Coefficients Of Partial Differential Equations Research Articles

An Eulerian–Lagrangian method of fundamental solutions for the advection–diffusion equation with time dependent coefficients

This paper investigates the application of the Eulerian–Lagrangian method of fundamental solutions (ELMFS), a truly meshless method that couples the Euler–Lagrange Method (ELM) with the time-dependent method of fundamental solution (MFS). It is applied to solve the transient advection–diffusion equation with transport coefficients exhibiting constant and temporally exponential behaviors. The stability and accuracy of the proposed method were analyzed in terms of condition numbers, location of source points, and error metrics. The stability results for constant coefficients of partial differential equations are presented in a map and compared with well-established stability criteria for explicit in-time mesh methods, such as the Courant–Friedrichs–Lewy and Von Neumann criteria. These comparisons demonstrate that ELMFS has larger stable regions.

A hierarchically normalized physics-informed neural network for solving differential equations: Application for solid mechanics problems

Physics-informed neural networks (PINNs) usually confront significant difficulties to accurately solve partial differential equations (PDEs) due to many pathologies caused by gradient failures during training process. In this paper, a gradient disease regarding the ill-conditioned loss function of the PINN methodology is intensively investigated in both theoretical and experimental aspects. Particularly, the regular PINN approaches can obtain good predictions for trivial problems but may fail to learn solutions when their PDE coefficients and/or physical domain sizes vary. Besides, the influences of neural network structure to the PINN training quality are inferred from the theoretical analyses and also confirmed by relating experiments, in which using large-width multi-layer perceptrons (MLPs) is proved to be beneficial in stabilizing PINN training process. To overcome the abovementioned restrictions, a novel PINN methodology using hierarchically normalized technique (hnPINN) motivated from the developed theory is devised. The key idea of the proposed hnPINN is, on the one hand, to transform the original PDE system into one of two proposed dimensionless forms to alleviate the negative effects of PDE coefficients and domain size; on the other hand, to use secondary output scalers to flexibly calibrate the gradient flow for training effectively and improving the solution preciseness. The determination of the secondary output scaler is formulated by a heuristic framework inspired from theoretical analyses on the hnPINN gradient flow. The obtained results from some typical PDEs and common problems in solid mechanics strongly confirm the high effectiveness of the hnPINN in terms of solution accuracy, superior convergence and performance stability in comparison to those of the vanilla PINN and the non-dimensionalized PINN (ndPINN). As a preprocessing procedure, the hnPINN method is independent on network architectures and also greatly potential to combine with other state-of-the-art PINN models for effectively solving challenged issues in practice.

Unified finite-volume physics informed neural networks to solve the heterogeneous partial differential equations

The emerging physics informed neural network (PINN) has been recently applied to a wide range of mathematical problems. It is promising to precisely solve the partial differential equations (PDEs) in a fast, flexible manner. Whereas, PINN struggles with poor accuracy and costly computation in case of heterogeneous PDE coefficients. To mitigate these issues, a new PINN, which is known as the unified finite volume PINN (UFV-PINN), is proposed to unify the sub-domain decomposition, finite volume discretization, and conventional numerical solvers. The output by neural network (NN) over the boundaries of agglomerated sub-domains functions as boundary conditions (BCs) for UFV-PINN training. In this connection, the customized differentiable conventional numerical solver further solves the PDEs. The discrepancy between NN prediction and the conventional numerical solution within the sub-domains is taken as the novel training loss, enforcing the conservation law of PDE. For illustration, the Poisson and advection–diffusion equations (ADE) are solved, which are classical but still challenging to PINN in the presence of heterogeneity. Numerical experiments are conducted to compare the performance of the proposed UFV-PINN and the standard PINN, both in terms of accuracy and efficiency. Results indicate that UFV-PINN attains remarkable accuracy improvement with less computation time. Hessian spectrum analysis indicates that the loss Hessian matrix of UFV-PINN is more inclined to be positive definite, highlighting its superior performance over the standard PINN. It is the first time that the FV numerical solver is seamlessly embedded into the PINN training for performance improvement.

A Benchmark for the Bayesian Inversion of Coefficients in Partial Differential Equations

A Benchmark for the Bayesian Inversion of Coefficients in Partial Differential Equations

Hyperparameter selection for physics-informed neural networks (PINNs) – Application to discontinuous heat conduction problems

In recent years, physics-informed neural networks (PINNs) have emerged as an alternative to conventional numerical techniques to solve forward and inverse problems involving partial differential equations (PDEs). Despite its success in problems with smooth solutions, implementing PINNs for problems with discontinuous boundary conditions (BCs) or discontinuous PDE coefficients is a challenge. The accuracy of the predicted solution is contingent upon the selection of appropriate hyperparameters. In this work, we performed hyperparameter optimization of PINNs to find the optimal neural network architecture, number of hidden layers, learning rate, and activation function for heat conduction problems with a discontinuous solution. Our aim was to obtain all the settings that achieve a relative L2 error of 10% or less across all the test cases. Results from five different heat conduction problems show that the optimized hyperparameters produce a mean relative L2 error of 5.60%.

Helmholtz-equation solution in nonsmooth media by a physics-informed neural network incorporating quadratic terms and a perfectly matching layer condition

Frequency-domain simulation of seismic waves plays an important role in seismic inversion, but it remains challenging in large models. The recently proposed physics-informed neural network (PINN), as an effective deep-learning method, has achieved successful applications in solving a wide range of partial differential equations (PDEs), and there is still room for improvement on this front. For example, PINN can lead to inaccurate solutions when PDE coefficients are nonsmooth and describe structurally complex media. Thus, we solve the acoustic and visco-acoustic scattered-field (Lippmann-Schwinger) wave equation in the frequency domain with PINN instead of the wave equation to remove the source singularity. We first illustrate that nonsmooth velocity models lead to inaccurate wavefields when no boundary conditions are implemented in the loss function. Then, we add the perfectly matched layer (PML) conditions in the loss function to better couple the real and imaginary parts of the wavefield. Moreover, we design new neurons by replacing the classical affine function with a quadratic function in the argument of the activation function to better capture the nonsmooth features of the wavefields. We find that the PML condition and the quadratic functions improve the results including handling attenuation and discuss the reason for this improvement. We also illustrate that a network trained to predict a wavefield for a specific medium can be used as an initial model of the neural network for predicting other wavefields corresponding to PDE-coefficient alterations and improving the convergence speed accordingly. This pretraining strategy should find applications in iterative full-waveform inversion and time-lag target-oriented imaging when the model perturbation between two consecutive iterations or two consecutive experiments is small.

An improved sparse identification of nonlinear dynamics with Akaike information criterion and group sparsity.

A crucial challenge encountered in diverse areas of engineering applications involves speculating the governing equations based upon partial observations. On this basis, a variant of the sparse identification of nonlinear dynamics (SINDy) algorithm is developed. First, the Akaike information criterion (AIC) is integrated to enforce model selection by hierarchically ranking the most informative model from several manageable candidate models. This integration avoids restricting the number of candidate models, which is a disadvantage of the traditional methods for model selection. The subsequent procedure expands the structure of dynamics from ordinary differential equations (ODEs) to partial differential equations (PDEs), while group sparsity is employed to identify the nonconstant coefficients of partial differential equations. Of practical consideration within an integrated frame is data processing, which tends to treat noise separate from signals and tends to parametrize the noise probability distribution. In particular, the coefficients of a species of canonical ODEs and PDEs, such as the Van der Pol, Rössler, Burgers’ and Kuramoto–Sivashinsky equations, can be identified correctly with the introduction of noise. Furthermore, except for normal noise, the proposed approach is able to capture the distribution of uniform noise. In accordance with the results of the experiments, the computational speed is markedly advanced and possesses robustness.

A Combination of Deep Neural Networks and Physics to Solve the Inverse Problem of Burger's Equation.

One of the most basic nonlinear Partial Differential Equations (PDEs) to model the effects of propagation and diffusion is Burger's equation. This puts great emphasize on seeking efficient versatile methods for finding a solution to the forward and inverse problems of this equation. The focus of this paper is to introduce a method for solving the inverse problem of Burger's equation using neural networks. With recent advances in the area of deep learning, a Physics-Informed Neural Network (PINN) is a category of neural networks that proved efficient for handling PDEs. In our work, the 1D and 2D Burger's equations are simulated by applying a PINN to a set of domain points. The training process of PINNs is governed by the PDE formula, the initial conditions (ICs), the Boundary Conditions (BCs), and the loss minimization algorithm. After training the network to predict the coefficients of the nonlinear PDE, the inverse problem of the 1D and 2D Burger's equations are solved with an error as low as 0.047 and 0.2 for 1D and 2D case studies, respectively. The wave propagation model is accomplished with an approximate training loss value of 1×e-4. The utilization of PINNs for modeling Burger's equation is a mesh-free approach that competes with the commonly used numerical methods as it overcomes the curse of dimensionality. Training the PINN model to predict the propagation and diffusion effects can also be generalized to address further detailed applications of Burger's equation with complex domains. This contributes to clinical applications such as ultrasound therapeutics.

Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

The matched interface and boundary method (MIB) and ghost fluid method (GFM) are two well-known methods for solving elliptic interface problems. Moreover, they can be coupled with efficient time advancing methods, such as the alternating direction implicit (ADI) methods, for solving time-dependent partial differential equations (PDEs) with interfaces. However, to our best knowledge, all existing interface ADI methods for solving parabolic interface problems concern only constant coefficient PDEs, and no efficient and accurate ADI method has been developed for variable coefficient PDEs. In this work, we propose to incorporate the MIB and GFM in the framework of the ADI methods for generalized methods to solve two-dimensional parabolic interface problems with variable coefficients. Various numerical tests are conducted to investigate the accuracy, efficiency, and stability of the proposed methods. Both the semi-implicit MIB-ADI and fully-implicit GFM-ADI methods can recover the accuracy reduction near interfaces while maintaining the ADI efficiency. In summary, the GFM-ADI is found to be more stable as a fully-implicit time integration method, while the MIB-ADI is found to be more accurate with higher spatial and temporal convergence rates.

On elliptic partial differential equations in bioimpedance

This paper deals with mathematical models in electrical bioimpedance fields that are described by means of elliptic partial differential equations (PDEs). To find solutions that have practical significance and value, it is necessary to gain a deep understanding of the underlying physical phenomena with the parameter details of PDE models as well as the data acquisition systems. This paper describes practical details of boundary conditions and effective coefficients of elliptic PDEs occurring in bioimpedance.

Determination of Time-Dependent Coefficients in Time-Fractional Diffusion Equations by Variational Iteration Method

Inverse problems of determining time-dependent coefficients in partial differential equations are difficult to deal with in general cases. The variational iteration method is introduced to determine the time-dependent coefficient in the fractional diffusion equation as well as the solution of the forward problem. By utilizing the additional condition and the property of the fractional derivative, an expression of the unknown is derived by which a nonlinear dynamical differential equation is obtained. The variational iteration method is applied to solve the nonlinear system and the time-dependent coefficient can be reconstructed in a semi-analytical form. Such method can give explicit expression of the solution in the meaning of approximation, or exact solution to the inverse problem in some cases. Several examples are presented to demonstrate feasibility and effectiveness of the proposed method for inverse time-dependent coefficient problems in the fractional diffusion equations.

Determination of Time-Dependent Coefficients in Time-Fractional Diffusion Equations by Variational Iteration Method

Inverse problems of determining time-dependent coefficients in partial differential equations are difficult to deal with in general cases. The variational iteration method is introduced to determine the time-dependent coefficient in the fractional diffusion equation as well as the solution of the forward problem. By utilizing the additional condition and the property of the fractional derivative, an expression of the unknown is derived by which a nonlinear dynamical differential equation is obtained. The variational iteration method is applied to solve the nonlinear system and the time-dependent coefficient can be reconstructed in a semi-analytical form. Such method can give explicit expression of the solution in the meaning of approximation, or exact solution to the inverse problem in some cases. Several examples are presented to demonstrate feasibility and effectiveness of the proposed method for inverse time-dependent coefficient problems in the fractional diffusion equations.

Spatial localization of the error of constitutive law for the identification of defects in elastic bodies

The error on constitutive law (ECL) is a cost functional currently used in inverse problems for identifying interior distribution of coefficients of partial differential equations from overspecified boundary conditions. In previous works, different authors have shown that the ECL permits a good spatial localization of the perturbations of the coefficients. The purpose of this paper is to justify this spatial localization property. The result is obtained for the elliptic equations of elasticity using boundary integral representations of the solutions and comparing the linear and the perturbed solution of the problem.

Extending the unified transform: curvilinear polygons and variable coefficient PDEs

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs

Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-to-state stability (ISS) properties and finite gains cannot be established with respect to disturbances at the boundary. This intuition has been reinforced by many unsuccessful attempts, as well as by the success in establishing ISS only with respect to the derivative of the disturbance. Contrary to this intuition, we establish such a result for parabolic PDEs. Our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time-varying subsets of the state space. The obtained results are used for the comparison of the gain coefficients of transport PDEs with respect to inlet disturbances and for the establishment of the ISS property with respect to control actuator errors for parabolic systems under boundary feedback control.

A preliminary study on the meshless local exponential basis functions method for nonlinear and variable coefficient PDEs

Purpose The purpose of this paper is to extend the meshless local exponential basis functions (MLEBF) method to the case of nonlinear and linear, variable coefficient partial differential equations (PDEs).

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.

Spatially Partitioned Embedded Runge--Kutta Methods

We study spatially partitioned embedded Runge--Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory.

An Algebraic Multigrid Method with Guaranteed Convergence Rate

We consider the iterative solution of large sparse symmetric positive definite linear systems. We present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric M-matrices with nonnegative row sum. The coarsening is based on the aggregation of the unknowns. A key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two-grid convergence rate is bounded by a user-defined parameter. For a sensible choice of this parameter, it is shown that the recursive use of the two-grid procedure yields a convergence independent of the number of levels, provided that one uses a proper AMLI-cycle. On the other hand, the computational cost per iteration step is of optimal order if the mean aggregate size is large enough. This cannot be guaranteed in all cases but is analytically shown to hold for the model Poisson problem. For more general problems, a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method. The experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations (PDEs). The set includes two- and three-dimensional problems, with both structured and unstructured grids, some of them with local refinement and/or reentering corner, and possible jumps or anisotropies in the PDE coefficients.

Exponential basis functions in solution of static and time harmonic elastic problems in a meshless style

AbstractIn this paper, exponential basis functions (EBFs) are used in a boundary collocation style to solve engineering problems whose governing partial differential equations (PDEs) are of constant coefficient type. Complex‐valued exponents are considered for the EBFs. Two‐dimensional elasto‐static and time harmonic elasto‐dynamic problems are chosen in this paper. The solution procedure begins with first finding a set of appropriate EBFs and then considering the solution as a summation of such EBFs with unknown coefficients. The unknown coefficients are determined by the satisfaction of the boundary conditions through a collocation method with the aid of a consistent and complex discrete transformation technique. The basis and various forms of the transformation have been addressed and discussed. We shall propose several strategies for selection of EBFs with the aid of the basis explained for the transformation. While using the transformation, the number of EBFs should not necessarily be equal to (or less than) the number of boundary information data. A library of EBFs has also been presented for further use. The effect of body forces is included in the solution via construction of particular solution by the use of the discrete transformation and another series of EBFs. A number of sample problems are solved to demonstrate the capabilities of the method. It has been shown that the time harmonic problems with high wave number can be solved without much effort. The method, categorized in meshless methods, can be applied to many other problems in engineering mechanics and general physics since EBFs can easily be found for almost all problems with constant coefficient PDEs. Copyright © 2009 John Wiley & Sons, Ltd.