Solution Of Time-dependent Differential Equations Research Articles

Machine learning for fast and reliable solution of time-dependent differential equations

We propose a data-driven Model Order Reduction (MOR) technique, based on Artificial Neural Networks (ANNs), applicable to dynamical systems arising from Ordinary Differential Equations (ODEs) or time-dependent Partial Differential Equations (PDEs). Unlike model-based approaches, the proposed approach is non-intrusive since it just requires a collection of input-output pairs generated through the high-fidelity (HF) ODE or PDE model. We formulate our model reduction problem as a maximum-likelihood problem, in which we look for the model that minimizes, in a class of candidate models, the error on the available input-output pairs. Specifically, we represent candidate models by means of ANNs, which we train to learn the dynamics of the HF model from the training input-output data. We prove that ANN models are able to approximate every time-dependent model described by ODEs with any desired level of accuracy. We test the proposed technique on different problems, including the model reduction of two large-scale models. Two of the HF systems of ODEs here considered stem from the spatial discretization of a parabolic and an hyperbolic PDE respectively, which sheds light on a promising field of application of the proposed technique.

High-order numerical solution of time-dependent differential equations with quasi-interpolation

Numerical solution of time-dependent differential equations with quasi-interpolation usually uses derivatives of a quasi-interpolant directly to approximate corresponding spatial derivatives (called the direct technique) in equations. This in turn requires that the quasi-interpolant should possess high-order derivatives for solving high-order differential equations. In addition, the resulting numerical solution usually gives a lower approximation accuracy. To circumvent these limitations, the paper proposes a new scheme for getting high-order numerical solutions of time-dependent differential equations based on quasi-interpolation. The scheme uses an iterated technique instead of the direct quasi-interpolation technique for approximating spatial derivatives. Moreover, it requires only computing the first-order derivative of the involved quasi-interpolant. Numerical examples of solving several benchmark equations using the proposed scheme are provided at the end of the paper to demonstrate these features vividly.

A generalized conjugate model for forced convection drying based on an evaporative kinetics

A model describing the heat and mass transfer involved in food drying is presented. The aim is to determine the effect of air temperature on the performance of the drying process applied to fresh-cut vegetable slices, but other effects can be easily incorporated in the model. The model allows to disregard one of the most limiting parameters in such modeling, i.e. the average heat and mass transfer coefficients at the food/drying substrate interface, which are generally taken from the literature. Such assumptions are limiting in the sense that they are referred to average transfer conditions and general geometries. The presented model relies upon a finite-element solution of time-dependent differential equations for simultaneous and conjugate heat and moisture transfer in a two-dimensional domain, without any inference in such empiricism. A special formulation for drying kinetic of the substrate is also exploited, and a treatment of the dependence of the properties upon the residual moisture content is included. After proper validation with the available experimental measurements, the numerical solution is discussed by presenting each involved field variables, emphasizing on the conjugate nature of the drying process. Due to its flexibility and generality, the model can be used in common industrial driers’ optimization, even in the assumption of a laminar flow field.

Spectral element method in time for rapidly actuated systems

In this paper, the spectral element (SE) method is applied in time to find the entire time-periodic or transient solution of time-dependent differential equations. The time-periodic solution is com...

Spectral element method in time for rapidly actuated systems

In this paper, the spectral element (SE) method is applied in time to find the entire time-periodic or transient solution of time-dependent differential equations. The time-periodic solution is computed by enforcing periodicity of the element set. Of particular interest are periodic forcing functions possessing high frequency content. To maintain the spectral accuracy for such forcing functions, an h-refinement scheme is employed near the semi-discontinuity without increasing the number of degrees of freedom.Time discretization by spectral elements is applied initially to a standard form of a set of linear, first-order differential equations subject to harmonic excitation and an excitation admitting rapid variation. Other case studies include the application of the SE approach to parabolic and hyperbolic partial differential equations. The first-order form of these equations is obtained through semi-discretization using conventional finite-element, spectral element and finite-difference schemes. Element clustering (h-refinement) is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation. The convergence in time of the method is demonstrated. In some cases, machine precision is obtained with 25 degrees of freedom per cycle. Finally the method is applied to a weakly nonlinear problem with time-periodic solution to demonstrate its future applicability to the analysis of limit-cycle oscillations in aeroelastic systems.

Optimal Prediction for Hamiltonian Partial Differential Equations

Optimal prediction methods compensate for a lack of resolution in the numerical solution of time-dependent differential equations through the use of prior statistical information. We present a new derivation of the basic methodology, show that field-theoretical perturbation theory provides a useful device for dealing with quasi-linear problems, and provide a nonlinear example that illuminates the difference between a pseudo-spectral method and an optimal prediction method with Fourier kernels. Along the way, we explain the differences and similarities between optimal prediction, the representer method in data assimilation, and duality methods for finding weak solutions. We also discuss the conditions under which a simple implementation of the optimal prediction method can be expected to perform well.

Solution of Time-Dependent Diffusion Equations with Variable Coefficients Using Multiwavelets

A new numerical algorithm is developed for the solution of time-dependent differential equations of diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities, and general boundary conditions. For space discretization we use the multiwavelet bases introduced by Alpert (1993,SIAM J. Math. Anal.24, 246–262), and then applied to the representation of differential operators and functions of operators presented by Alpert, Beylkin, and Vozovoi (Representation of operators in the multiwavelet basis, in preparation). An important advantage of multiwavelet basis functions is the fact that they are supported only on non-overlapping subdomains. Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of Hesthaven and Gottlieb (1996,SIAM J. Sci. Comput., 579–612) which can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem. This penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains. Consequently the matrices representing the differential operators (on the finest scale) have block-diagonal structure. For a fixed order of multiwavelets (i.e., a fixed number of vanishing moments) the computational complexity of the present algorithm is proportional to the number of subdomains. The time discretization method of Beylkin, Keiser, and Vozovoi (1998, PAM Report 347) is used in view of its favorable stability properties. Numerical results are presented for evolution equations with variable coefficients in one and two dimensions.

A Two-Dimensional Mesh Refinement Method for Problems with One-Dimensional Singularities

This paper introduces a method for resolving internal layers that can occur in the solutions of time-dependent differential equations in two space dimensions. Singular features in these solutions that are essentially one-dimensional in nature but are not oriented with the computational mesh are resolved using one-dimensional mesh refinement techniques with a procedure that is similar to an ADI method. A careful interpolation procedure assures that the resolution obtained in each ADI step is not lost in the succeeding ADI step.