近几年来,深度学习无所不在,赋能于各个领域.尤其是人工智能与传统科学的结合(AI for science,AI4Science)引发广泛关注.在AI4Science领域,利用人工智能算法求解PDEs(AI4PDEs)已成为计算力学研究的焦点.AI4PDEs的核心是将数据与方程相融...近几年来,深度学习无所不在,赋能于各个领域.尤其是人工智能与传统科学的结合(AI for science,AI4Science)引发广泛关注.在AI4Science领域,利用人工智能算法求解PDEs(AI4PDEs)已成为计算力学研究的焦点.AI4PDEs的核心是将数据与方程相融合,并且几乎可以求解任何偏微分方程问题,由于其融合数据的优势,相较于传统算法,其计算效率通常提升数万倍.因此,本文全面综述了AI4PDEs的研究,总结了现有AI4PDEs算法、理论,并讨论了其在固体力学中的应用,包括正问题和反问题,展望了未来研究方向,尤其是必然会出现的计算力学大模型.现有AI4PDEs算法包括基于物理信息神经网络(physicsinformed neural network,PINNs)、深度能量法(deep energy methods,DEM)、算子学习(operator learning),以及基于物理神经网络算子(physics-informed neural operator,PINO).AI4PDEs在科学计算中有许多应用,本文聚焦于固体力学,正问题包括线弹性、弹塑性,超弹性、以及断裂力学;反问题包括材料参数,本构,缺陷的识别,以及拓朴优化.AI4PDEs代表了一种全新的科学模拟方法,通过利用大量数据在特定问题上提供近似解,然后根据具体的物理方程进行微调,避免了像传统算法那样从头开始计算,因此AI4PDEs是未来计算力学大模型的雏形,能够大大加速传统数值算法.我们相信,利用人工智能助力科学计算不仅仅是计算领域的未来重要方向,同时也是计算力学的未来,即是智能计算力学。展开更多
The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework i...The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework is the automatic way of getting arbitrarily high order methods,which can be put in the Runge-Kutta(RK)form.The drawback is the larger computational cost with respect to the most used RK methods.To reduce such cost,in an explicit setting,we propose an efcient modifcation:we introduce interpolation processes between the DeC iterations,decreasing the computational cost associated to the low order ones.We provide the Butcher tableaux of the new modifed methods and we study their stability,showing that in some cases the computational advantage does not afect the stability.The fexibility of the novel modifcation allows nontrivial applications to PDEs and construction of adaptive methods.The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.展开更多
This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations(FODEs)which have been widely used in modeling various phenomena in engineering a...This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations(FODEs)which have been widely used in modeling various phenomena in engineering and science.An approximate solution of the system is sought in the formof the finite series over the Müntz polynomials.By using the collocation procedure in the time interval,one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure.This technique also serves as the basis for solving the time-fractional partial differential equations(PDEs).The modified radial basis functions are used for spatial approximation of the solution.The collocation in the solution domain transforms the equation into a system of fractional ordinary differential equations similar to the one mentioned above.Several examples have verified the performance of the proposed novel technique with high accuracy and efficiency.展开更多
In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple...In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.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
In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs wit...In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.展开更多
This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilin...This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilinear elliptic equations. This strategy not only grently reduces the expensive computation, but also is successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems with non-odd nonlinearity on some convex or nonconvex domains. Numerical solutions illustrated by their graphics for visualization will show the efficiency of the approach.展开更多
In this paper the authors consider the summability of formal solutions for some first order singular PDEs with irregular singularity. They prove that in this case the formal solutions will be divergent, but except a e...In this paper the authors consider the summability of formal solutions for some first order singular PDEs with irregular singularity. They prove that in this case the formal solutions will be divergent, but except a enumerable directions, the formal solutions are Borel summable.展开更多
文摘近几年来,深度学习无所不在,赋能于各个领域.尤其是人工智能与传统科学的结合(AI for science,AI4Science)引发广泛关注.在AI4Science领域,利用人工智能算法求解PDEs(AI4PDEs)已成为计算力学研究的焦点.AI4PDEs的核心是将数据与方程相融合,并且几乎可以求解任何偏微分方程问题,由于其融合数据的优势,相较于传统算法,其计算效率通常提升数万倍.因此,本文全面综述了AI4PDEs的研究,总结了现有AI4PDEs算法、理论,并讨论了其在固体力学中的应用,包括正问题和反问题,展望了未来研究方向,尤其是必然会出现的计算力学大模型.现有AI4PDEs算法包括基于物理信息神经网络(physicsinformed neural network,PINNs)、深度能量法(deep energy methods,DEM)、算子学习(operator learning),以及基于物理神经网络算子(physics-informed neural operator,PINO).AI4PDEs在科学计算中有许多应用,本文聚焦于固体力学,正问题包括线弹性、弹塑性,超弹性、以及断裂力学;反问题包括材料参数,本构,缺陷的识别,以及拓朴优化.AI4PDEs代表了一种全新的科学模拟方法,通过利用大量数据在特定问题上提供近似解,然后根据具体的物理方程进行微调,避免了像传统算法那样从头开始计算,因此AI4PDEs是未来计算力学大模型的雏形,能够大大加速传统数值算法.我们相信,利用人工智能助力科学计算不仅仅是计算领域的未来重要方向,同时也是计算力学的未来,即是智能计算力学。
文摘The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework is the automatic way of getting arbitrarily high order methods,which can be put in the Runge-Kutta(RK)form.The drawback is the larger computational cost with respect to the most used RK methods.To reduce such cost,in an explicit setting,we propose an efcient modifcation:we introduce interpolation processes between the DeC iterations,decreasing the computational cost associated to the low order ones.We provide the Butcher tableaux of the new modifed methods and we study their stability,showing that in some cases the computational advantage does not afect the stability.The fexibility of the novel modifcation allows nontrivial applications to PDEs and construction of adaptive methods.The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.
基金funded by the National Key Research and Development Program of China(No.2021YFB2600704)the National Natural Science Foundation of China(No.52171272)the Significant Science and Technology Project of the Ministry of Water Resources of China(No.SKS-2022112).
文摘This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations(FODEs)which have been widely used in modeling various phenomena in engineering and science.An approximate solution of the system is sought in the formof the finite series over the Müntz polynomials.By using the collocation procedure in the time interval,one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure.This technique also serves as the basis for solving the time-fractional partial differential equations(PDEs).The modified radial basis functions are used for spatial approximation of the solution.The collocation in the solution domain transforms the equation into a system of fractional ordinary differential equations similar to the one mentioned above.Several examples have verified the performance of the proposed novel technique with high accuracy and efficiency.
基金supported by the National Natural Science Foundation of China (10571053, 10871066, 10811120282)Programme for New Century Excellent Talents in University(NCET-06-0712)
文摘In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.
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.
基金This work was supported by the Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment,Postdoctoral Startup Fund of Taiyuan University of Science and Technology(20152034)the Natural Science Foundation of Shanxi Province(201701D221135)National College Students Innovation and Entrepreneurship Project(201710109003)and(201610109007).
文摘In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.
基金This research was supported by the National Natural Science Foundation of China (10571053)Scientific Research Fund of Hunan Provincial Education Department (0513039)the Special Funds of State Major Basic Research Projects (G1999032804)
文摘This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilinear elliptic equations. This strategy not only grently reduces the expensive computation, but also is successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems with non-odd nonlinearity on some convex or nonconvex domains. Numerical solutions illustrated by their graphics for visualization will show the efficiency of the approach.
基金supported by the NSFC and the 973 key project of the MOST
文摘In this paper the authors consider the summability of formal solutions for some first order singular PDEs with irregular singularity. They prove that in this case the formal solutions will be divergent, but except a enumerable directions, the formal solutions are Borel summable.
