We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters ...in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss–Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.
•A rigorous and unified framework for solving and learning nonlinear PDEs.•The proposed framework is based on techniques derived from Gaussian process regression and kernel methods.•It is provably convergent and inherits complexity vs accuracy guarantees of state of the art dense kernel matrix solvers.•It is interpretable and amenable to numerical analysis.
The properties of Tzitzéica equations in non-linear optics have been the subject of many recent studies. In this article, a new and effective modification of Kudryashov method is adopted to study ...this class of non-linear evolution equations. As an achievement, new exact traveling wave solutions of Tzitzéica, Dodd-Bullough-Mikhailov (DBM) and Tzitzéica-Dodd-Bullough (TDB) equations are formally extracted. It is believed that the modified Kudryashov method along with the symbolic computation package suggests a promising technique to handle non-linear evolution equations in non-linear optics.
This article describes a numerical method based on the dual reciprocity boundary element method (DRBEM) for solving some well-known nonlinear parabolic partial differential equations (PDEs). The ...equations include the classic and generalized Fisher’s equations, Allen–Cahn equation, Newell–Whithead equation, FitzHugh–Nagumo equation, and generalized FitzHugh–Nagumo equation with time-dependent coefficients. The concept of the dual reciprocity is used to convert the domain integral to the boundary that leads to an integration-free method. We employ the time stepping scheme to approximate the time derivative, and the linear radial basis functions (RBFs) are used as approximate functions in the presented method. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in some numerical test examples. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.
Differential equations with random perturbations are the mathematical models of real-world processes that cannot be described via deterministic laws, and their evolution depends on random factors. ...The modern theory of differential equations with random perturbations is on the edge of two mathematical disciplines: random processes and ordinary differential equations. Consequently, the sources of these methods come both from the theory of random processes and from the classic theory of differential equations.
The present paper aims to define three new notions: Θ e -contraction, a Hardy–Rogers-type Θ -contraction, and an interpolative Θ -contraction in the framework of extended b-metric space. Further, ...some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique.
This book is devoted to impulsive functional differential equations which are a natural generalization of impulsive ordinary differential equations (without delay) and of functional differential ...equations (without impulses). At the present time the qualitative theory of such equations is under rapid development. After a presentation of the fundamental theory of existence, uniqueness and continuability of solutions, a systematic development of stability theory for that class of problems is given which makes the book unique. It addresses to a wide audience such as mathematicians, applied researches and practitioners.
We propose a discretization‐free approach based on the physics‐informed neural network (PINN) method for solving the coupled advection‐dispersion equation (ADE) and Darcy flow equation with ...space‐dependent hydraulic conductivity K(x). In this approach, K(x), hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that K(x) is given by its values on a grid, and we use these values to train the K DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one‐ and two‐dimensional forward ADEs, where its performance for various Péclet numbers (Pe) is compared with the analytical and numerical solutions. We find that the PINN method is accurate with errors of less than 1% and outperforms some conventional discretization‐based methods for large Pe. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50% in the considered cases) the accuracy of the PINN solution of the backward ADE.
Key Points
Physics‐informed neural network (PINN) method is proposed for forward and backward advection‐dispersion equations
The physics‐informed neural network (PINN) method has several advantages over some grid‐based discretization methods for high Péclet number problems
The physics‐informed neural network (PINN) method is accurate for the considered backward advection‐dispersion equations (ADEs) that otherwise must be treated as computationally expensive inverse problems
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.
The conformable fractional derivative was proposed by R. Khalil et al. in 2014, which is natural and obeys the Leibniz rule and chain rule. Based on the properties, a class of time-fractional partial ...differential equations can be reduced into ODEs using traveling wave transformation. Then the simplest equation method is applied to find exact solutions of some time-fractional partial differential equations. The exact solutions (solitary wave solutions, periodic function solutions, rational function solutions) of time-fractional generalized Burgers equation, time-fractional generalized KdV equation, time-fractional generalized Sharma–Tasso–Olver (FSTO) equation and time-fractional fifth-order KdV equation, (3+1)-dimensional time-fractional KdV–Zakharov–Kuznetsov (KdV–ZK) equation are constructed. This method presents a wide applicability to solve some nonlinear time-fractional differential equations with conformable derivative.
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