In this article, we consider some type of q-difference equations, which have meromorphic solutions with Borel exceptional zeros and poles. We also give a precise result in the finite order case and some further result...In this article, we consider some type of q-difference equations, which have meromorphic solutions with Borel exceptional zeros and poles. We also give a precise result in the finite order case and some further results in a particular case where qi = qi.展开更多
Let q be a finite nonzero complex number,let the q-difference equation f(qz)f(z/p)=R(z,f(z))=P(z,f(z))/Q(z,f(z))=Σ_(j=0)^(p)=0aj(z)f^(j)(z)/Σ_(k=0)b_(k)(z)f^(k)(z)(+)admit a nonconstant meromorphic solution f,where ...Let q be a finite nonzero complex number,let the q-difference equation f(qz)f(z/p)=R(z,f(z))=P(z,f(z))/Q(z,f(z))=Σ_(j=0)^(p)=0aj(z)f^(j)(z)/Σ_(k=0)b_(k)(z)f^(k)(z)(+)admit a nonconstant meromorphic solution f,where p and q are nonnegative integers,a_(j)with 0≤j≤p and b_(k)with 0≤k≤q polynomials in zwith a_(p)≠0 and b_(q)≠0 such that P(z,f(z))and Q(z,f(z))are relatively prime polynomials in f(z)and let m=p-q≥3.Then,(+)has no transcendental meromorphic solution when|q|=1,and the lower bound of the lower order of f is obtained when m≥3 and |q|≠1.展开更多
In this paper,we mainly investigate some properties of meromorphic solutions for several q-difference equations,which can be seen as the q-difference analogues of Painlevéequations.Some results about the existenc...In this paper,we mainly investigate some properties of meromorphic solutions for several q-difference equations,which can be seen as the q-difference analogues of Painlevéequations.Some results about the existence and the estimates of growth of meromorphic solution f for q-difference equations are obtained,especially for some estimates for the exponent of convergence of poles of△qf(z):=f(qz)-f(z),which extends some previous results by Qi and Yang.展开更多
In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principl...In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principle are used. At the end of the manuscript, we have an example that illustrates the key findings.展开更多
We consider the properties on solutions of some q-difference equations of the form∑n j=0 aj(z)f(qj z)=an+1(z),where a0(z),...,an+1(z)are meromorphic functions,a0(z)an(z)≠0 and q∈C such that 0〈|q|≤1.We give estima...We consider the properties on solutions of some q-difference equations of the form∑n j=0 aj(z)f(qj z)=an+1(z),where a0(z),...,an+1(z)are meromorphic functions,a0(z)an(z)≠0 and q∈C such that 0〈|q|≤1.We give estimates on the upper bound for the length of the gap in the power series of entire solutions of(*)when the coefficients a0(z),...,an+1(z)are polynomials and 0〈|q|〈1.For some special cases,we give estimates of growth of f(z).And we also show that the case 0〈|q|〈1 is different from the case|q|=1.展开更多
In this paper, we consider the value distribution of meromorphic solutions of order zero of some kind of q-difference equations and examples are also given to elaborate our results.
This is a set of lecture notes for the first author’s lectures on the difference equations in2019 at the Institute of Advanced Study for Mathematics at Zhejiang University.We focus on explicit computations and exampl...This is a set of lecture notes for the first author’s lectures on the difference equations in2019 at the Institute of Advanced Study for Mathematics at Zhejiang University.We focus on explicit computations and examples.The convergence of local solutions is discussed.展开更多
In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence...In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence,asymptotic behavior and blow-up of solutions with initial energy J(u_(0))≤d.Moreover,we estimate the upper bound of the blow-up time for J(u_(0))≤0.展开更多
This paper deals with Mckean-Vlasov backward stochastic differential equations with weak monotonicity coefficients.We first establish the existence and uniqueness of solutions to Mckean-Vlasov backward stochastic diff...This paper deals with Mckean-Vlasov backward stochastic differential equations with weak monotonicity coefficients.We first establish the existence and uniqueness of solutions to Mckean-Vlasov backward stochastic differential equations.Then we obtain a comparison theorem in one-dimensional situation.展开更多
In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to ...In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.展开更多
Let A be a 3×3 singular or diagonalizable matrix,all solutions to the Yang-Baxter-like matrix equation have been determined.However,finding all solutions for full rank,non-diagonalizable matrices remains challeng...Let A be a 3×3 singular or diagonalizable matrix,all solutions to the Yang-Baxter-like matrix equation have been determined.However,finding all solutions for full rank,non-diagonalizable matrices remains challenging.By utilizing classification techniques,we establish all solutions of the Yang-Baxter-like matrix equation in this paper when the coefficient matrix A is similar to non-diagonalizable matrix diag(λ,J_(2)(λ))withλ̸=0.More specifically,we divide the non-diagonal elements of the solution into 10 different cases.By discussing each situation,we establish all solutions of the Yang-Baxter-like matrix equation.The results of this work enrich the existing ones.展开更多
The existence of a global attractor is established for generalized Navier-Stokes equations incorporating damping term within the periodic domainΩ=[−π,π]^(n).Initially,we show the existence and uniqueness of strong ...The existence of a global attractor is established for generalized Navier-Stokes equations incorporating damping term within the periodic domainΩ=[−π,π]^(n).Initially,we show the existence and uniqueness of strong solutions.Subsequently,we verify the continuity of the associated semigroup when max{2n+1/n-1,5n+2/3n-2} < β <3n+2/n-2.Finally,we establish the existence of both H^(α)-global attractor and H^(2α)-global attractor.展开更多
The goal of this paper is to investigate the long-time dynamics of solutions to a Kirchhoff type suspension bridge equation with nonlinear damping and memory term.For this problem we establish the well-posedness and e...The goal of this paper is to investigate the long-time dynamics of solutions to a Kirchhoff type suspension bridge equation with nonlinear damping and memory term.For this problem we establish the well-posedness and existence of uniform attractor under some suitable assumptions on the nonlinear term g(u),the nonlinear damping f(u_(t))and the external force h(x,t).Specifically,the asymptotic compactness of the semigroup is verified by the energy reconstruction method.展开更多
A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue prob...A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue problem,the method is spectral-correct and spurious-free.Stability and error estimates are obtained,including the interpolation error estimates and the error estimates between the finite element solution and the exact solution.The method is suitable for singular solution as well as smooth solution,and consequently,the method is valid for nonconvex domains which may have a number of reentrant corners.Of course,the method is suitable for arbitrary quadrilaterals(under the usual shape-regular condition).展开更多
The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to ...The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to contractive mapping principle,Strichartz estimates,Caffarelli-Kohn-Nirenberg-type inequality and the continuity method.展开更多
We investigate the constrained fractional Choquard equation■where m>0,N>2s with s∈(0,1)being the order of the fractional Laplacian operator and I_(α)forα∈(0,N)denotes the Riesz potential.The parameterμ∈ℝa...We investigate the constrained fractional Choquard equation■where m>0,N>2s with s∈(0,1)being the order of the fractional Laplacian operator and I_(α)forα∈(0,N)denotes the Riesz potential.The parameterμ∈ℝappears as a Lagrange multiplier.By imposing general mass-supercritical conditions on F,we confirm the existence of normalized solutions that characterize the global minimizer on the Pohozaev manifold.Our proof does not depend on the assumption that all weak solutions satisfy the Pohozaev identity,a challenge that remains unsolved for this doubly nonlocal equation.展开更多
In this paper,we present a finite volume trigonometric weighted essentially non-oscillatory(TWENO)scheme to solve nonlinear degenerate parabolic equations that may exhibit non-smooth solutions.The present method is de...In this paper,we present a finite volume trigonometric weighted essentially non-oscillatory(TWENO)scheme to solve nonlinear degenerate parabolic equations that may exhibit non-smooth solutions.The present method is developed using the trigonometric scheme,which is based on zero,first,and second moments,and the direct discontinuous Galerkin(DDG)flux is used to discretize the diffusion term.Moreover,the DDG method directly applies the weak form of the parabolic equation to each computational cell,which can better capture the characteristics of the solution,especially the discontinuous solution.Meanwhile,the third-order TVD-Runge-Kutta method is applied for temporal discretization.Finally,the effectiveness and stability of the method constructed in this paper are evaluated through numerical tests.展开更多
The neutron diffusion equation plays a pivotal role in nuclear reactor analysis.Nevertheless,employing the physics-informed neural network(PINN)method for its solution entails certain limitations.Conventional PINN app...The neutron diffusion equation plays a pivotal role in nuclear reactor analysis.Nevertheless,employing the physics-informed neural network(PINN)method for its solution entails certain limitations.Conventional PINN approaches generally utilize a fully connected network(FCN)architecture that is susceptible to overfitting,training instability,and gradient vanishing as the network depth increases.These challenges result in accuracy bottlenecks in the solution.In response to these issues,the residual-based resample physics-informed neural network(R2-PINN)is proposed.It is an improved PINN architecture that replaces the FCN with a convolutional neural network with a shortcut(S-CNN).It incorporates skip connections to facilitate gradient propagation between network layers.Additionally,the incorporation of the residual adaptive resampling(RAR)mechanism dynamically increases the number of sampling points.This,in turn,enhances the spatial representation capabilities and overall predictive accuracy of the model.The experimental results illustrate that our approach significantly improves the convergence capability of the model and achieves high-precision predictions of the physical fields.Compared with conventional FCN-based PINN methods,R 2-PINN effectively overcomes the limitations inherent in current methods.Thus,it provides more accurate and robust solutions for neutron diffusion equations.展开更多
AIM:To build a functional generalized estimating equation(GEE)model to detect glaucomatous visual field progression and compare the performance of the proposed method with that of commonly employed algorithms.METHODS:...AIM:To build a functional generalized estimating equation(GEE)model to detect glaucomatous visual field progression and compare the performance of the proposed method with that of commonly employed algorithms.METHODS:Totally 716 eyes of 716 patients with primary open angle glaucoma(POAG)with at least 5 reliable 24-2 test results and 2y of follow-up were selected.The functional GEE model was used to detect perimetric progression in the training dataset(501 eyes).In the testing dataset(215 eyes),progression was evaluated the functional GEE model,mean deviation(MD)and visual field index(VFI)rates of change,Advanced Glaucoma Intervention Study(AGIS)and Collaborative Initial Glaucoma Treatment Study(CIGTS)scores,and pointwise linear regression(PLR).RESULTS:The proposed method showed the highest proportion of eyes detected as progression(54.4%),followed by the VFI rate(34.4%),PLR(23.3%),and MD rate(21.4%).The CIGTS and AGIS scores had a lower proportion of eyes detected as progression(7.9%and 5.1%,respectively).The time to detection of progression was significantly shorter for the proposed method than that of other algorithms(adjusted P≤0.019).The VFI rate displayed moderate pairwise agreement with the proposed method(k=0.47).CONCLUSION:The functional GEE model shows the highest proportion of eyes detected as perimetric progression and the shortest time to detect perimetric progression in patients with POAG.展开更多
The augmented evolution equation is established under the framework of the Variation Evolving Method(VEM)that seeks optimal solutions by solving the transformed Initial-Value Problems(IVPs).To improve the numerical pe...The augmented evolution equation is established under the framework of the Variation Evolving Method(VEM)that seeks optimal solutions by solving the transformed Initial-Value Problems(IVPs).To improve the numerical performance,its compact form is developed herein.Through replacing the states and costates variation evolution with that of the controls,the dimension-reduced Evolution Partial Differential Equation(EPDE)only solves the control variables along the variation time to get the optimal solution,and the initial conditions for the definite solution may be arbitrary.With this equation,the scale of the resulting IVPs,obtained via the semi-discrete method,is significantly reduced and they may be solved with common Ordinary Differential Equation(ODE)integration methods conveniently.Meanwhile,the state and the costate dynamics share consistent stability in the numerical computation and this avoids the intrinsic numerical difficulty as in the indirect methods.Numerical examples are solved and it is shown that the compact form evolution equation outperforms the primary form in the precision,and the efficiency may be higher for the dense discretization.Actually,it is uncovered that the compact form of the augmented evolution equation is a continuous realization of the Newton type iteration mechanism.展开更多
基金supported by the National Natural Science Foundation of China (11171119,11126145,11026096)the Nature Science Foundation of Jiangxi Province in China(20114BAB211003)
文摘In this article, we consider some type of q-difference equations, which have meromorphic solutions with Borel exceptional zeros and poles. We also give a precise result in the finite order case and some further results in a particular case where qi = qi.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1226102311861023)the Foundation of Science and Technology Project of Guizhou Province of China(Grant No.[2018]5769-05).
文摘Let q be a finite nonzero complex number,let the q-difference equation f(qz)f(z/p)=R(z,f(z))=P(z,f(z))/Q(z,f(z))=Σ_(j=0)^(p)=0aj(z)f^(j)(z)/Σ_(k=0)b_(k)(z)f^(k)(z)(+)admit a nonconstant meromorphic solution f,where p and q are nonnegative integers,a_(j)with 0≤j≤p and b_(k)with 0≤k≤q polynomials in zwith a_(p)≠0 and b_(q)≠0 such that P(z,f(z))and Q(z,f(z))are relatively prime polynomials in f(z)and let m=p-q≥3.Then,(+)has no transcendental meromorphic solution when|q|=1,and the lower bound of the lower order of f is obtained when m≥3 and |q|≠1.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11561033,11761035)the Natural Science Foundation of Jiangxi Province(Grant No.20181BAB201001)the Foundation of Education Department of Jiangxi Province(Grant Nos.GJJ190876,GJJ191042,GJJ190895)。
文摘In this paper,we mainly investigate some properties of meromorphic solutions for several q-difference equations,which can be seen as the q-difference analogues of Painlevéequations.Some results about the existence and the estimates of growth of meromorphic solution f for q-difference equations are obtained,especially for some estimates for the exponent of convergence of poles of△qf(z):=f(qz)-f(z),which extends some previous results by Qi and Yang.
文摘In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principle are used. At the end of the manuscript, we have an example that illustrates the key findings.
基金Supported by National Natural Science Foundation of China(Grant No.10871076)
文摘We consider the properties on solutions of some q-difference equations of the form∑n j=0 aj(z)f(qj z)=an+1(z),where a0(z),...,an+1(z)are meromorphic functions,a0(z)an(z)≠0 and q∈C such that 0〈|q|≤1.We give estimates on the upper bound for the length of the gap in the power series of entire solutions of(*)when the coefficients a0(z),...,an+1(z)are polynomials and 0〈|q|〈1.For some special cases,we give estimates of growth of f(z).And we also show that the case 0〈|q|〈1 is different from the case|q|=1.
基金Supported by the National Natural Science Foundation of China (Grant No.10871076)the Youth Science Foundation of Education Bureau of Jiangxi Province (Grant No.GJJ11072)the Natural Science Foundation of Jiangxi Province (Grant No.2009GQS0013)
文摘In this paper, we consider the value distribution of meromorphic solutions of order zero of some kind of q-difference equations and examples are also given to elaborate our results.
基金supported by a KIAS Individual Grant(MG083901)at Korea Institute for Advanced Studya POSCO Science fellowship。
文摘This is a set of lecture notes for the first author’s lectures on the difference equations in2019 at the Institute of Advanced Study for Mathematics at Zhejiang University.We focus on explicit computations and examples.The convergence of local solutions is discussed.
基金Supported by NSFC(No.12101482)the Natural Science Foundation of Shaanxi Province,China(No.2018JQ1052)。
文摘In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence,asymptotic behavior and blow-up of solutions with initial energy J(u_(0))≤d.Moreover,we estimate the upper bound of the blow-up time for J(u_(0))≤0.
基金Supported by the National Natural Science Foundation of China(12001074)the Research Innovation Program of Graduate Students in Hunan Province(CX20220258)+1 种基金the Research Innovation Program of Graduate Students of Central South University(1053320214147)the Key Scientific Research Project of Higher Education Institutions in Henan Province(25B110025)。
文摘This paper deals with Mckean-Vlasov backward stochastic differential equations with weak monotonicity coefficients.We first establish the existence and uniqueness of solutions to Mckean-Vlasov backward stochastic differential equations.Then we obtain a comparison theorem in one-dimensional situation.
基金Supported in part by Natural Science Foundation of Guangxi(2023GXNSFAA026246)in part by the Central Government's Guide to Local Science and Technology Development Fund(GuikeZY23055044)in part by the National Natural Science Foundation of China(62363003)。
文摘In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.
基金Supported by National Natural Science Foundation of China(Grant No.62173161).
文摘Let A be a 3×3 singular or diagonalizable matrix,all solutions to the Yang-Baxter-like matrix equation have been determined.However,finding all solutions for full rank,non-diagonalizable matrices remains challenging.By utilizing classification techniques,we establish all solutions of the Yang-Baxter-like matrix equation in this paper when the coefficient matrix A is similar to non-diagonalizable matrix diag(λ,J_(2)(λ))withλ̸=0.More specifically,we divide the non-diagonal elements of the solution into 10 different cases.By discussing each situation,we establish all solutions of the Yang-Baxter-like matrix equation.The results of this work enrich the existing ones.
文摘The existence of a global attractor is established for generalized Navier-Stokes equations incorporating damping term within the periodic domainΩ=[−π,π]^(n).Initially,we show the existence and uniqueness of strong solutions.Subsequently,we verify the continuity of the associated semigroup when max{2n+1/n-1,5n+2/3n-2} < β <3n+2/n-2.Finally,we establish the existence of both H^(α)-global attractor and H^(2α)-global attractor.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11961059,1210502)the University Innovation Project of Gansu Province(Grant No.2023B-062)the Gansu Province Basic Research Innovation Group Project(Grant No.23JRRA684).
文摘The goal of this paper is to investigate the long-time dynamics of solutions to a Kirchhoff type suspension bridge equation with nonlinear damping and memory term.For this problem we establish the well-posedness and existence of uniform attractor under some suitable assumptions on the nonlinear term g(u),the nonlinear damping f(u_(t))and the external force h(x,t).Specifically,the asymptotic compactness of the semigroup is verified by the energy reconstruction method.
基金supported by the National Natural Science Foundation of China(12401482)the second author was supported by the National Natural Science Foundation of China(12371371,12261160361,11971366)supported by the Open Research Fund of Hubei Key Laboratory of Computational Science,Wuhan University.
文摘A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue problem,the method is spectral-correct and spurious-free.Stability and error estimates are obtained,including the interpolation error estimates and the error estimates between the finite element solution and the exact solution.The method is suitable for singular solution as well as smooth solution,and consequently,the method is valid for nonconvex domains which may have a number of reentrant corners.Of course,the method is suitable for arbitrary quadrilaterals(under the usual shape-regular condition).
基金Supported by National Natural Science Foundation of China(Grant No.11601122).
文摘The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to contractive mapping principle,Strichartz estimates,Caffarelli-Kohn-Nirenberg-type inequality and the continuity method.
基金supported by the Guangdong Basic and Applied Basic Research Foundation(2022A1515012138)the NSFC(12271436,12371119)supported by the Natural Science Basic Research Program of Shaanxi(2022JC-04).
文摘We investigate the constrained fractional Choquard equation■where m>0,N>2s with s∈(0,1)being the order of the fractional Laplacian operator and I_(α)forα∈(0,N)denotes the Riesz potential.The parameterμ∈ℝappears as a Lagrange multiplier.By imposing general mass-supercritical conditions on F,we confirm the existence of normalized solutions that characterize the global minimizer on the Pohozaev manifold.Our proof does not depend on the assumption that all weak solutions satisfy the Pohozaev identity,a challenge that remains unsolved for this doubly nonlocal equation.
基金The Natural Science Foundation of Xinjiang Uygur Autonomous Region of China“RBF-Hermite difference scheme for the time-fractional kdv-Burgers equation”(2024D01C43)。
文摘In this paper,we present a finite volume trigonometric weighted essentially non-oscillatory(TWENO)scheme to solve nonlinear degenerate parabolic equations that may exhibit non-smooth solutions.The present method is developed using the trigonometric scheme,which is based on zero,first,and second moments,and the direct discontinuous Galerkin(DDG)flux is used to discretize the diffusion term.Moreover,the DDG method directly applies the weak form of the parabolic equation to each computational cell,which can better capture the characteristics of the solution,especially the discontinuous solution.Meanwhile,the third-order TVD-Runge-Kutta method is applied for temporal discretization.Finally,the effectiveness and stability of the method constructed in this paper are evaluated through numerical tests.
基金supported by the Science and Technology on Reactor System Design Technology Laboratory(No.LRSDT12023108)supported in part by the Chongqing Postdoctoral Science Foundation(No.cstc2021jcyj-bsh0252)+2 种基金the National Natural Science Foundation of China(No.12005030)Sichuan Province to unveil the list of marshal industry common technology research projects(No.23jBGOV0001)Special Program for Stabilizing Support to Basic Research of National Basic Research Institutes(No.WDZC-2023-05-03-05).
文摘The neutron diffusion equation plays a pivotal role in nuclear reactor analysis.Nevertheless,employing the physics-informed neural network(PINN)method for its solution entails certain limitations.Conventional PINN approaches generally utilize a fully connected network(FCN)architecture that is susceptible to overfitting,training instability,and gradient vanishing as the network depth increases.These challenges result in accuracy bottlenecks in the solution.In response to these issues,the residual-based resample physics-informed neural network(R2-PINN)is proposed.It is an improved PINN architecture that replaces the FCN with a convolutional neural network with a shortcut(S-CNN).It incorporates skip connections to facilitate gradient propagation between network layers.Additionally,the incorporation of the residual adaptive resampling(RAR)mechanism dynamically increases the number of sampling points.This,in turn,enhances the spatial representation capabilities and overall predictive accuracy of the model.The experimental results illustrate that our approach significantly improves the convergence capability of the model and achieves high-precision predictions of the physical fields.Compared with conventional FCN-based PINN methods,R 2-PINN effectively overcomes the limitations inherent in current methods.Thus,it provides more accurate and robust solutions for neutron diffusion equations.
基金Supported by the Korea Health Technology R&D Project through the Korea Health Industry Development Institute(KHIDI),funded by the Ministry of Health&Welfare,Republic of Korea(No.HR20C0026)the National Research Foundation of Korea(NRF)(No.RS-2023-00247504)the Patient-Centered Clinical Research Coordinating Center,funded by the Ministry of Health&Welfare,Republic of Korea(No.HC19C0276).
文摘AIM:To build a functional generalized estimating equation(GEE)model to detect glaucomatous visual field progression and compare the performance of the proposed method with that of commonly employed algorithms.METHODS:Totally 716 eyes of 716 patients with primary open angle glaucoma(POAG)with at least 5 reliable 24-2 test results and 2y of follow-up were selected.The functional GEE model was used to detect perimetric progression in the training dataset(501 eyes).In the testing dataset(215 eyes),progression was evaluated the functional GEE model,mean deviation(MD)and visual field index(VFI)rates of change,Advanced Glaucoma Intervention Study(AGIS)and Collaborative Initial Glaucoma Treatment Study(CIGTS)scores,and pointwise linear regression(PLR).RESULTS:The proposed method showed the highest proportion of eyes detected as progression(54.4%),followed by the VFI rate(34.4%),PLR(23.3%),and MD rate(21.4%).The CIGTS and AGIS scores had a lower proportion of eyes detected as progression(7.9%and 5.1%,respectively).The time to detection of progression was significantly shorter for the proposed method than that of other algorithms(adjusted P≤0.019).The VFI rate displayed moderate pairwise agreement with the proposed method(k=0.47).CONCLUSION:The functional GEE model shows the highest proportion of eyes detected as perimetric progression and the shortest time to detect perimetric progression in patients with POAG.
基金supported by the National Nature Science Foundation of China under Grant No.11902332。
文摘The augmented evolution equation is established under the framework of the Variation Evolving Method(VEM)that seeks optimal solutions by solving the transformed Initial-Value Problems(IVPs).To improve the numerical performance,its compact form is developed herein.Through replacing the states and costates variation evolution with that of the controls,the dimension-reduced Evolution Partial Differential Equation(EPDE)only solves the control variables along the variation time to get the optimal solution,and the initial conditions for the definite solution may be arbitrary.With this equation,the scale of the resulting IVPs,obtained via the semi-discrete method,is significantly reduced and they may be solved with common Ordinary Differential Equation(ODE)integration methods conveniently.Meanwhile,the state and the costate dynamics share consistent stability in the numerical computation and this avoids the intrinsic numerical difficulty as in the indirect methods.Numerical examples are solved and it is shown that the compact form evolution equation outperforms the primary form in the precision,and the efficiency may be higher for the dense discretization.Actually,it is uncovered that the compact form of the augmented evolution equation is a continuous realization of the Newton type iteration mechanism.
