Methods For Partial Differential Equations Research Articles

Machine-learning-based spectral methods for partial differential equations

Spectral methods are an important part of scientific computing’s arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet) to identify candidate functions on which to expand the solution of PDEs. We have devised an approach that uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions that have the following properties: (1) they constitute a basis, (2) they are orthonormal, and (3) they are hierarchical, i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs. The proposed approach advances the state of the art and versatility of spectral methods and, more generally, promotes the synergy between traditional scientific computing and machine learning.

One-stage explicit trigonometric integrators for effectively solving quasilinear wave equations

In this paper, one-stage explicit trigonometric integrators for solving quasilinear wave equations are formulated and studied. For solving wave equations, we first introduce trigonometric integrators as the semidiscretization in time and then consider a spectral Galerkin method for the discretization in space. We show that one-stage explicit trigonometric integrators in time have second-order convergence and the result is also true for the fully discrete scheme without requiring any CFL-type coupling of the discretization parameters. The results are proved by using energy techniques, which are widely applied in the numerical analysis of methods for partial differential equations.

Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory

We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution to a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.

Space-time error estimates for deep neural network approximations for differential equations

Over the last few years deep artificial neural networks (ANNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of high-dimensional partial differential equations (PDEs) by means of stochastic learning problems involving deep ANNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for ANN approximations for PDEs but do not provide error estimates for the entire space-time error for the considered ANN approximations. It is the subject of the main result of this article to provide space-time error estimates for deep ANN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain ANN calculus and (ii) on ANN approximation results for products of the form [0,T]times mathbb {R}^{d}ni (t,x){kern -.5pt}mapsto {kern -.5pt} tx{kern -.5pt}in {kern -.5pt} mathbb {R}^{d} where T{kern -.5pt}in {kern -.5pt} (0,infty ), d{kern -.5pt}in {kern -.5pt} mathbb {N}, which we both develop within this article.

Stability and convergence of a new hybrid method for fractional partial differential equations

Stability and convergence of a new hybrid method for fractional partial differential equations

Fourier transform method for partial differential equations. Part 2. Existence and uniqueness of solutions to the Cauchy problem for linear equations

The article proposes a method for analyzing the Cauchy problem for a wide class of evolutionary linear partial differential equations with variable coefficients. By applying the (inverse) Fourier transform, the original equation is reduced to an integro-differential equation, which can be considered as an ordinary differential equation in the corresponding Banach space. The selection of this space is carried out in such a way that the principle of contraction mappings can be used. To carry out the corresponding estimates for the operators generated by the transformed equation, we impose the conditions of finiteness in the space variable for the inverse Fourier transform of the coefficients, and the spaces of the coefficients of the original equation are determined from the Paley-Wiener Fourier transform theorems. In this case, the apparatus of the theory of the Bochner integral in pseudo-normed spaces, countably-normed spaces and Sobolev spaces is used. Classes of functions are distinguished in which the existence and uniqueness of solutions are proved. For equations with coefficients with separated variables, exact solutions are obtained in the form of a Fourier transform of finite sums for operator exponentials.

An overview on deep learning-based approximation methods for partial differential equations

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.

Closed-form approximate solutions for stop-loss and Russian options with multiscale stochastic volatility

<abstract><p>In general, derivation of closed-form analytic formulas for the prices of path-dependent exotic options is a challenging task when the underlying asset price model is chosen to be a stochastic volatility model. Pricing stop-loss and Russian options is studied under a multiscale stochastic volatility model in this paper. Both options are commonly perpetual American-style derivatives with a lookback provision. We derive closed-form formulas explicitly for the approximate prices of these two exotic options by using multiscale asymptotic analysis and partial differential equation method. The formulas can be efficiently computed starting with the Black-Scholes option prices. The accuracy of the analytic approximation is verified via Monte-Carlo simulations and the impacts of the multiscale stochastic volatility on the corresponding Black-Scholes option prices are revealed. Also, the performance of the model is compared with that of other models.</p></abstract>

A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations

This paper proposes a new high-resolution fuzzy transform algorithm for solving two-dimensional nonlinear elliptic partial differential equations (PDEs). The underlying new computational method implements the method of so-called approximating fuzzy components, which evaluate the solution values with fourth-order accuracy at internal mesh points. Triangular basic functions and fuzzy components are locally determined by linear combinations of solution values at nine points. Such a scheme connects the proposed method of approximating fuzzy components with the exact values of the solution using a linear system of equations. Compact approximations of high-resolution fuzzy components using nine points give a block tridiagonal Jacobi matrix. Apart from the numerical solution, it is easy to construct closed-form approximate solutions using a 2D spline interpolation polynomial from the available data with fuzzy components. The upper bounds of the approximation errors are estimated, as well as the convergence of the approximating solutions. Simulations with linear and nonlinear elliptical PDEs arising from quantum mechanics and convection-dominated diffusion phenomena are presented to confirm the usefulness of the new scheme and fourth-order convergence. To summarize:•The paper presents a high-resolution numerical method for the two-dimensions elliptic PDEs with nonlinear terms.•The combined effect of fuzzy transform and compact discretizations yields almost fourth-order accuracies to Schro¨dinger equation, convection-diffusion equation, and Burgers equation.•The high-order numerical scheme is computationally efficient and employs minimal data storage.

Conformable finite element method for conformable fractional partial differential equations

<abstract><p>The finite element (FE) method is a widely used numerical technique for approximating solutions to various problems in different fields such as thermal diffusion, mechanics of continuous media, electromagnetism and multi-physics problems. Recently, there has been growing interest among researchers in the application of fractional derivatives. In this paper, we present a generalization of the FE method known as the conformable finite element method, which is specifically designed to solve conformable fractional partial differential equations (CF-PDE). We introduce the basis functions that are used to approximate the solution of CF-PDE and provide error estimation techniques. Furthermore, we provide an illustrative example to demonstrate the effectiveness of the proposed method. This work serves as a starting point for tackling more complex problems involving fractional derivatives.</p></abstract>

Reduced-order modeling using the frequency-domain method for parabolic partial differential equations

<abstract><p>This paper suggests reduced-order modeling using the Galerkin proper orthogonal decomposition (POD) to find approximate solutions for parabolic partial differential equations. We first transform a parabolic partial differential equation to the frequency-dependent elliptic equations using the Fourier integral transform in time. Such a frequency-domain method enables efficiently implementing a parallel computation to approximate the solutions because the frequency-variable elliptic equations have independent frequencies. Then, we introduce reduced-order modeling to determine approximate solutions of the frequency-variable elliptic equations quickly. A set of snapshots consists of the finite element solutions of the frequency-variable elliptic equations with some selected frequencies. The solutions are approximated using the general basis of the high-dimensional finite element space in a Hilbert space. reduced-order modeling employs the Galerkin POD for the snapshot subspace spanned by a set of snapshots. An orthonormal basis for the snapshot space can be easily computed using the spectral decomposition of the correlation matrix of the snapshots. Additionally, using an appropriate low-order basis of the snapshot space allows approximating the solutions of the frequency-variable elliptic equations quickly, where the approximate solutions are used for the inverse Fourier transforms to determine the approximated solutions in the time variable. Several numerical tests based on the finite element method are presented to asses the efficient performances of the suggested approaches.</p></abstract>

Adaptive finite element method for wick stochastic partial differential equations

Adaptive finite element method for wick stochastic partial differential equations

Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations

Abstract This article combines the fractional differential theory with the total variation method and applies it to defining cultural industry image color matching. At the same time, we propose a new image color matching denoising model based on fractional partial differential equations. The model achieves simultaneous denoising in the time direction and the spaceplane. Experiments have proved that the fractional partial differential equations method has more advantages than integer-order partial differential equations in denoising and reducing step effects. It can effectively improve the contrast and clarity of image color matching in the cultural industry.

Domain Structure Formation in Designing of the Opened Informative Measuring Systems

The opened systems possess an increasing significance and possibilities of applying in designing of measuring devices. Now an essentially nonlinear models are used for such systems. The perturbation approach is not enough for these purposes. Models of new types have solutions in a form of soliton or kink and similar objects. The equation of Fisher–Kolmogorov–Petrovskii–Piskunov is one of such equations. This equation is used for description of convection-reaction-diffusion processes. Such processes are used for studying of a self-organisation and formation of a structure in non-equilibrium opened systems. The aim of this work was to construct of a new solution for the modified equation of Fisher–Kolmogorov– Petrovskii–Piskunov in which a space inhomogeneity is accounted.To solve this problem the direct Hirota method for nonlinear partial differential equation is applied.Some modifications into this method were introduced.The new topologically non-trivial solution of the modified Fisher–Kolmogorov–Petrovskii–Piskunov equation is constructed explicitly. This solution has a kink-like form. Some arguments on the stability of such solution are considered.A possibility of domain structure formation in the systems which describe by the Fisher–Kolmogorov– Petrovskii–Piskunov equation is demonstrated.

A generalized midpoint-based boundary value method for unstable partial differential equations

We present a class of Boundary Value Methods (BVMs) that can be applied to semi-stable and unstable Partial Differential Equation (PDE) models. Step-by-step (initial value) methods, such as Runge–Kutta and linear multistep methods have numerical stability regions which do not intersect with certain regions of the complex plane that are significant to the time-integration of unstable PDEs. BVMs, which need extra numerical conditions at the final time, are global methods and are, in some sense, free of such barriers. We discuss BVMs based on generalized midpoint methods, combined with appropriate numerical initial and final conditions. The stability regions of these methods intersect with a significant part of the complex plane. In several numerical experiments we will illustrate the usefulness of such methods when applied to PDE models, such as a dispersive wave equation, a space-fractional PDE and the backward heat equation.

A localized Fourier collocation method for 2D and 3D elliptic partial differential equations: Theory and MATLAB code

AbstractA localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems. The method first discretizes the entire domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated using the pseudo‐spectral Fourier collocation method. The key idea of the present method is to combine the merits of the quick convergence of the pseudo‐spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large‐scale simulations. The present method can be viewed as a competitive alternative for solving numerically large‐scale boundary value problems with complex‐shape geometries. Preliminary numerical experiments involving Poisson, Helmholtz, and modified‐Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.

Learning Groundwater Contaminant Diffusion‐Sorption Processes With a Finite Volume Neural Network

AbstractImproved understanding of complex hydrosystem processes is key to advance water resources research. Nevertheless, the conventional way of modeling these processes suffers from a high conceptual uncertainty, due to almost ubiquitous simplifying assumptions used in model parameterizations/closures. Machine learning (ML) models are considered as a potential alternative, but their generalization abilities remain limited. For example, they normally fail to predict accurately across different boundary conditions. Moreover, as a black box, they do not add to our process understanding or to discover improved parameterizations/closures. To tackle this issue, we propose the hybrid modeling framework FINN (finite volume neural network). It merges existing numerical methods for partial differential equations (PDEs) with the learning abilities of artificial neural networks (ANNs). FINN is applied on discrete control volumes and learns components of the investigated system equations, such as numerical stencils, model parameters, and arbitrary closure/constitutive relations. Consequently, FINN yields highly interpretable results. We demonstrate FINN's potential on a diffusion‐sorption problem in clay. Results on numerically generated data show that FINN outperforms other ML models when tested under modified boundary conditions, and that it can successfully differentiate between the usual, known sorption isotherms. Moreover, we also equip FINN with uncertainty quantification methods to lay open the total uncertainty of scientific learning, and then apply it to a laboratory experiment. The results show that FINN performs better than calibrated PDE‐based models as it is able to flexibly learn and model sorption isotherms without being restricted to choose among available parametric models.

A deep domain decomposition method based on Fourier features

In this paper we present a Fourier feature based deep domain decomposition method (F-D3M) for partial differential equations (PDEs). Currently, deep neural network based methods are actively developed for solving PDEs, but their efficiency can degenerate for problems with high frequency modes. In this new F-D3M strategy, overlapping domain decomposition is conducted for the spatial domain, such that high frequency modes can be reduced to relatively low frequency ones. In each local subdomain, multi Fourier feature networks (MFFNets) are constructed, where efficient boundary and interface treatments are applied for the corresponding loss functions. We present a general mathematical framework of F-D3M, validate its accuracy and demonstrate its efficiency with numerical experiments.

Strong optimal error estimates of discontinuous Galerkin method for multiplicative noise driving nonlinear SPDEs

AbstractThis paper investigates the strong convergence of a fully discrete numerical method for the stochastic partial differential equations driven by multiplicative noise. The fully discrete space–time approximation consists of the symmetric interior penalty discontinuous Galerkin method for the spatial discretization and the implicit Euler method for the temporal discretization. Rather than the usual semi group analysis techniques, in this paper, we present an analysis framework in the variational formulation by introducing new weak variational approximation techniques. Some error estimates in a strong sense are established for the proposed fully discrete scheme. The optimal convergence rates are then obtained in both space and time. Numerical results for the nonlinear stochastic partial differential equations are finally presented to confirm the theoretical results.

A Legendre-spectral method for Hadamard fractional partial differential equations

A Legendre-spectral method for Hadamard fractional partial differential equations