Many linear-in-parameters models arising in identification and control can be expressed as singlelayer artificial neural networks(ANNs)with linear activation,enabling online learning viafirst-order optimization.In pra...Many linear-in-parameters models arising in identification and control can be expressed as singlelayer artificial neural networks(ANNs)with linear activation,enabling online learning viafirst-order optimization.In practice,however,standard gradient descent ofien exhibits slow convergence,large intermediate weights,and stagnation when the regressor data are ill-conditioned or computations are performed underfinite precision.This paper proposes Gradient Descent with Time-Decaying Regularization(GD-TDR),a training algorithm that augments the quadratic loss with a regularization term whose weight decays exponentially in time.fie proposed schedule enforces uniform strong convexity during early iterations,efiectively mitigating neural-paralysis-like behavior associated withfiat directions,while asymptotically vanishing so that the unregularized least-squares solution is recovered.A convergence theorem for GD-TDR is established and a concise pseudocode implementation is provided.Numerical and embedded experiments on an online identification problem of a Chua-type chaotic oscillator demonstrate that GD-TDR converges faster and avoids stagnation compared to standard gradient descent,without introducing the steady-state bias characteristic offixed quadratic regularization.展开更多
In this note it is shown that the Vlasov-Poisson-Fokker-Planck system in the three-dimensional whole space driven by a time-periodic background profile near a positive constant state admits a time-periodic small-ampli...In this note it is shown that the Vlasov-Poisson-Fokker-Planck system in the three-dimensional whole space driven by a time-periodic background profile near a positive constant state admits a time-periodic small-amplitude solution with the same period. The proof follows by the Serrin＇s method on the basis of the exponential time-decay property of the linearized system in the case of the constant background profile.展开更多
This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaus...This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaussian kernel,the tensor kernel,and the Stokes semigroup,we prove that under the assumption∫0∞∫ΩT[u,p](y,t).νdSydt=0 for the body pressure tensor T[u,p],if u0∈L1(Ω)∩Lσ3(Ω)∩W2/5,5/4(Ω)with‖u0‖3≤ηfor some sufficiently small numberη>0,then rapid time-decay phenomenon of the Navier-Stokes flow appears.If additionally|x|αu0∈Lr0(Ω)for some0<α<1 and 1<r0<(1-α/3)-1 orα=1 and r0=1,then the flow exhibits higher decay rates as t→∞.展开更多
We establish the global well-posedness for the multidimensional chemotaxis model with some classes of large initial data,especially the case when the rate of variation of ln v0(v0 is the chemical concentration)contain...We establish the global well-posedness for the multidimensional chemotaxis model with some classes of large initial data,especially the case when the rate of variation of ln v0(v0 is the chemical concentration)contains high oscillation and the initial density near the equilibrium is allowed to have large oscillation in 3D.Besides,we show the optimal time-decay rates of the strong solutions under an additional perturbation assumption,which include specially the situations of d=2,3 and improve the previous time-decay rates.Our method mainly relies on the introduce of the effective velocity and the application of the localization in Fourier spaces.展开更多
In this paper,we investigate the initial value problem for the two-dimensiona magneto-micropolar fluid equations with partial viscosity.We prove that global existence of smooth large solutions by the energy method.Fur...In this paper,we investigate the initial value problem for the two-dimensiona magneto-micropolar fluid equations with partial viscosity.We prove that global existence of smooth large solutions by the energy method.Furthermore,with aid of the Fourier splitting methods,optimal time-decay rates of global smooth large solutions are also established.展开更多
In this paper, we pay attention to the time-decay rate of the viscous bipolar quantum hydrodynamic （QHD） models for semiconductors. By applying the entropy method, we prove that the solution of the viscous bipolar Q...In this paper, we pay attention to the time-decay rate of the viscous bipolar quantum hydrodynamic （QHD） models for semiconductors. By applying the entropy method, we prove that the solution of the viscous bipolar QHD models tends to the equilibrium state at an exponential decay rate for the multi-dimensional cases. The arguments is based on a series of a priori estimates.展开更多
基金Funding was provided by the IPN-SIP(SIP 20250023,20250424,20251300,20251721,20253411,MULTI-2026-0035)SECIHTI(CF-2023-I-1635)the Sistema Nacional de Investigadores e Investigadoras(SNII)of Mexico.
文摘Many linear-in-parameters models arising in identification and control can be expressed as singlelayer artificial neural networks(ANNs)with linear activation,enabling online learning viafirst-order optimization.In practice,however,standard gradient descent ofien exhibits slow convergence,large intermediate weights,and stagnation when the regressor data are ill-conditioned or computations are performed underfinite precision.This paper proposes Gradient Descent with Time-Decaying Regularization(GD-TDR),a training algorithm that augments the quadratic loss with a regularization term whose weight decays exponentially in time.fie proposed schedule enforces uniform strong convexity during early iterations,efiectively mitigating neural-paralysis-like behavior associated withfiat directions,while asymptotically vanishing so that the unregularized least-squares solution is recovered.A convergence theorem for GD-TDR is established and a concise pseudocode implementation is provided.Numerical and embedded experiments on an online identification problem of a Chua-type chaotic oscillator demonstrate that GD-TDR converges faster and avoids stagnation compared to standard gradient descent,without introducing the steady-state bias characteristic offixed quadratic regularization.
基金supported by the General Research Fund(Project No.409913)from RGC of Hong Kongsupported by grants from the National Natural Science Foundation of China(11101188 and 11271160)
文摘In this note it is shown that the Vlasov-Poisson-Fokker-Planck system in the three-dimensional whole space driven by a time-periodic background profile near a positive constant state admits a time-periodic small-amplitude solution with the same period. The proof follows by the Serrin＇s method on the basis of the exponential time-decay property of the linearized system in the case of the constant background profile.
基金National Natural Science Foundation of China(Grant No.11771223)。
文摘This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaussian kernel,the tensor kernel,and the Stokes semigroup,we prove that under the assumption∫0∞∫ΩT[u,p](y,t).νdSydt=0 for the body pressure tensor T[u,p],if u0∈L1(Ω)∩Lσ3(Ω)∩W2/5,5/4(Ω)with‖u0‖3≤ηfor some sufficiently small numberη>0,then rapid time-decay phenomenon of the Navier-Stokes flow appears.If additionally|x|αu0∈Lr0(Ω)for some0<α<1 and 1<r0<(1-α/3)-1 orα=1 and r0=1,then the flow exhibits higher decay rates as t→∞.
基金Supported by the National Natural Science Foundation of China(Grant No.12071043)the National Key Research and Development Program of China(Grant No.2020YFA0712900)。
文摘We establish the global well-posedness for the multidimensional chemotaxis model with some classes of large initial data,especially the case when the rate of variation of ln v0(v0 is the chemical concentration)contains high oscillation and the initial density near the equilibrium is allowed to have large oscillation in 3D.Besides,we show the optimal time-decay rates of the strong solutions under an additional perturbation assumption,which include specially the situations of d=2,3 and improve the previous time-decay rates.Our method mainly relies on the introduce of the effective velocity and the application of the localization in Fourier spaces.
基金supported in part by the NNSF of China(Grant No.11871212)the Basic Research Project of Key Scientific Research Project Plan of Universities in Henan Province(Grant No.20ZX002).
文摘In this paper,we investigate the initial value problem for the two-dimensiona magneto-micropolar fluid equations with partial viscosity.We prove that global existence of smooth large solutions by the energy method.Furthermore,with aid of the Fourier splitting methods,optimal time-decay rates of global smooth large solutions are also established.
文摘In this paper, we pay attention to the time-decay rate of the viscous bipolar quantum hydrodynamic （QHD） models for semiconductors. By applying the entropy method, we prove that the solution of the viscous bipolar QHD models tends to the equilibrium state at an exponential decay rate for the multi-dimensional cases. The arguments is based on a series of a priori estimates.
