College of Information Engineering, Shanghai Maritime University, Shanghai, China.
College of Arts and Sciences, Shanghai Maritime University, Shanghai, China.
PLoS One. 2022 Apr 6;17(4):e0265992. doi: 10.1371/journal.pone.0265992. eCollection 2022.
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.
为了将前馈神经网络 (FNN) 和微分方程 (DE) 的李群 (对称) 理论相结合，提出了一种替代的人工神经网络方法来解决常微分方程 (ODE) 的初值问题 (IVP)。通过引入解的李群表达式，将 ODE 的试探解分成两部分。第一部分是具有原始 IVP 的初始值的其他 ODE 的解。这可以使用李群和已知的符号或数值方法轻松解决，而无需任何网络参数（权重和偏差）。第二部分由具有可调参数的 FNN 组成。这是通过使用误差反向传播方法通过最小化误差（损失）函数并更新参数来训练的。与现有类似方法相比，该方法大大减少了可训练参数的数量，并且可以更快、更准确地学习真实解。该数值方法应用于包括物理振荡问题在内的多个案例。结果以图形表示，并得出了一些结论。
