Artificial neural networks for solving ordinary and partialdifferential equations

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE), to systems of coupled ODE and also to partial differential equations (PDE). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galerkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.     

A PDE breach to the SDRE

The state‐dependent Riccati equation (SDRE) is a nonlinear optimal controller derived from applying optimality conditions on a Hamiltonian equation. A co‐state vector is involved in the derivation process. This has been commonly considered a function of time only, despite the existence of states in the co‐state vector. This has resulted in a series of nonlinear coupled ordinary differential equations (ODEs) with a final boundary condition, known as the SDRE. In this work, for the first time, the co‐state vector is regarded as a function of time and states that results in a partial differential equation (PDE) instead of an ODE. The new governing equation is named partial differential state‐dependent Riccati equation (PDSDRE), and the PDE provides a tensor for gain over domains of time and states. Since the generated PDE is highly nonlinear, the solution to the PDSDRE is proposed based on the method of lines (MOL), which is an extension to the finite difference method (FDM). The proposed approach is implemented on both scalar and second order systems and is compared with an SDRE technique to validate the results and show the advantages of proposed structure.     

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1?d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.     

Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems

This paper addresses the problem of feedback control design for a class of linear cascaded ordinary differential equation (ODE)–partial differential equation (PDE) systems via a boundary interconnection, where the ODE system is linear time-invariant and the PDE system is described by an Euler–Bernoulli beam (EBB) equation with variable coefficients. The objective of this paper is to design a static output feedback (SOF) controller via EBB boundary and ODE measurements such that the resulting closed-loop cascaded system is exponentially stable. The Lyapunov’s direct method is employed to derive the stabilization condition for the cascaded ODE–beam system, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, in order to compute the gain matrices of SOF controllers, a two-step procedure is presented to solve the BMI feasibility problem via the existing linear matrix inequality (LMI) optimization techniques. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed design method.     

The coupling of boundary integral and finite element methods for the exterior steady Oseen problem

In this article, we present a new numerical method for solving the steady Oseen equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one. The solution in exterior region is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocity-pressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. Finally, the optimal error estimates of the numerical solution are derived.Computer results will be discussed in a forthcoming paper.     

Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains

The numerical simulation of coupled nonlinear Schrödinger equations on unbounded domains is considered in this paper. By using the operator splitting technique, the original problem is decomposed into linear and nonlinear subproblems in a small time step. The linear subproblem turns out to be two decoupled linear Schrödinger equations on unbounded domains, where artificial boundaries are introduced to truncate the unbounded physical domains into finite ones. Local absorbing boundary conditions are imposed on the artificial boundaries. On the other hand, the coupled nonlinear subproblem is an ODE system, which can be solved exactly. To demonstrate the effectiveness of our method, some comparisons in terms of accuracy and computational cost are made between the PML approach and our method in numerical examples.     

Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains

The numerical simulation of coupled nonlinear Schrödinger equations on unbounded domains is considered in this paper. By using the operator splitting technique, the original problem is decomposed into linear and nonlinear subproblems in a small time step. The linear subproblem turns out to be two decoupled linear Schrödinger equations on unbounded domains, where artificial boundaries are introduced to truncate the unbounded physical domains into finite ones. Local absorbing boundary conditions are imposed on the artificial boundaries. On the other hand, the coupled nonlinear subproblem is an ODE system, which can be solved exactly. To demonstrate the effectiveness of our method, some comparisons in terms of accuracy and computational cost are made between the PML approach and our method in numerical examples.     

Alternating Schwarz methods for partial differential equation-based mesh generation

To solve boundary value problems with moving fronts or sharp variations, moving mesh methods can be used to achieve reasonable solution resolution with a fixed, moderate number of mesh points. Such meshes are obtained by solving a nonlinear elliptic differential equation in the steady case, and a nonlinear parabolic equation in the time-dependent case. To reduce the potential overhead of adaptive partial differential equation-(PDE) based mesh generation, we consider solving the mesh PDE by various alternating Schwarz domain decomposition methods. Convergence results are established for alternating iterations with classical and optimal transmission conditions on an arbitrary number of subdomains. An analysis of a colouring algorithm is given which allows the subdomains to be grouped for parallel computation. A first result is provided for the generation of time-dependent meshes by an alternating Schwarz algorithm on an arbitrary number of subdomains. The paper concludes with numerical experiments illustrating the relative contraction rates of the iterations discussed.     

An adaptive Rothe method for the wave equation

The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved virtually using an adaptive state-of-the-art integrator. The actual realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus perturbing the virtual function space method. The admissible size of that perturbation can be computed a priori and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle. Accepted: 12 August 1997     

Solidification and control of a liquid metal column

In this paper, two different 1D mechanistic models for the solidification of a pure substance are presented. The first model is based on the two-domain approach, resulting in 2 partial differential equations (PDEs) and one ordinary differential equation (ODE) with 2 boundary conditions, 2 interface conditions, and one initial condition: the Stefan problem.In the second model, the metal column is considered as one-domain, and one PDE is valid for the whole domain. The result is one PDE with two boundary conditions.The models are implemented in MATLAB, and the ODE solver ode23s is used for solving the systems of equations. The models are developed in order to simulate and control the dynamic response of the solidification rate. The control scheme is based on a linear PI controller.     

Artificial boundary method for two-dimensional Burgers’ equation

The numerical solution of the two-dimensional Burgers equation in unbounded domains is considered. By introducing a circular artificial boundary, we consider the initial-boundary problem on the disc enclosed by the artificial boundary. Based on the Cole–Hopf transformation and Fourier series expansion, we obtain the exact boundary condition and a series of approximating boundary conditions on the artificial boundary. Then the original problem is reduced to an equivalent problem on the bounded domain. Furthermore, the stability of the reduced problem is obtained. Finally, the finite difference method is applied to the reduced problem, and some numerical examples are given to demonstrate the feasibility and effectiveness of the approach.     

A heat equation for freezing processes with phase change: stability analysis and applications

In this work, the stability properties as well as possible applications of a partial differential equation (PDE) with state-dependent parameters are investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (potential) Burgers’ equation. We show that for certain forms of coefficient functions, the PDE converges to a stationary solution given by (fixed) boundary conditions that make physical sense. These boundary conditions are either symmetric or asymmetric of Dirichlet type. Furthermore, we present an observer design based on the PDE model for estimation of inner-domain temperatures in block-frozen fish and for monitoring freezing time. We illustrate the results with numerical simulations.     

Boundary control for a constrained two-link rigid–flexible manipulator with prescribed performance

In this paper, we consider a boundary control problem for a constrained two-link rigid–flexible manipulator. The nonlinear system is described by hybrid ordinary differential equation–partial differential equation (ODE–PDE) dynamic model. Based on the coupled ODE–PDE model, boundary control is proposed to regulate the joint positions and eliminate the elastic vibration simultaneously. With the help of prescribed performance functions, the tracking error can converge to an arbitrarily small residual set and the convergence rate is no less than a certain pre-specified value. Asymptotic stability of the closed-loop system is rigorously proved by the LaSalle's Invariance Principle extended to infinite-dimensional system. Numerical simulations are provided to demonstrate the effectiveness of the proposed controller.     

Modeling and vibration control for a flexible pendulum inverted system based on a PDE observer

This study addresses the problem of trajectory control of a flexible pendulum inverted system on the basis of the partial differential equation (PDE) and ordinary differential equation (ODE) dynamic model. One of the key contributions of this study is that a new model is proposed to simplify the complex system. In addition, this study proposed a nonlinear PDE observer to estimate distributed positions and velocities along flexible pendulum. Singular perturbation method is proposed to solve the coupling system of nonlinear PDE observer. The nonlinear PDE observer is divided into a fast subsystem and a slow subsystem by the use of the singular perturbation method. To stabilise this fast subsystem, a boundary controller is proposed at the free end of the beam. The sliding-mode control method is proposed to design controller for slow subsystems. The asymptotic stability of both the proposed nonlinear PDE observer and controller is validated by theoretical analysis. The results are illustrated by simulation.     

Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems

This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial solution of differential equation has been used in the regression-based neural network (RBNN) model for single input and single output system. The artificial neural network (ANN) trial solution of ODE is written as sum of two terms, first one satisfies initial/boundary conditions and contains no adjustable parameters. The second part involves a RBNN model containing adjustable parameters. Network has been trained using the initial weights generated by the coefficients of regression fitting. We have used feed-forward neural network and error back propagation algorithm for minimizing error function. Proposed model has been tested for first, second and fourth-order ODEs. We also compare the results of proposed algorithm with the traditional ANN algorithm. The idea may very well be extended to other complicated differential equations.     

The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method

In this paper, we study a nonlinear two-point boundary value problem on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation (ODE) is investigated by means of the radial basis function (RBF) collocation method. The RBF reduces the solution of the above-mentioned problem to the solution of a system of algebraic equations and finds its numerical solution. To examine the accuracy and stability of the approach, we transform the mentioned problem into another nonlinear ODE which simplifies the original problem. The comparisons are made between the results of the present work and the numerical method by shooting method combined with the Runge–Kutta technique. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work.     

Absolute stability via boundary control of a semilinear parabolic PDE

We consider the problem of achieving global absolute stability of an unstable equilibrium solution of a semilinear dissipative parabolic partial differential equation (PDE) through boundary control. The state space of the system is extended in order to write the action of the boundary control as an unbounded operator in an abstract evolution equation. Absolute stability via boundary control is accomplished by analyzing a control Lyapunov function based on the infinite-dimensional dynamics and applying a finite-dimensional linear quadratic regulator (LQR) controller. Sufficient conditions for absolute stability of the infinite-dimensional system are established by the feasibility of two finite-dimensional linear matrix inequalities (LMIs). Numerical results are presented for a Dirichlet boundary controlled system, however the analysis in this work applies to Nuemann and Robin type boundary controllers as well.     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations

In this article, we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations. A series of test examples was solved with these two schemes, different problems with different type of governing equations, and boundary conditions. Particular emphasis was paid to the ability of these schemes to solve the steady-state convection-diffusion equation at high values of the Péclet number. From the examples tested in this work, it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better. However, the unsymmetric method has the advantage of being simpler to implement. Two main features about the results obtained in this work are worthy of special attention: First, with the symmetric method it was possible to solve convection-diffusion problems at a very high Péclet number without the need of any artificial damping term, and second, with these two approaches, symmetric and unsymmetric, it is possible to impose free boundary conditions for problems in unbounded domains.     

Analysis and control of parabolic PDE systems with input constraints

This paper develops a general framework for the analysis and control of parabolic partial differential equations (PDE) systems with input constraints. Initially, Galerkin's method is used for the derivation of ordinary differential equation (ODE) system that capture the dominant dynamics of the PDE system. This ODE systems are then used as the basis for the synthesis, via Lyapunov techniques, of stabilizing bounded nonlinear state and output feedback control laws that provide an explicit characterization of the sets of admissible initial conditions and admissible control actuator locations that can be used to guarantee closed-loop stability in the presence of constraints. Precise conditions that guarantee stability of the constrained closed-loop parabolic PDE system are provided in terms of the separation between the fast and slow eigenmodes of the spatial differential operator. The theoretical results are used to stabilize an unstable steady-state of a diffusion-reaction process using constrained control action.