General Partial Differential Equations Research Articles

Solving high-dimensional partial differential equations using deep learning

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality." This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.

Rapid Modeling and Parameter Estimation of Partial Differential Algebraic Equations by a Functional Spreadsheet Paradigm

We present a systematic spreadsheet method for modeling and optimizing general partial differential algebraic equations (PDAE). The method exploits a pure spreadsheet PDAE solver function design that encapsulates the Method of Lines and permits seamless integration with an Excel spreadsheet nonlinear programming solver. Two alternative least-square dynamical minimization schemes are devised and demonstrated on a complex parameterized PDAE system with discontinues properties and coupled time derivatives. Applying the method involves no more than defining a few formulas that closely parallel the original mathematical equations, without any programming skills. It offers a simpler alternative to more complex environments which require nontrivial programming skill and effort.

An effective numerical algorithm for intra-granular fission gas release during non-equilibrium trapping and resolution

Fission gas release and gaseous swelling in nuclear fuel are driven by the transport of fission gas from within the fuel grains to grain boundaries (intra-granular fission gas release). The process involves gas atom diffusion in conjunction with trapping in and resolution from intra-granular bubbles, and is described mathematically by a system of two partial differential equations (PDE). Under the assumption of equilibrium between trapping and resolution (quasi-stationary approximation) the system can be reduced to a single diffusion equation with an effective diffusion coefficient. Numerical solutions used in engineering fuel performance calculations invariably rely on this simplification. First, we investigate the validity of the quasi-stationary approximation compared to the solution of the general system of PDEs. Results demonstrate that the approximation is valid under most conditions of practical interest, but is inadequate to describe intra-granular fission gas release during rapid transients to relatively high temperatures such as postulated reactivity-initiated accidents (RIA). Then, we develop a novel numerical algorithm for the solution of the general PDE system in time-varying conditions. We verify the PolyPole-2 algorithm against a reference finite difference solution for a large number of randomly generated operation histories including prototypical RIA transients. Results demonstrate that PolyPole-2 captures the solution of the general system with a high accuracy and a low computational cost. The PolyPole-2 algorithm overcomes the quasi-stationary approximation and the concept of an effective diffusion coefficient for the solution of the intra-granular fission gas release problem in nuclear fuel analysis.

A general memristor-based partial differential equation solver

Memristive devices have been extensively studied for data-intensive tasks such as artificial neural networks. These types of computing tasks are considered to be ‘soft’ as they can tolerate low computing precision without suffering from performance degradation. However, ‘hard’ computing tasks, which require high precision and accurate solutions, dominate many applications and are difficult to implement with memristors because the devices normally offer low native precision and suffer from high device variability. Here we report a complete memristor-based hardware and software system that can perform high-precision computing tasks, making memristor-based in-memory computing approaches attractive for general high-performance computing environments. We experimentally implement a numerical partial differential equation solver using a tantalum oxide memristor crossbar system, which we use to solve static and time-evolving problems. We also illustrate the practical capabilities of our memristive hardware by using it to simulate an argon plasma reactor. A memristor-based hardware and software system that uses a tantalum oxide memristor crossbar can be used to solve static and time-evolving partial differential equations at high precision, and to simulate an argon plasma reactor.

A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are $L^2$ stable even without interior penalty. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal $L^2$ error estimate of $O(h^{k+1})$ for polynomials of degree $k$ for semi-discrete DG schemes, and the $L^2$ error of $O(h^{k+1} +(\Delta t)^2)$ for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift-Hohenberg equation endowed with a decay free energy is presented.

Solutions of Some Linear Fractional Partial Differential Equations in Mathematical Physics

In this article, we use double Laplace transform method to find solution of general linear fractional partial differential equation in terms of Mittag-Leffler function subject to the initial and boundary conditions. The efficiency of the method is illustrated by considering fractional wave and diffusion equations, Klein-Gordon equation, Burger’s equation, Fokker-Planck equation, KdV equation, and KdV-Burger’s equation of mathematical physics.

Local improvements in numerical forecasts of relative humidity using polynomial solutions of general differential equations

Large‐scale forecast models are based on the numerical integration of differential equation systems, which can describe atmospheric processes in light of global meteorological observations. Mesoscale forecast systems need to define the initial and lateral boundary conditions, which may be carried out with robust global numerical models. Their overall solutions are able to describe the dynamic weather system on the Earth scale using a large number of complete globe 3D matrix variables in several atmospheric layers. Post‐processing methods using local measurements were developed in order to clarify surface weather details and adapt numerical weather prediction model outputs for local conditions. Differential polynomial network is a new type of neural network that can model local weather using spatial data observations to process forecasts of the input variables and revise the target 24 h prognosis. It defines and solves general partial differential equations, being able to describe unknown dynamic systems. The proposed forecast correction method uses a differential network to estimate the optimal numbers of training days and form derivative prediction models. It can improve final numerical forecasts, processed with additional data analysis and statistical techniques, in the majority of cases. The two‐stage procedure presented is analogous to the perfect‐prognosis method using real observations to derive a model, which is applied to the forecasts of the predictors to calculate output predictions.

Adaptive vertex-centered finite volume methods for general second-order linear elliptic partial differential equations

We prove optimal convergence rates for the discretization of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54 (2016), pp. 2228--2255] was restricted to symmetric problems, the present analysis also covers non-symmetric problems and hence the important case of present convection.

Numerical Investigation on Magnetohydrodynamics (MHD) Free Convection Fluid Flow over a Vertical Porous Plate with induced Magnetic Field

In this paper, investigate a two dimensional unsteady Magneto hydro dynamics (MHD) free convection flow of viscous incompressible and electrically conducting fluid flow past an vertical plate in the presence of Grashof Number, Modified Grashof Number, Prandtl Number, Schamidt Number as well as Dufour effects. The governing equations of the problem contain a system of non-linear partial differential equations; have been transformed into a set of coupled non-linear ordinary differential equations which is solved numerically by applying well known explicit finite difference method. The Finite-difference method is an enormously used technique to investigate of the general non linear partial differential equation. Partial differential equations occur in many branches of applied mathematics for example, in hydrodynamics, elasticity, quantum mechanics. Hence, the proposed study is to investigate the numerical results which are performed for various physical parameters such as velocity profiles, temperature distribution and concentration profiles within the boundary layer are separately discussed in graphically.

A General Model for Estimating Macroevolutionary Landscapes.

The evolution of quantitative characters over long timescales is often studied using stochastic diffusion models. The current toolbox available to students of macroevolution is however limited to two main models: Brownian motion and the Ornstein-Uhlenbeck process, plus some of their extensions. Here, we present a very general model for inferring the dynamics of quantitative characters evolving under both random diffusion and deterministic forces of any possible shape and strength, which can accommodate interesting evolutionary scenarios like directional trends, disruptive selection, or macroevolutionary landscapes with multiple peaks. This model is based on a general partial differential equation widely used in statistical mechanics: the Fokker-Planck equation, also known in population genetics as the Kolmogorov forward equation. We thus call the model FPK, for Fokker-Planck-Kolmogorov. We first explain how this model can be used to describe macroevolutionary landscapes over which quantitative traits evolve and, more importantly, we detail how it can be fitted to empirical data. Using simulations, we show that the model has good behavior both in terms of discrimination from alternative models and in terms of parameter inference. We provide R code to fit the model to empirical data using either maximum-likelihood or Bayesian estimation, and illustrate the use of this code with two empirical examples of body mass evolution in mammals. FPK should greatly expand the set of macroevolutionary scenarios that can be studied since it opens the way to estimating macroevolutionary landscapes of any conceivable shape. [Adaptation; bounds; diffusion; FPK model; macroevolution; maximum-likelihood estimation; MCMC methods; phylogenetic comparative data; selection.].

A dynamic subgrid scale model for Large Eddy Simulations based on the Mori–Zwanzig formalism

The development of reduced models for complex multiscale problems remains one of the principal challenges in computational physics. The optimal prediction framework of Chorin et al. [1], which is a reformulation of the Mori–Zwanzig (M–Z) formalism of non-equilibrium statistical mechanics, provides a framework for the development of mathematically-derived reduced models of dynamical systems. Several promising models have emerged from the optimal prediction community and have found application in molecular dynamics and turbulent flows. In this work, a new M–Z-based closure model that addresses some of the deficiencies of existing methods is developed. The model is constructed by exploiting similarities between two levels of coarse-graining via the Germano identity of fluid mechanics and by assuming that memory effects have a finite temporal support. The appeal of the proposed model, which will be referred to as the ‘dynamic-MZ-τ’ model, is that it is parameter-free and has a structural form imposed by the mathematics of the coarse-graining process (rather than the phenomenological assumptions made by the modeler, such as in classical subgrid scale models). To promote the applicability of M–Z models in general, two procedures are presented to compute the resulting model form, helping to bypass the tedious error-prone algebra that has proven to be a hindrance to the construction of M–Z-based models for complex dynamical systems. While the new formulation is applicable to the solution of general partial differential equations, demonstrations are presented in the context of Large Eddy Simulation closures for the Burgers equation, decaying homogeneous turbulence, and turbulent channel flow. The performance of the model and validity of the underlying assumptions are investigated in detail.

A robust higher order compact scheme for solving general second order partial differential equation with derivative source terms on nonuniform curvilinear meshes

A fourth order compact finite difference scheme is proposed for solving general second order steady partial differential equation (PDE) in two-dimension (2D) on geometries having nonuniform curvilinear grids. In this work, the main efforts are focused not only on nonorthogonal curvilinear grids but also on the presence of mixed derivative term and nonhomogeneous derivative source terms in the governing equation. This is in turn suitable for solving fluid flow and heat transfer problems governed by Navier–Stokes(N–S) equations on geometries having nonuniform and nonorthogonal curvilinear grids. The newly proposed scheme has been applied to solve general second order partial differential equation having analytical solution and some pertinent fluid flow problems, namely, viscous flows in a lid driven cavity such as trapezoidal cavity using nonorthogonal grid, square cavity using distorted grid, complicated enclosures using curvilinear grid, and mixed convection flow in a bottom wavy wall cavity. It is seen to efficiently capture steady-state solutions of the N–S equations with Dirichlet as well as Neumann boundary conditions. Detailed comparison data produced by the proposed scheme for all the test cases are provided and compared with existing analytical as well as established numerical results available in the literature. Excellent comparison is obtained in all the cases.

More New Exact Solutions for Bose Einstein Condensate in Weak Bias Magnetic and Time Dependent Laser Field

With the aid of Mathematica software, in this paper, based on Wu computation algorithms, we offer a direct and systematic extended function transform method of solving general variable coefficient nonlinear partial differential equation or equations. By applying this effectiveness method, more new exact solutions are successfully obtained for variable coefficient nonlinear schrǒdinger equation which described Bose Einstein Condensate in weak bias magnetic and time-dependent laser field.

English

English

Synchronization and Impulsive Control of Some Parabolic Partial Differential Equations

Novel equi-attractivity in large generalized non-linear partial differential equations were performed for the impulsive control of spatiotemporal chaotic. Attractive solutions of these general partial differential equations were determined. A proof for existence of a certain kind of impulses for synchronization such that the small error dynamics that is equi-attractive in the large is established. A comparative study between these general non-linear partial differential equations and the existent reported numerical theoretical models was developed. Several boundary conditions were given to confirm the theoretical results of the general non-linear partial differential equations. Moreover, the equations were applied to Kuramoto–Sivashinsky PDE′s equation; Grey–Scott models, and Lyapunov exponents for stabilization of the large chaotic systems with elimination of the dynamic error.

Error Estimates for Semi-Discrete and Fully Discrete Galerkin Finite Element Approximations of the General Linear Second-Order Hyperbolic Equation

ABSTRACTIn this article, we derive error estimates for the semi-discrete and fully discrete Galerkin approximations of a general linear second-order hyperbolic partial differential equation with general damping (which includes boundary damping). The results can be applied to a variety of cases (e.g. vibrating systems of linked elastic bodies). The results generalize pioneering work of Dupont and complement a recent article by Basson and Van Rensburg.

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

A novel parameterized non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) has been developed. This P-NIROM is a generic and efficient approach for model reduction of parameterized partial differential equations (P-PDEs). Over existing parameterized reduced order models (P-ROM) (most of them are based on the reduced basis method), it is non-intrusive and independent on partial differential equations and computational codes. During the training process, the Smolyak sparse grid method is used to select a set of parameters over a specific parameterized space (Ωp∈RP). For each selected parameter, the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model. More generally, the snapshots and basis function sets for any parameters over Ωp can be obtained using an interpolation method. The P-NIROM can then be constructed by using our recently developed technique (Xiao et al., 2015 [41,42]) where either the Smolyak or radial basis function (RBF) methods are used to generate a set of hyper-surfaces representing the underlying dynamical system over the reduced space.The new P-NIROM technique has been applied to parameterized Navier–Stokes equations and implemented with an unstructured mesh finite element model. The capability of this P-NIROM has been illustrated numerically by two test cases: flow past a cylinder and lock exchange case. The prediction capabilities of the P-NIROM have been evaluated by varying the viscosity, initial and boundary conditions. The results show that this P-NIROM has captured the quasi-totality of the details of the flow with CPU speedup of three orders of magnitude. An error analysis for the P-NIROM has been carried out.

Implementation of the Multiscale Stochastic Finite Element Method on Elliptic PDE Problems

In this study, a multi-scale finite element method was proposed to solve two linear scale-coupling stochastic elliptic PDE problems, a tightly stretched wire and flow through porous media. At microscopic level, the main idea was to form coarse-scale equations with a prescribed analytic form that may differ from the underlying fine-scale equations. The relevant stochastic homogenization theory was proposed to model the effective global material coefficient matrix. At the macroscopic level, the Karhunen–Loeve decomposition was coupled with a Polynomial Chaos expansion in conjunction with a Galerkin projection to achieve an efficient implementation of the randomness into the solution procedure. Various stochastic methods were used to plug the microscopic cell to the global system. Strategy and relevant algorithms were developed to boost computational efficiency and to break the curse of dimension. The results of numerical examples were shown consistent with ones from literature. It indicates that the proposed numerical method can act as a paradigm for general stochastic partial differential equations involving multi-scale stochastic data. After some modification, the proposed numerical method could be extended to diverse scientific disciplines such as geophysics, material science, biological systems, chemical physics, oceanography, and astrophysics, etc.

MODELLING THE PHOTOVOLTAIC OUTPUT POWER USING THE DIFFERENTIAL POLYNOMIAL NETWORK AND EVOLUTIONARY FUZZY RULES

The unstable production of renewable energy sources, which is difficult to model using conventional computational techniques, may be predicted to advantage by means of biologically inspired soft-computing methods. The photovoltaic output power is primarily dependent on the solar direct or global radiation, which short-term numerical forecasts are possible to apply for daily power predictions. The study compares two methods, which can successfully model dynamic fluctuant variances of the solar irradiance and corresponding output power time-series. Differential polynomial network is a new neural network class, which defines and substitutes for the general partial differential equation to model an unknown system function. Its total output is composed from selected neurons, i.e. relative polynomial substitution terms, formed in all network layers of a multi-layer structure. The proposed derivative polynomial regression using relative dimensionless fraction units, formed according to the Similarity analysis, can describe and generalize data relations on a wider range of values than defined by the training interval when using standard soft-computing composing techniques that apply only absolute data. 1-variable time-series observations are possible to model by time derivatives of a converted ordinary differential equation, solved analogously with partial derivative substitution terms of several time-point variables.

Explicit Bézier control net of a PDE surface

The PDE under study here is a general fourth-order linear elliptic Partial Differential Equation. Having prescribed the boundary control points, we provide the explicit expression of the whole control net of the associated PDE Bézier surface.In other words, we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points. The set of scalar coefficients of these combinations works like a mould for PDE surfaces. Thus, once this mould has been computed for a given degree, real-time manipulation of the resulting surfaces becomes possible by modifying the prescribed information.