Problems For Partial Differential Equations Research Articles

Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems

Deep learning has been shown to be an effective tool in solving partial differential equations (PDEs) through physics-informed neural networks (PINNs). PINNs embed the PDE residual into the loss function of the neural network, and have been successfully employed to solve diverse forward and inverse PDE problems. However, one disadvantage of the first generation of PINNs is that they usually have limited accuracy even with many training points. Here, we propose a new method, gradient-enhanced physics-informed neural networks (gPINNs), for improving the accuracy of PINNs. gPINNs leverage gradient information of the PDE residual and embed the gradient into the loss function. We tested gPINNs extensively and demonstrated the effectiveness of gPINNs in both forward and inverse PDE problems. Our numerical results show that gPINN performs better than PINN with fewer training points. Furthermore, we combined gPINN with the method of residual-based adaptive refinement (RAR), a method for improving the distribution of training points adaptively during training, to further improve the performance of gPINN, especially in PDEs with solutions that have steep gradients.

Neumann problem for backward SPDEs with singular terminal conditions and application in constrained stochastic control under target zone

In this paper, a Neumann problem for the backward stochastic partial differential equation (BSPDE) with singular terminal condition is studied, which characterizes the value function for a constrained stochastic control problem (also called optimal liquidation problem) in target zone models. The existence and the uniqueness of strong solutions to such BSPDEs are addressed. Furthermore, the uniqueness and existence of strong solutions to the Neumann problem for general semilinear BSPDEs in finer function spaces, a comparison theorem, and a new link between forward–backward stochastic differential equations and BSPDEs are proved as well. In addition, we apply the strong solution theory to the associated optimal liquidation problems and derive the optimal feedback control.

Difference Approach to Solving Boundary Value Problems for Elliptic Partial Differential Equations in the Sense of Generalized Functions

Difference Approach to Solving Boundary Value Problems for Elliptic Partial Differential Equations in the Sense of Generalized Functions

Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo gH-Type Derivatives

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo gH-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of gH-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and ε-approximate solution of the coupled system. Finally, we propose some work for future research.

Fractional Calculus and Time-Fractional Differential Equations: Revisit and Construction of a Theory

For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the Riemann–Liouville derivatives within Sobolev spaces of fractional orders, including negative ones. Our approach enables a unified treatment for fractional calculus and time-fractional differential equations. We formulate initial value problems for fractional ordinary differential equations and initial boundary value problems for fractional partial differential equations to prove well-posedness and other properties.

DZHUMABAEV PARAMETERIZATION METHOD FOR SOLVING AN INITIAL–BOUNDARY VALUE PROBLEMFOR HIGHER ORDER PARTIAL DIFFERENTIAL EQUATIONS

Weconsideran application ofthe Dzhumabaev parameterization method for solving initial-boundary value problems for higher order partial differential equations with two variables. These problems are reduced to nonlocal problems for system of hyperbolic equations of second order with mixed derivatives, or to the family of boundary value problems for hybrid systems consisting of first order partial differential equations, or systems of ordinary differential equations with a parameter and functional relations. A family of multipoint boundary value problems for higher order differential equations is solved by the Dzhumabaev parameterization method. The methods and results are developed to nonlocal problems for higher order par-tial differential equations with loading and delay arguments, nonlocal problems with integral conditionsand impulse effects for higher order partial differential equations.

Learning functional priors and posteriors from data and physics

We develop a new Bayesian framework based on deep neural networks to be able to extrapolate in space-time using historical data and to quantify uncertainties arising from both noisy and gappy data in physical problems. Specifically, the proposed approach has two stages: (1) prior learning and (2) posterior estimation. At the first stage, we employ the physics-informed Generative Adversarial Networks (PI-GAN) to learn a functional prior either from a prescribed function distribution, e.g., Gaussian process, or from historical data and physics. At the second stage, we employ the Hamiltonian Monte Carlo (HMC) method to estimate the posterior in the latent space of PI-GANs. In addition, we use two different approaches to encode the physics: (1) automatic differentiation, used in the physics-informed neural networks (PINNs) for scenarios with explicitly known partial differential equations (PDEs), and (2) operator regression using the deep operator network (DeepONet) for PDE-agnostic scenarios. We then test the proposed method for (1) meta-learning for one-dimensional regression, and forward/inverse PDE problems (combined with PINNs); (2) PDE-agnostic physical problems (combined with DeepONet), e.g., fractional diffusion as well as saturated stochastic (100-dimensional) flows in heterogeneous porous media; and (3) spatial-temporal regression problems, i.e., inference of a marine riser displacement field using experimental data from the Norwegian Deepwater Programme (NDP). The results demonstrate that the proposed approach can provide accurate predictions as well as uncertainty quantification given very limited scattered and noisy data, since historical data could be available to provide informative priors. In summary, the proposed method is capable of learning flexible functional priors, e.g., both Gaussian and non-Gaussian process, and can be readily extended to big data problems by enabling mini-batch training using stochastic HMC or normalizing flows since the latent space is generally characterized as low dimensional.

Fourier Neural Operator for Solving Subsurface Oil/Water Two-Phase Flow Partial Differential Equation

SummaryWhile deep learning has achieved great success in solving partial differential equations (PDEs) that accurately describe engineering systems, it remains a big challenge to obtain efficient and accurate solutions for complex problems instead of traditional numerical simulation. In the field of reservoir engineering, the current mainstream machine learning methods have been successfully applied. However, these popular methods cannot directly solve the problem of 2D two-phase oil/water PDEs well, which is the core of reservoir numerical simulation. Fourier neural operator (FNO) is a recently proposed high-efficiency PDE solution architecture that overcomes the shortcomings of the above popular methods, which can handle this type of PDE problem well in our work. In this paper, a deep-learning-based model is developed to solve three categories of problems controlled by the subsurface 2D oil/water two-phase flow PDE based on the FNO. For this complex engineering equation, we consider many factors, select characteristic variables, increase the dimension channel, expand the network structure, and realize the solution of the engineering problem. The first category is to predict the distribution of saturation and pressure fields by PDE parameters. The second category is the prediction of time series. The third category is for the inverse problem. It has achieved good results on both forward and inverse problems. The network uses fast Fourier transform (FFT) to extract PDE information in Fourier space to approximate differential operators, making the network faster and with greater physics significance. The model is mesh-independent and has good generalization, which also shows superresolution. Compared to the original FNO, we improve the network structure, add physical constraints to deal with boundary conditions (BCs), and use a shape matrix to control irregular boundaries. Also, we have improved the FFT module to make the transformation smoother. Compared with advanced deep learning-based solvers at different resolutions, the results show that this model overcomes some shortcomings of popular algorithms such as physics-informed neural networks (PINNs) and fully convolutional network (FCN) and has stronger accuracy and applicability. Our work has great potential in the replacement of traditional numerical methods with neural networks for reservoir numerical simulation.

USER-ORIENTED INTELLIGENCE MINING UNDER THE EXISTENCE OF SOLUTIONS TO INTEGRAL BOUNDARY VALUE PROBLEMS FOR FUZZY PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

Functional A Posteriori Error Estimates for the Parabolic Obstacle Problem

Abstract The paper is concerned with functional-type a posteriori estimates for the initial boundary value problem for a parabolic partial differential equation with an obstacle. We deduce a guaranteed and computable bound of the distance between the exact minimizer and any function from the admissible (energy) class of functions. Applications to the analysis of modeling errors caused by data implification are discussed. An important case of time incremental approximations is specially studied. Numerical examples presented in the last section show how the estimates work in practice.

Solvability of Dirichlet Problem For a Fractional Partial Differential equation by using energy inequality and Faedo-Galerkin method

In this paper, we establish sufficient conditions for the existence and uniqueness of the solution for a class of initial-boundary value problems with Dirichlet condition for a class of fractional partial differential equations. The results are established by a method based on a priori estimate "energy inequality" and the Faedo-Galerkin method.

Estimates on the generalization error of physics-informed neural networks for approximating PDEs

Abstract Physics-informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of partial differential equations (PDEs). We provide upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.

A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks

In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original equations as variational problems with the help of semigroup operators and then solve the variational problems with neural network (NN) parameterization. The main advantages are that no mixed second-order derivative computation is needed during the stochastic gradient descent training and that the boundary conditions are taken into account automatically by the semigroup operator. Unlike popular methods like PINN [16] and Deep Ritz [5] where the Dirichlet boundary condition is enforced solely through penalty functions and thus changes the true solution, the proposed method is able to address the boundary conditions without penalty functions and it gives the correct true solution even when penalty functions are added, thanks to the semigroup operator. For eigenvalue problems, a primal-dual method is proposed, efficiently resolving the constraint with a simple scalar dual variable and resulting in a faster algorithm compared with the BSDE solver [8] in certain problems such as the eigenvalue problem associated with the linear Schrödinger operator. Numerical results are provided to demonstrate the performance of the proposed methods.

Learning time-dependent PDEs with a linear and nonlinear separate convolutional neural network

Partial differential equations (PDEs) that describe many physical phenomena are discovered or derived based on professional knowledge or empirical observations. However, the rapid development of machine learning technology interests us to discover the PDEs with a data-driven approach based on neural networks. We introduce the convolution neural network (CNN) to learn the time-dependent PDEs. Instead of customizing a complex regularization term in loss function, we introduce a series of constraints on the structure of neural network model to avoid the overfitting problems. Firstly, since the linear PDEs can be accurately approximated by a linear CNN, we construct a network consisting of a linear CNN and a nonlinear CNN. The linear CNN helps to mitigate the overfitting problems of learning nonlinear PDEs. Secondly, the boundary conditions are hard encoded into the neural networks by custom padding operations. Finally, the time-series data are learned by an auto-regressive framework corresponding with the Euler scheme. We test the proposed framework with a series of PDEs, including heat equation, Burgers equation, reaction-diffusion equation and Kuramoto–Sivashinsky equation. Moreover, we also test the two-dimensional equations, and it shows similar accuracy as one-dimensional case. These numerical results demonstrate that the proposed framework is able to learn the PDEs from few data more accurately and stably.

The effect of a new two-point nonlocal condition on the eigenvalue problem of the difference scheme for an elliptic partial differential equation

In this paper, a new form of two-point nonlocal boundary conditions is presented. This condition generalizes a Dirichlet condition at one boundary and a mixed condition at the other one. The focus of this work is not the numerical solution of the considered problem. Rather, we study the effect of the proposed nonlocal boundary condition on the difference eigenvalue problem for an elliptic partial differential equation in one and two dimensions.

Fourier transform method for partial differential equations: Formulas for representing solutions to the Cauchy problem

The paper proposes a method for solving the Cauchy problem for linear partial differential equations with variable coefficients of a special form, allowing, after applying the (inverse) Fourier transform, to rewrite the original problem as a Cauchy problem for first-order partial differential equations. The resulting problem is solved by the method of characteristics and the (direct) Fourier transform is applied to its solution. And for this it is necessary to know the solution of the Cauchy problem for a first-order equation in the entire domain of definition. This leads to the requirement that the support of the (inverse) Fourier transform of the initial function of the original problem be compact, and to describe the class of initial functions, it is necessary to use Paley -Wiener - Schwarz-type theorems on Fourierimages, including distributions. The presentation of solutions in the form of the Fourier transform of some function (distribution), determined by the initial function, is presented. A general form of the evolutionary equation is written down, which, when the described method is applied, leads to the consideration of a homogeneous first-order equation, and a formula for the solution of the Cauchy problem in this general case is derived. The general form of the equation is written down, which leads to the consideration of a first-order inhomogeneous equation, and a formula for solutions it is derived. Particular cases of these equations are the well-known equations that are encountered in the description of various processes in physics, chemistry, and biology.

Finite line method and its application in coupled heat transfer between fluid-solid domains

In this paper, a completely new numerical method, called finite line method, is proposed and is used to solve fluid-solid coupled heat transfer problems. The extensively used finite element method is a method based on volume discretization; the finite volume method is a method operated on the surface of the control volume; the boundary element method is the one based on boundary surface discretization; the meshless method is the one constructing the computational algorithm using surrounding scatter points at a collocation point. The method proposed in the work is based on the use of finite number of lines, in which an arbitrarily high-order computational scheme can be established by using only two or three straight or curved lines at each point. The creative idea of the method is that by using a directional derivative technique along a line, high-order two- and three-dimensional spatial partial derivatives with respective to the global coordinates can be derived from the Lagrange polynomial interpolation formulation, based on which the discretized system of equations can be directly formed by the problem-governing partial differential equation and relevant boundary conditions. The proposed finite line method is very simple in theory and robust in universality, by using which the boundary value problems of partial differential equations in solid and fluid mechanics problems can be solved in a unified way. In solving fluid mechanics problems, the diffusion term is simulated by using the central line set to maintain a high efficiency, and the convection term is computed by using an upwind line set to embody its directional characteristic. A few of numerical examples will be given in this paper for fluid-solid coupled heat transfer problems for verifying the correctness and efficiency of the proposed method.

NONLOCAL BOUNDARY VALUE PROBLEM IN SPACES OF EXPONENTIAL TYPE OF DIRICHLET-TAYLOR SERIES FOR THE EQUATION WITH COMPLEX DIFFERENTIATION OPERATOR

Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Such problems are mainly ill possed in the Hadamard sence, and their solvability is connected with the problem of small denominators. A specific feature of the present work is the study of a nonlocal boundary-value problem for partial differential equations with the operator of the generalized differentiation $B=zd/dz$, which operate on functions of scalar complex variable $z$. A criterion for the unique solvability of these problems and a sufficient conditions for the existence of its solutions are established in the spaces of functions, which are Dirichlet-Taylor series. The unity theorem and existence theorems of the solution of problem in these spaces are proved. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the article shown that in the case of one variable the corresponding denominators are not small and are estimated from below by some constants. Correctness after Hadamard of the problem is shown. It distinguishes it from an illconditioned after Hadamard problem with many spatial variables.

Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations

Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations

Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations

Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations