Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions

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.

纳米甲壳素的快速制备及其乳化性能研究

简化磷酸溶解再生制备纳米甲壳素的工艺,避免大量溶剂的使用、缩短制备时间以及提高产率。采用X射线衍射、透射电镜和电感耦合等离子体发射光谱仪对甲壳素及再生纳米甲壳素进行了表征。以辛酸/癸酸甘油三酯(GTCC)作为油相,用纳米甲壳素作为Pickering乳液稳定剂,制得水包油(O/W)型Pickering乳液。通过各类表征证明该方法制备的所有纳米甲壳素均可以乳化GTCC得到稳定的Pickering乳液(纳米甲壳素含量为0.2%)。随后其乳液的流变行为表明纳米甲壳素能很好地吸附在油水界面上,并且稳定Pickering乳液。

纳米甲壳素基Pickering乳液辅助构筑相变微胶囊

相变微胶囊可以通过增加传热面积和防止泄漏来提高相变材料的热性能和机械性能。本文首先通过一种简单、高效的方法制备了纳米甲壳素,然后利用纳米甲壳素稳定石蜡制备Pickering乳液,最后通过乳液模板法得到相变微胶囊。文章着重研究了盐酸汽蒸甲壳素的时间,纳米甲壳素悬浮液的质量分数,核壳比等变量对微胶囊表面形貌和粒径的影响,实验结果表明,当汽蒸时间为5 h,纳米甲壳素悬浮液质量分数为0.5%,核壳比为3∶1时,所制得的相变微胶囊粒径为29.6μm,其熔融焓高达187.4 J/g,包覆率为71.6%。通过热重分析和加热/冷却循环测试对相变微胶囊的热稳定性和热循环性能进行表征,结果表明微胶囊具有良好的热循环稳定性。

Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations

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.

Analysis and Approximation of Gradient Flows Associated with a Fractional Order Gross-Pitaevskii Free Energy

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.

Polyoxometalate-loaded Nanocellulose Sponge as a Novel Catalyst for the Regioselective Hydroboration of Phenylacetylene

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.

高灵敏度,超宽响应范围,环境耐受性高的多模式离子凝胶基传感器

近年来,离子类皮肤传感器因其高性能和良好的兼容性等优点而备受关注.然而,开发一种多功能、稳定、高灵敏度和耐用的离子类皮肤传感器仍面临挑战.本文通过简单的紫外引发聚合制备了具有良好耐用性、环境(抗冻、耐真空)稳定性、离子导电性、自愈性、高粘附性和拉伸性的复合离子凝胶.该离子凝胶可以组装为应变、压力和温度传感器,用于检测外部环境的变化.无论是作为应变传感器还是压力传感器,离子凝胶基传感器都具有高灵敏度(GF=14.7)、宽响应范围(1%-1600%)、快速响应时同(95.8 ms)、优异的稳定性和可重复性(1000次).因此,它不仅可以追踪关节运动,还可以监测细微的表情变化(皱眉).该离子凝胶还可以组装成敲击传感器和高精度书写板传感器,用于信息传递.此外,该传感器对温度变化具有较高的灵敏度,温度感知范围在0-120℃之间,且检测阈值较低(0.1℃).因此,基于离子凝胶的传感器有望应用于多功能电子和传感设备.

Physics-informed neural networks(PINNs)for fluidmechanics:a review

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.

Thermally conductive poly(lactic acid)/boron nitride composites via regenerated cellulose assisted Pickering emulsion approach

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.

Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations

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.

High Order Conservative Finite Difference/Fourier Spectral Methods for Inviscid Surface Quasi-Geostrophic Flows

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.