of a result of Kenig and Staubach on pseudo-differential operators and it improves a result of Dos Santos Ferreira and Staubach on Fourier integral operators.Moreover,the Fourier integral operator with rough symbols and inhomogeneous phase functions we study in this paper can be used to obtain the almost everywhere convergence of the fractional Schr odinger operator.
基金supported by the National Natural Science Foundation of China(Grant No.12101222)National Natural Science Foundation of China(Grant No.12271041)+2 种基金Scientific Research Fund of Hunan Provincial Education Department(Grant No.23B0458)supported by the Scientific Research Fund of Hunan Provincial Education Department(Grant No.22B0155)partly supported by the National Key R&D Program of China(Grant No.2020YFA0712900)。
文摘Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions,which may not fall under the scope of the theory of standard Calder¬on–Zygmund classes.For anyb^(→)=(b_(1),b_(2))∈(CMO(R^(n)))^(2),let[T,b_(j)]e_(j)(j=1,2),[T,→b]_(α)be the commutators in the j-th entry and the iterated commutators of T,respectively.In this paper,for all p_(0)>1,p0/2<p<∞,and p0≤p1,p2<∞with 1/p=1/p1+1/p2,we prove that[T,b_(j)]_(ej) and[T,b^(→)]αare weighted compact operators from L^(p1)(w1)×L^(p2)(w2)to L^(p)(νw^(→)),wherew^(→)=(w1,w2)∈A_(p^(→)/p0) andνw^(→)=w_(1)^(p/p1) w_()2)^(p/p2).As applications,we obtain the weighted compactness of commutators in the j-th entry and the iterated commutators of several kinds of bilinear Littlewood–Paley square operators with some mild kernel regularity,including bilinear g function,bilinear gλ^(∗)function and bilinear Lusin’s area integral.In addition,we also get the weighted compactness of commutators in the j-th entry and the iterated commutators of bilinear Fourier multiplier operators,and bilinear square Fourier multiplier operators associated with bilinear g function,bilinear gλ^(∗) function and bilinear Lusin’s area integral,respectively.
基金funding from the project“SPEED”(03XP0585)funded by the German Federal Ministry of ResearchTech-nology and Space(BMFTR)and the project“ADMirABLE”(03ETE053E)funded by the German Federal Ministry for Economic Affairs and Energy(BMWE)support of Shell Research UK Ltd.for the Ph.D.studentship of Amir Ali Panahi and the EPSRC Faraday Institution Multi-Scale Modelling Project(FIRG084).
文摘Digital twins of lithium-ion batteries are increasingly used to enable predictive monitoring,control,and design at system scale.Increasing their capabilities involves improving their physical fidelity while maintaining sub-millisecond computational speed.In this work,we introduce machine learning surrogates that learn physical dynamics.Specifically,we benchmark three operator-learning surrogates for the Single Particle Model(SPM):Deep Operator Networks(DeepONets),Fourier Neural Operators(FNOs)and a newly proposed parameter-embedded Fourier Neural Operator(PE-FNO),which conditions each spectral layer on particle radius and solid-phase diffusivity.We extend the comparison to classical machine-learning baselines by including U-Nets.Models are trained on simulated trajectories spanning four current families(constant,triangular,pulse-train,and Gaussian-random-field)and a full range of State-of-Charge(SOC)(0%to 100%).DeepONet accurately replicates constant-current behaviour but struggles with more dynamic loads.The basic FNO maintains mesh invariance and keeps concentration errors below 1%,with voltage mean-absolute errors under 1.7mV across all load types.Introducing parameter embedding marginally increases error but enables generalisation to varying radii and diffusivities.PE-FNO executes approximately 200 times faster than a 16-thread SPM solver.Consequently,PE-FNO’s capabilities in inverse tasks are explored in a parameter estimation task with Bayesian optimisation,recovering anode and cathode diffusivities with 1.14%and 8.4%mean absolute percentage error,respectively,and 0.5918 percentage points higher error in comparison with classical methods.These results pave the way for neural operators to meet the accuracy,speed and parametric flexibility demands of real-time battery management,design-of-experiments and large-scale inference.PE-FNO outperforms conventional neural surrogates,offering a practical path towards high-speed and high-fidelity electrochemical digital twins.
基金the State Key Laboratory of Software Development Environmentthe Grant No.SKLSDE-07-004 under the National Basic Research Program of China (the 973 Program Grant No.2005CB321901)
文摘This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve the singularity of such kind of operators： a delicate method to decompose the operator and balance the L^2 norm estimates; and a method for resolution of singularity of the convolution type. The operator is decomposed into four major pieces instead of infinite dyadic pieces, which reveals that Cotlar＇s Lemma is not essential for the L^2 estimate of the operator. In the end the conclusion is further improved from the degenerate C^∞ phase function to the degenerate C^4 phase function.
基金supported by the NSFC(12331007)the National Key Research and Development Program of China(2020YFA0713803)。
文摘In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as well as the global existence of small data weak solution u when the nonlinearity power p is larger than some critical power p_(crit)(μ).Our proof is based on a class of new weighted Strichartz estimates with the weight t^(θ)|(1-μ)^(2)t^(2/|1-μ|)-x^(2)|^(γ)(θ>0andγ>0 are appropriate constants)for the solution of linear generalized Tricomi equation∂_(t)^(2)φ-t^(m)∂_(x)^(2)φ=0 with m being any fixed positive number.
基金supported by the Hong Kong Research Grants under Competitive Earmarked Research Grant No.16206320.
文摘Recent progress in topology optimization(TO)has seen a growing integration of machine learning to accelerate computation.Among these,online learning stands out as a promising strategy for large-scale TO tasks,as it eliminates the need for pre-collected training datasets by updating surrogate models dynamically using intermediate optimization data.Stress-constrained lightweight design is an important class of problem with broad engineering relevance.Most existing frameworks use pixel or voxel-based representations and employ the finite element method(FEM)for analysis.The limited continuity across finite elements often compromises the accuracy of stress evaluation.To overcome this limitation,isogeometric analysis is employed as it enables smooth representation of structures and thus more accurate stress computation.However,the complexity of the stress-constrained design problem together with the isogeometric representation results in a large computational cost.This work proposes a multi-grid,single-mesh online learning framework for isogeometric topology optimization(ITO),leveraging the Fourier Neural Operator(FNO)as a surrogate model.Operating entirely within the isogeometric analysis setting,the framework provides smooth geometry representation and precise stress computation,without requiring traditional mesh generation.A localized training approach is employed to enhance scalability,while a multi-grid decomposition scheme incorporates global structural context into local predictions to boost FNO accuracy.By learning the mapping from spatial features to sensitivity fields,the framework enables efficient single-resolution optimization,avoiding the computational burden of two-resolution simulations.The proposed method is validated through 2D stress-constrained design examples,and the effect of key parameters is studied.
