论文标题
通过基于谎言组的神经网络方法解决普通微分方程的初始价值问题
Solving the Initial Value Problem of Ordinary Differential Equations by Lie Group based Neural Network Method
论文作者
论文摘要
为了结合微分方程（DES）的馈电神经网络（FNN）和谎言组（对称）理论，提出了一种替代性人工NN方法来解决普通DES（ODES）的初始值问题（IVP）。引入解决方案的谎言组表达式，ODE的试验解决方案分为两个部分。第一部分是具有原始IVP初始值的其他ODE的解决方案。使用Lie组以及已知的符号或数值方法可以轻松解决这一点，而没有任何网络参数（权重和偏见）。第二部分由具有可调参数的FNN组成。通过最小化错误（丢失）函数并更新参数，使用错误背回传播方法对此进行了训练。与现有的类似方法相比，该方法大大减少了可训练参数的数量，并且可以更快，更准确地学习实际解决方案。数值方法应用于几种情况，包括物理振荡问题。结果已被图形表示，并得出了一些结论。
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.
