To combine a feedforward neural network (FNN) and Lie group (symmetry) theory of differential equations (DEs), an alternative artificial NN approach is proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs). Introducing the Lie group expressions of the solution, the trial solution of ODEs is split into two parts. The first part is a solution of other ODEs with initial values of original IVP. This is easily solved using the Lie group and known symbolic or numerical methods without any network parameters (weights and biases). The second part consists of an FNN with adjustable parameters. This is trained using the error back propagation method by minimizing an error (loss) function and updating the parameters. The method significantly reduces the number of the trainable parameters and can more quickly and accurately learn the real solution, compared to the existing similar methods. The numerical method is applied to several cases, including physical oscillation problems. The results have been graphically represented, and some conclusions have been made.
University of Tehran（德黑兰大学）；Utrecht University（乌得勒支大学）；Eindhoven University of Technology（埃因霍温理工大学）
The main aim of this study is to introduce a 2-layered Artificial Neural Network (ANN) for solving the Black-Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of Ordinary Differential Equations (ODE). Then each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine t…
Many eigenvalue problems arising in practice are often of the generalized form $A\x=λB\x$. One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than…
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the other to gradient matching. All methods have greater accuracy than previously suggested Gaussian Process approaches. We also suggest a general approach that can yield error estimates from any standard ODE solver.
A new method for solving non-autonomous ordinary differential equations is proposed, the method achieves spectral accuracy. It is based on a new result which expresses the solution of such ODEs as an element in the so called $\star$-algebra. This algebra is equipped with a product, the $\star$-product, which is the integral over the usual product of two bivariate distributions. Expanding the bivariate distributions in bases of Legendre polynomials leads to a discretization of the $\star$-product and this allows for the solution to be approximated by a vector that is obtained by solving a linea…
There exists a huge number of numerical methods that iteratively construct approximations to the solution $y(x)$ of an ordinary differential equation (ODE) $y'(x)=f(x,y)$ starting from an initial value $y_0=y(x_0)$ and using a finite approximation step $h$ that influences the accuracy of the obtained approximation. In this paper, a new framework for solving ODEs is presented for a new kind of a computer -- the Infinity Computer (it has been patented and its working prototype exists). The new computer is able to work numerically with finite, infinite, and infinitesimal numbers giving so the pos…
In this paper we consider utilizing a residual neural network (ResNet) to solve ordinary differential equations. Stochastic gradient descent method is applied to obtain the optimal parameter set of weights and biases of the network. We apply forward Euler, Runge-Kutta2 and Runge-Kutta4 finite difference methods to generate three sets of targets training the ResNet and carry out the target study. The well trained ResNet behaves just as its counterpart of the corresponding one-step finite difference method. In particular, we carry out (1) the architecture study in terms of number of hidden layer…
We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the exact solution. We study the precision, in terms of the local error, of the method by applying it to different well known examples. The advantage of the method over others widely used lies on the simplicity of its implementation.
Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order $p$ and stage order $q=p$, some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.
Optimal Solution of Linear Ordinary Differential Equations by Conjugate Gradient Method
Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences（中国科学院重庆绿色智能技术研究院）；Department of Applied Mathematics, Western University（韦仕敦大学应用数学系）
In this paper we introduce a procedure, based on the method of equivariant moving frames, for formulating continuous Galerkin finite element schemes that preserve the Lie point symmetries of initial value problems for ordinary differential equations. Our methodology applies to projectable and non-projectable symmetry group actions, to ordinary differential equations of arbitrary order, and finite element approximations of arbitrary polynomial degree. Several examples are included to illustrate various features of the symmetry-preserving process. We summarise extensive numerical experiments sho…
We consider the following inverse problem for an ordinary differential equation (ODE): given a set of data points $P=\{(t_i,x_i),\; i=1,\dots,N\}$, find an ODE $x^\prime(t) = v (x)$ that admits a solution $x(t)$ such that $x_i \approx x(t_i)$ as closely as possible. The key to the proposed method is to find approximations of the recursive or discrete propagation function $D(x)$ from the given data set. Afterwards, we determine the field $v(x)$, using the conjugate map defined by Schröder's equation and the solution of a related Julia's equation. Moreover, our approach also works for the invers…
In this paper, the performance characteristics of different solution techniques and program packages to solve a large number of independent ordinary differential equation systems is examined. The employed hardware are an Intel Core i7-4820K CPU with 30.4 GFLOPS peak double-precision performance per cores and an Nvidia GeForce Titan Black GPU that has a total of 1707 GFLOPS peak double-precision performance. The tested systems (Lorenz equation, Keller--Miksis equation and a pressure relief valve model) are non-stiff and have low dimension. Thus, the performance of the codes are not limited by m…
A new approach for solving stiff boundary value problems for systems of ordinary differential equations is presented. Its idea essentially generalizes and extends that from arXiv:1601.04272v8. The approach can be viewed as a methodology framework that allows to enhance "stiffness resistance" capabilities of pretty much all the known numerical methods for solving two-point BVPs. The latter is demonstrated on the example of the {\it trapezoidal scheme} with the corresponding C++ source code available at \url{https://github.com/imathsoft/MathSoftDevelopment}. Results of numerical experiments are …
Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages the multitude of numerical algorithms in OrdinaryDiffEq for solving both stiff and non-stiff ODEs, and manages to solve challenging stiff DDEs. We describe how compiling the ODE integrator within itself, and accounting for discontinuity propagation,…
We present a novel numerical routine (oscode) with a C++ and Python interface for the efficient solution of one-dimensional, second-order, ordinary differential equations with rapidly oscillating solutions. The method is based on a Runge-Kutta-like stepping procedure that makes use of the Wentzel-Kramers-Brillouin (WKB) approximation to skip regions of integration where the characteristic frequency varies slowly. In regions where this is not the case, the method is able to switch to a made-to-measure Runge-Kutta integrator that minimises the total number of function evaluations. We demonstrate…
Improving the local solution of the DG predictor of the ADER-DG method for solving systems of ordinary differential equations and its applicability to systems of differential-algebraic equations
Department of Theoretical Physics, Dostoevsky Omsk State University（陀思妥耶夫斯基鄂木斯克国立大学理论物理系）
The coincidence between polynomial neural networks and matrix Lie maps is discussed in the article. The matrix form of Lie transform is an approximation of the general solution of the nonlinear system of ordinary differential equations. It can be used for solving systems of differential equations more efficiently than traditional step-by-step numerical methods. Implementation of the Lie map as a polynomial neural network provides a tool for both simulation and data-driven identification of dynamical systems. If the differential equation is provided, training a neural network is unnecessary. Th…
