scholarly journals Transfer learning for deep neural network-based partial differential equations solving

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Xinhai Chen ◽  
Chunye Gong ◽  
Qian Wan ◽  
Liang Deng ◽  
Yunbo Wan ◽  
...  
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Transfer Learning ◽  
Stokes Equations ◽  
Surrogate Models ◽  
Training Data ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations

AbstractDeep neural networks (DNNs) have recently shown great potential in solving partial differential equations (PDEs). The success of neural network-based surrogate models is attributed to their ability to learn a rich set of solution-related features. However, learning DNNs usually involves tedious training iterations to converge and requires a very large number of training data, which hinders the application of these models to complex physical contexts. To address this problem, we propose to apply the transfer learning approach to DNN-based PDE solving tasks. In our work, we create pairs of transfer experiments on Helmholtz and Navier-Stokes equations by constructing subtasks with different source terms and Reynolds numbers. We also conduct a series of experiments to investigate the degree of generality of the features between different equations. Our results demonstrate that despite differences in underlying PDE systems, the transfer methodology can lead to a significant improvement in the accuracy of the predicted solutions and achieve a maximum performance boost of 97.3% on widely used surrogate models.

scholarly journals Wick–Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulations

2013 ◽  
Vol 469 (2158) ◽  
pp. 20130001 ◽  
Author(s):  
D. Venturi ◽  
X. Wan ◽  
R. Mikulevicius ◽  
B. L. Rozovskii ◽  
G. E. Karniadakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Stokes Equations ◽  
Wick Product ◽  
Navier Stokes ◽  
Quantum Field ◽  
Partial Differential ◽  
Stochastic Approximations ◽  
Navier Stokes Equations

Approximating nonlinearities in stochastic partial differential equations (SPDEs) via the Wick product has often been used in quantum field theory and stochastic analysis. The main benefit is simplification of the equations but at the expense of introducing modelling errors. In this paper, we study the accuracy and computational efficiency of Wick-type approximations to SPDEs and demonstrate that the Wick propagator, i.e. the system of equations for the coefficients of the polynomial chaos expansion of the solution, has a sparse lower triangular structure that is seemingly universal, i.e. independent of the type of noise. We also introduce new higher-order stochastic approximations via Wick–Malliavin series expansions for Gaussian and uniformly distributed noises, and demonstrate convergence as the number of expansion terms increases. Our results are for diffusion, Burgers and Navier–Stokes equations, but the same approach can be readily adopted for other nonlinear SPDEs and more general noises.

Symmetry-invariant conservation laws of partial differential equations

2017 ◽  
Vol 29 (1) ◽  
pp. 78-117 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
ABDUL H. KARA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations ◽  
General Method

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, theb-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

Exact Solutions of the Unsteady Navier-Stokes Equations

1989 ◽  
Vol 42 (11S) ◽  
pp. S269-S282 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations

The unsteady Navier-Stokes equations are a set of nonlinear partial differential equations with very few exact solutions. This paper attempts to classify and review the existing unsteady exact solutions. There are three main categories: parallel, concentric and related solutions, Beltrami and related solutions, and similarity solutions. Physically significant examples are emphasized.

Uniqueness of the exact solutions of the Navier—Stokes equations having null nonlinearity

2006 ◽  
Vol 136 (6) ◽  
pp. 1303-1315 ◽  
Author(s):  
Sun-Chul Kim ◽  
Hisashi Okamoto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Euler Equations ◽  
Stokes Equations ◽  
Elliptic Partial Differential Equations ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Overdetermined System ◽  
Navier Stokes Equations

We consider an overdetermined system of elliptic partial differential equations arising in the Navier–Stokes equations. This analysis enables us to prove that the well-known classical solutions such as Couette flows and others are the only solutions that satisfy both the stationary Navier–Stokes and Euler equations.

scholarly journals On combining the techniques for convergence acceleration of iteration processes during the numerical solution of Navier-Stokes equations

2017 ◽  
pp. 80-102
Author(s):  
Е.В. Ворожцов ◽  
В.П. Шапеев
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Stokes Equations ◽  
Iteration Process ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Combined Use ◽  
Iteration Processes

Рассматривается проблема ускорения итерационного процесса численного решения методом коллокаций и наименьших невязок (КНН) краевых задач для уравнений с частными производными. Для ее решения в методе КНН предложено применять одновременно три способа ускорения итерационного процесса: предобуславливатель, многосеточный алгоритм и метод Крылова. Исследован двухпараметрический предобуславливатель. Предложено находить оптимальные значения его параметров путем численного решения относительно нетрудоемкой задачи минимизации числа обусловленности системы линейных алгебраических уравнений приближенной задачи. Использование найденного предобуславливателя существенно ускоряет итерационный процесс. Исследовано влияние на итерационный процесс всех трех способов его ускорения: каждого по отдельности, а также при их комбинированном применении. При этом наибольший вклад дает применение алгоритма, использующего подпространства Крылова. Комбинированное применение одновременно всех трех способов ускорения итерационного процесса решения краевых задач для двумерных уравнений Навье-Стокса уменьшило время их решения на компьютере до 160 раз по сравнению со случаем, когда ни один из них не применялся. Предложенная комбинация способов ускорения итерационных процессов может быть реализована также в рамках применения других численных методов решения уравнений с частными производными. The problem of accelerating the iteration process of the numerical solution of boundary value problems for partial differential equations by the method of collocations and least residuals (CLR) is considered. In the CLR method it is proposed to simultaneously apply three techniques for accelerating the iteration process: a preconditioner, a multigrid algorithm, and the Krylov method. A two-parameter preconditioner is studied. It is proposed to find the optimal values of its parameters by the numerical solution of a relatively computationally inexpensive problem of minimizing the condition number of the system of linear algebraic equations for the approximate problem. The use of the found preconditioner substantially speeds up the iteration process. The individual effect of each technique as well as the effect of their combined use on accelerating the entire iteration process acceleration are analyzed. The application of the algorithm based on the Krylov subspaces gives the most significant contribution. A simultaneous combined use of all the three techniques for accelerating the iteration process of solving the boundary value problems for the two-dimensional Navier-Stokes equations reduces the CPU time of their solution by a factor of up to 160 compared to the case when no such technique is applied. The proposed combination of the above techniques for accelerating the iteration processes may also be implemented in the framework of other numerical methods for solving the partial differential equations.

scholarly journals Exact Solutions to the Navier–Stokes Equations with Couple Stresses

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1355
Author(s):  
Evgenii S. Baranovskii ◽  
Natalya V. Burmasheva ◽  
Evgenii Yu. Prosviryakov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Navier Stokes ◽  
Shear Stresses ◽  
Overdetermined System ◽  
Navier Stokes Equations

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

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