A Novel Method for Solving Ordinary Differential Equations with Artificial Neural Networks

This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.

PROJECTION METHODS AND APPROXIMATIONS FOR ORDINARY DIFFERENTIAL EQUATIONS

A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in the L2 norm and nodal supercnnvergence. These results generalize those obtained earlier by Hulme for continuous piecevjise polynomials and by Delfour-Dubeau for discontinuous pieceuiise polynomials. A duality relationship for the two types of approximations is also given.

An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

This is the first paper on symmetry classification for ordinary differential equations(ODEs)based on Wu’s method.We carry out symmetry classification of two ODEs,named the generalizations of the Kummer-Schwarz equations which involving arbitrary function.First,Lie algorithm is used to give the determining equations of symmetry for the given equations,which involving arbitrary functions.Next,differential form Wu’s method is used to decompose determining equations into a union of a series of zero sets of differential characteristic sets,which are easy to be solved relatively.Each branch of the decomposition yields a class of symmetries and associated parameters.The algorithm makes the classification become direct and systematic.Yuri Dimitrov Bozhkov,and Pammela Ramos da Conceição have used the Lie algorithm to give the symmetry classifications of the equations talked in this paper in 2020.From this paper,we can find that the differential form Wu’s method for symmetry classification of ODEs with arbitrary function(parameter)is effective,and is an alternative method.

An Approach to Parallel Simulation of Ordinary Differential Equations

Cyber-physical systems (CPS) represent a class of complex engineered systems where functionality and behavior emerge through the interaction between the computational and physical domains. Simulation provides design engineers with quick and accurate feedback on the behaviors generated by their designs. However, as systems become more complex, simulating their behaviors becomes computation all complex. But, most modern simulation environments still execute on a single thread, which does not take advantage of the processing power available on modern multi-core CPUs. This paper investigates methods to partition and simulate differential equation-based models of cyber-physical systems using multiple threads on multi-core CPUs that can share data across threads. We describe model partitioning methods using fixed step and variable step numerical in-tegration methods that consider the multi-layer cache structure of these CPUs to avoid simulation performance degradation due to cache conflicts. We study the effectiveness of each parallel simu-lation algorithm by calculating the relative speedup compared to a serial simulation applied to a series of large electric circuit models. We also develop a series of guidelines for maximizing performance when developing parallel simulation software intended for use on multi-core CPUs.

An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models

A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

Local parameter identification with neural ordinary differential equations

The data-driven methods extract the feature information from data to build system models, which enable estimation and identification of the systems and can be utilized for prognosis and health management(PHM). However, most data-driven models are still black-box models that cannot be interpreted. In this study, we use the neural ordinary differential equations(ODEs), especially the inherent computational relationships of a system added to the loss function calculation, to approximate the governing equations. In addition, a new strategy for identifying the local parameters of the system is investigated, which can be utilized for system parameter identification and damage detection. The numerical and experimental examples presented in the paper demonstrate that the strategy has high accuracy and good local parameter identification. Moreover, the proposed method has the advantage of being interpretable. It can directly approximate the underlying governing dynamics and be a worthwhile strategy for system identification and PHM.

Tau-Collocation Approximation Approach for Solving First and Second Order Ordinary Differential Equations

This paper presents Tau-collocation approximation approach for solving first and second orders ordinary differential equations. We use the method in the stimulation of numerical techniques for the approximate solution of linear initial value problems (IVP) in first and second order ordinary differential equations. The resulting numerical evidences show the method is adequate and effective.

Continuous-Time Channel Prediction Based on Tensor Neural Ordinary Differential Equation

Channel prediction is critical to address the channel aging issue in mobile scenarios.Existing channel prediction techniques are mainly designed for discrete channel prediction,which can only predict the future channel in a fixed time slot per frame,while the other intra-frame channels are usually recovered by interpolation.However,these approaches suffer from a serious interpolation loss,especially for mobile millimeter-wave communications.To solve this challenging problem,we propose a tensor neural ordinary differential equation(TN-ODE)based continuous-time channel prediction scheme to realize the direct prediction of intra-frame channels.Specifically,inspired by the recently developed continuous mapping model named neural ODE in the field of machine learning,we first utilize the neural ODE model to predict future continuous-time channels.To improve the channel prediction accuracy and reduce computational complexity,we then propose the TN-ODE scheme to learn the structural characteristics of the high-dimensional channel by low-dimensional learnable transform.Simulation results show that the proposed scheme is able to achieve higher intra-frame channel prediction accuracy than existing schemes.

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

This paper considers the class of autonomous algebraic ordinary differential equations(AODEs)of order one,and studies their Liouvillian general solutions.In particular,let F(y,w)=0 be a rational algebraic curve over C.The authors give necessary and sufficient conditions for the autonomous first-order AODE F(y,y′)=0 to have a Liouvillian solution over C.Moreover,the authors show that a Liouvillian solutionαof this equation is either an algebraic function over C(x)or an algebraic function over C(exp(ax)).As a byproduct,these results lead to an algorithm for determining a Liouvillian general solution of an autonomous AODE of order one of genus zero.Rational parametrizations of rational algebraic curves play an important role on this method.

Solution of Laguerre’s Differential Equations via Modified Adomian Decomposition Method

This paper presents a technique for obtaining an exact solution for the well-known Laguerre’s differential equations that arise in the modeling of several phenomena in quantum mechanics and engineering. We utilize an efficient procedure based on the modified Adomian decomposition method to obtain closed-form solutions of the Laguerre’s and the associated Laguerre’s differential equations. The proposed technique makes sense as the attitudes of the acquired solutions towards the neighboring singular points are correctly taken care of.

ChemNODE: A neural ordinary differential equations framework for efficient chemical kinetic solvers

Solving for detailed chemical kinetics remains one of the major bottlenecks for computational fluid dynamics simulations of reacting flows using a finite-rate-chemistry approach.This has motivated the use of neural networks to predict stiff chemical source terms as functions of the thermochemical state of the combustion system.However,due to the nonlinearities and multi-scale nature of combustion,the predicted solution often diverges from the true solution when these machine learning models are coupled with a computational fluid dynamics solver.This is because these approaches minimize the error during training without guaranteeing successful integration with ordinary differential equation solvers.In the present work,a novel neural ordinary differential equations approach to modeling chemical kinetics,termed as ChemNODE,is developed.In this machine learning framework,the chemical source terms predicted by the neural networks are integrated during training,and by computing the required derivatives,the neural network weights are adjusted accordingly to minimize the difference between the predicted and ground-truth solution.A proof-of-concept study is performed with ChemNODE for homogeneous autoignition of hydrogen-air mixture over a range of composition and thermodynamic conditions.It is shown that ChemNODE accurately captures the chemical kinetic behavior and reproduces the results obtained using the detailed kinetic mechanism at a fraction of the computational cost.

Numerical Method for Singularly Perturbed Third Order Ordinary Differential Equations of Convection-Diffusion Type

In this paper,we have proposed a numerical method for Singularly Perturbed Boundary Value Problems(SPBVPs)of convection-diffusion type of third order Ordinary Differential Equations(ODEs)in which the SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions.The numerical method combines boundary value technique,asymptotic expansion approximation,shooting method and finite difference scheme.In order to get a numerical solution for the derivative of the solution,the domain is divided into two regions namely inner region and outer region.The shooting method is applied to the inner region while standard finite difference scheme(FD)is applied for the outer region.Necessary error estimates are derived for the method.Computational efficiency and accuracy are verified through numerical examples.The method is easy to implement and suitable for parallel computing.

Rational Solutions of High-Order Algebraic Ordinary Differential Equations

This paper considers algebraic ordinary differential equations(AODEs)and study their polynomial and rational solutions.The authors first prove a sufficient condition for the existence of a bound on the degree of the possible polynomial solutions to an AODE.An AODE satisfying this condition is called noncritical.Then the authors prove that some common classes of low-order AODEs are noncritical.For rational solutions,the authors determine a class of AODEs,which are called maximally comparable,such that the possible poles of any rational solutions are recognizable from their coefficients.This generalizes the well-known fact that any pole of rational solutions to a linear ODE is contained in the set of zeros of its leading coefficient.Finally,the authors develop an algorithm to compute all rational solutions of certain maximally comparable AODEs,which is applicable to 78.54%of the AODEs in Kamke's collection of standard differential equations.

A superlinear numerical scheme for multi-term fractional nonlinear ordinary differential equations

In this paper,we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations(MTFNODEs).First,the presented problem is equivalently transformed into its integral form with multi-term Riemann-Liouville integrals.Second,the compound product trapezoidal rule is used to approximate the fractional integrals.Then,the unconditional stability and convergence with the order 1+αN−1−αN−2 of the proposed scheme are strictly established,whereαN−1 andαN−2 are the maximum and the second maximum fractional indexes in the considered MTFNODEs,respectively.Finally,two numerical examples are provided to support the theoretical results.

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial,a variable-order and variable-step-size numerical method for solving differential equations is designed.The stability properties of the formulas are discussed and the stability regions are analyzed.The deduced methods are applied to a simulation problem.The results show that the numerical method can satisfy calculation accuracy,reduce the number of calculation steps and accelerate calculation speed.

Descent of Ordinary Differential Equations with Rational General Solutions

Let F be an irreducible differential polynomial over k(t)with k being an algebraically closed field of characteristic zero.The authors prove that F=0 has rational general solutions if and only if the differential algebraic function field over k(t)associated to F is generated over k(t)by constants,i.e.,the variety defined by F descends to a variety over k.As a consequence,the authors prove that if F is of first order and has movable singularities then F has only finitely many rational solutions.

EXISTENCE OF POSITIVE SOLUTION TO TWO-POINT BOUNDARY VALUE PROBLEM FOR A SYSTEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we consider a two-point boundary value problem for a system of second order ordinary differential equations. Under some conditions, we show the existence of positive solution to the system of second order ordinary differential equa-tions.

Derivation of the Reduction Formula of Sixth Order and Seven Stages Runge-Kutta Method for the Solution of an Ordinary Differential Equation

This paper is describing in detail the way we define the equations which give the formulas in the methods Runge-Kutta 6th order 7 stages with the incorporated control step size in the numerical solution of Ordinary Differential Equations (ODE). The purpose of the present work is to construct a system of nonlinear equations and then by solving this system to find the values of all set parameters and finally the reduction formula of the Runge-Kutta (6,7) method (6th order and 7 stages) for the solution of an Ordinary Differential Equation (ODE). Since the system of high order conditions required to be solved is complicated, all coefficients are found with respect to 7 free parameters. These free parameters, as well as some others in addition, are adjusted in such a way to furnish more efficient R-K methods. We use the MATLAB software to solve several of the created subsystems for the comparison of our results which have been solved analytically. Some examples for five different choices of the arbitrary values of the systems are presented in this paper.

Production of the Reduction Formula of Seventh Order Runge-Kutta Method with Step Size Control of an Ordinary Differential Equation

The purpose of the present work is to construct a nonlinear equation system (85 × 53) using Butcher’s Table and then by solving this system to find the values of all set parameters and finally the reduction formula of the Runge-Kutta (7,9) method (7th order and 9 stages) for the solution of an Ordinary Differential Equation (ODE). Since the system of high order conditions required to be solved is too complicated, we introduce a subsystem from the original system where all coefficients are found with respect to 9 free parameters. These free parameters, as well as some others in addition, are adjusted in such a way to furnish more efficient R-K methods. We use the MATLAB software to solve several of the created subsystems for the comparison of our results which have been solved analytically.