We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this work.In particular,to better capture the sharpness of the ...We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this work.In particular,to better capture the sharpness of the solution,we propose the adaptive sampling methods(ASMs)based on the residual and the gradient of the solution.We first present a residual only-based ASM denoted by ASMⅠ.In this approach,we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains,then we add new residual points in the sub-domain which has the largest mean absolute value of the residual,and those points which have the largest absolute values of the residual in this sub-domain as new residual points.We further develop a second type of ASM(denoted by ASMⅡ)based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the solution.The procedure of ASMⅡis almost the same as that of ASMⅠ,and we add new residual points which have not only large residuals but also large gradients.To demonstrate the effectiveness of the present methods,we use both ASMⅠand ASMⅡto solve a number of PDEs,including the Burger equation,the compressible Euler equation,the Poisson equation over an Lshape domain as well as the high-dimensional Poisson equation.It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASMⅠor ASMⅡ,and both methods deliver much more accurate solutions than the original PINNs with the same number of residual points.Moreover,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠalgorithm.This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution.Furthermore,we also employ the similar adaptive sampling technique for the data points of boundary conditions(BCs)if the sharpness of the solution is near the boundary.The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency,stability,and accuracy.展开更多
Flocking refers to collective behavior of a large number of interacting entities,where the interactions between discrete individuals produce collective motion on the large scale.We employ an agent-based model to descr...Flocking refers to collective behavior of a large number of interacting entities,where the interactions between discrete individuals produce collective motion on the large scale.We employ an agent-based model to describe the microscopic dynamics of each individual in a flock,and use a fractional partial differential equation(fPDE)to model the evolution of macroscopic quantities of interest.The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model.Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics,we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based simulations.We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one-and two-dimensional nonlocal flocking dynamics.In particular,a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual,while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities.The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally.They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization.We show in one-and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method.The proposed method offers new insights into how to scale the discrete agent-based models to the continuum-based PDE models,and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.展开更多
We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross-Pitaevskii free energy functional and some basic properties of the solution.The eq...We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross-Pitaevskii free energy functional and some basic properties of the solution.The equation reduces to the Allen-Cahn or Cahn-Hilliard equations in the case where the mass tends to zero and an integer order derivative is used in the energy.We study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard case.In particular,we show that,analogous to the Cahn-Hilliard case,the solutions consist of regions in which the solution is a piecewise constant(whose value depends on the mass and the fractional order)separated by an interface whose width is independent of the mass and the fractional derivative.However,if the average value of the initial data exceeds some threshold(which we determine explic让ly),then the solution will tend to a single constant steady state.展开更多
In this work,an amino-modified cellulose nanofiber sponge was prepared and used as a support for polyoxometalate(POM)catalysts with a high loading efficiency.Fourier transform infrared spectroscopy,thermogravimetric a...In this work,an amino-modified cellulose nanofiber sponge was prepared and used as a support for polyoxometalate(POM)catalysts with a high loading efficiency.Fourier transform infrared spectroscopy,thermogravimetric analysis,and energy-dispersive X-ray spectroscopy revealed that an Anderson-type POM,(NH4)4[CuMo6O18(OH)6]·5H2O was successfully immobilized on the sponge based on electrostatic interactions.Morphological analysis indicated that the POM-loaded sponge retained its porous structure and that the POM was homogeneously distributed on the sponge walls.The POM-loaded sponge exhibited excellent mechanical properties by recovering 79.9%of its original thickness following a 60%compression strain.The POM-loaded sponge was found to effectively catalyze the hydroboration of phenylacetylenes,yielding excellent conversion and regioselectivity of up to 96%and 99%,respectively.Its catalytic activity remained unchanged after five reuse cycles.These findings represent a scalable strategy for immobilizing POMs on porous supports.展开更多
Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier–Stokes equations(NSE),we still cannot incorporate seamlessly noisy data into existing a...Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier–Stokes equations(NSE),we still cannot incorporate seamlessly noisy data into existing algorithms,mesh-generation is complex,and we cannot tackle high-dimensional problems governed by parametrized NSE.Moreover,solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes.Here,we review flow physics-informed learning,integrating seamlessly data and mathematical models,and implement them using physics-informed neural networks(PINNs).We demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows,supersonic flows,and biomedical flows.展开更多
Polymer-based thermally conductive composites have attracted tremendous interest in thermal management of electronics.However,it remains challenging to achieve high thermal conductivity partly because the difficulty t...Polymer-based thermally conductive composites have attracted tremendous interest in thermal management of electronics.However,it remains challenging to achieve high thermal conductivity partly because the difficulty to obtain favorable distribution and orientation of conductive fillers within the polymer matrix.Herein,networked boron nitride(BN)conductive pathway was realized within the poly(lactic acid)(PLA)matrix,via regenerated cellulose(RC)-assisted assembly of BN on Pickering emulsion interface based on the noncovalent interaction,followed by solvent evaporation and hot-compressing.The strong noncovalent interactions between BN and RC were found critical to enhance the wettability and stability of BN in aqueous media with a lowest mass ratio of 1:40 of RC and BN.The obtained PLA/BN composites feature a thermal conductivity of 1.06 W/(m K)at 28.4 wt%BN loading,representing an enhancement of 430%comparing to neat PLA,and the crystallinity of the composites could increase significantly from11.7%(neat PLA)to 43.7%.This simple,environmentally friendly and effective strategy could be easily extended for effective construction of thermally conductive composites.展开更多
We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically...We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method.We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff problems.Furthermore,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost.Moreover,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.展开更多
In this paper,we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasigeostrophic equations,which include a spectral fractional Lap...In this paper,we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasigeostrophic equations,which include a spectral fractional Laplacian determining the vorticity for the transport velocity of the potential temperature.The fractional Laplacian is approximated by a Fourier-Galerkin spectral method,while the time evolution of the potential temperature is discretized by a high order conservative finite difference scheme.Weighted essentially non-oscillatory(WENO)reconstructions are also considered for comparison.Due to a low regularity of problems involving such a fractional Laplacian,especially in the critical or supercritical regime,directly applying the Fourier spectral method leads to a very oscillatory transport velocity associated with the gradient of the vorticity,e.g.around smooth extrema.Instead of using an artificial filter,we propose to reconstruct the velocity from the vorticity with central difference discretizations.Numerical results are performed to demonstrate the good performance of our proposed approach.展开更多
基金Project supported by the National Key R&D Program of China(No.2022YFA1004504)the National Natural Science Foundation of China(Nos.12171404 and 12201229)the Fundamental Research Funds for Central Universities of China(No.20720210037)。
文摘We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this work.In particular,to better capture the sharpness of the solution,we propose the adaptive sampling methods(ASMs)based on the residual and the gradient of the solution.We first present a residual only-based ASM denoted by ASMⅠ.In this approach,we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains,then we add new residual points in the sub-domain which has the largest mean absolute value of the residual,and those points which have the largest absolute values of the residual in this sub-domain as new residual points.We further develop a second type of ASM(denoted by ASMⅡ)based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the solution.The procedure of ASMⅡis almost the same as that of ASMⅠ,and we add new residual points which have not only large residuals but also large gradients.To demonstrate the effectiveness of the present methods,we use both ASMⅠand ASMⅡto solve a number of PDEs,including the Burger equation,the compressible Euler equation,the Poisson equation over an Lshape domain as well as the high-dimensional Poisson equation.It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASMⅠor ASMⅡ,and both methods deliver much more accurate solutions than the original PINNs with the same number of residual points.Moreover,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠalgorithm.This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution.Furthermore,we also employ the similar adaptive sampling technique for the data points of boundary conditions(BCs)if the sharpness of the solution is near the boundary.The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency,stability,and accuracy.
文摘Flocking refers to collective behavior of a large number of interacting entities,where the interactions between discrete individuals produce collective motion on the large scale.We employ an agent-based model to describe the microscopic dynamics of each individual in a flock,and use a fractional partial differential equation(fPDE)to model the evolution of macroscopic quantities of interest.The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model.Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics,we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based simulations.We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one-and two-dimensional nonlocal flocking dynamics.In particular,a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual,while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities.The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally.They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization.We show in one-and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method.The proposed method offers new insights into how to scale the discrete agent-based models to the continuum-based PDE models,and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.
文摘We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross-Pitaevskii free energy functional and some basic properties of the solution.The equation reduces to the Allen-Cahn or Cahn-Hilliard equations in the case where the mass tends to zero and an integer order derivative is used in the energy.We study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard case.In particular,we show that,analogous to the Cahn-Hilliard case,the solutions consist of regions in which the solution is a piecewise constant(whose value depends on the mass and the fractional order)separated by an interface whose width is independent of the mass and the fractional derivative.However,if the average value of the initial data exceeds some threshold(which we determine explic让ly),then the solution will tend to a single constant steady state.
基金financially supported by the Fundamental Research Funds for the Central Universities(No.2232018A3-04,No.2232018-02,and No.2232018G-043)the Program of Introducing Talents of Discipline to Universities(No.105-07-005735)
文摘In this work,an amino-modified cellulose nanofiber sponge was prepared and used as a support for polyoxometalate(POM)catalysts with a high loading efficiency.Fourier transform infrared spectroscopy,thermogravimetric analysis,and energy-dispersive X-ray spectroscopy revealed that an Anderson-type POM,(NH4)4[CuMo6O18(OH)6]·5H2O was successfully immobilized on the sponge based on electrostatic interactions.Morphological analysis indicated that the POM-loaded sponge retained its porous structure and that the POM was homogeneously distributed on the sponge walls.The POM-loaded sponge exhibited excellent mechanical properties by recovering 79.9%of its original thickness following a 60%compression strain.The POM-loaded sponge was found to effectively catalyze the hydroboration of phenylacetylenes,yielding excellent conversion and regioselectivity of up to 96%and 99%,respectively.Its catalytic activity remained unchanged after five reuse cycles.These findings represent a scalable strategy for immobilizing POMs on porous supports.
基金The research of the second author(ZM)was sup-539 ported by the National Natural Science Foundation of China(Grant 54012171404)The last author(GEK)would like to acknowledge support 541 by the Alexander von Humboldt fellowship.
文摘Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier–Stokes equations(NSE),we still cannot incorporate seamlessly noisy data into existing algorithms,mesh-generation is complex,and we cannot tackle high-dimensional problems governed by parametrized NSE.Moreover,solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes.Here,we review flow physics-informed learning,integrating seamlessly data and mathematical models,and implement them using physics-informed neural networks(PINNs).We demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows,supersonic flows,and biomedical flows.
基金supported by the One Belt and One Road Innovative Talent Exchange Program for Foreign Experts[Grant No.DL20200009005]the Fundamental Research Funds for the Central Universities[Grant No.2232021G-02]Fundamental Research Funds for the Central Universities[Grant No.2232020G-04]。
文摘Polymer-based thermally conductive composites have attracted tremendous interest in thermal management of electronics.However,it remains challenging to achieve high thermal conductivity partly because the difficulty to obtain favorable distribution and orientation of conductive fillers within the polymer matrix.Herein,networked boron nitride(BN)conductive pathway was realized within the poly(lactic acid)(PLA)matrix,via regenerated cellulose(RC)-assisted assembly of BN on Pickering emulsion interface based on the noncovalent interaction,followed by solvent evaporation and hot-compressing.The strong noncovalent interactions between BN and RC were found critical to enhance the wettability and stability of BN in aqueous media with a lowest mass ratio of 1:40 of RC and BN.The obtained PLA/BN composites feature a thermal conductivity of 1.06 W/(m K)at 28.4 wt%BN loading,representing an enhancement of 430%comparing to neat PLA,and the crystallinity of the composites could increase significantly from11.7%(neat PLA)to 43.7%.This simple,environmentally friendly and effective strategy could be easily extended for effective construction of thermally conductive composites.
基金Z.Mao was supported by the Fundamental Research Funds for the Central Universities(Grant 20720210037)G.E.Karniadakis was supported by the MURI/ARO on Fractional PDEs for Conservation Laws and Beyond:Theory,Numerics and Applications(Grant W911NF-15-1-0562)X.Chen was supported by the Fujian Provincial Natural Science Foundation of China(Grants 2022J01338,2020J01703).
文摘We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method.We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff problems.Furthermore,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost.Moreover,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.
基金partially supported by NSFC grant No.11971025,NSF grant of Fujian Province No.2019J06002the Fundamental Research Funds for the Central Universities(20720210037).
文摘In this paper,we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasigeostrophic equations,which include a spectral fractional Laplacian determining the vorticity for the transport velocity of the potential temperature.The fractional Laplacian is approximated by a Fourier-Galerkin spectral method,while the time evolution of the potential temperature is discretized by a high order conservative finite difference scheme.Weighted essentially non-oscillatory(WENO)reconstructions are also considered for comparison.Due to a low regularity of problems involving such a fractional Laplacian,especially in the critical or supercritical regime,directly applying the Fourier spectral method leads to a very oscillatory transport velocity associated with the gradient of the vorticity,e.g.around smooth extrema.Instead of using an artificial filter,we propose to reconstruct the velocity from the vorticity with central difference discretizations.Numerical results are performed to demonstrate the good performance of our proposed approach.
