Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A piecewise neural network method for solving large interval solution to initial value problem of ordinary differential equations

Dongpeng Han, Chaolu Temuer

TL;DR

This paper proposes a piecewise neural network approach to obtain a large interval numerical solution for initial value problems of differential equations, and proves the continuous differentiability of the solution over the entire interval, except for finite points.

Abstract

Various traditional numerical methods for solving initial value problems of differential equations often produce local solutions near the initial value point, despite the problems having larger interval solutions. Even current popular neural network algorithms or deep learning methods cannot guarantee yielding large interval solutions for these problems. In this paper, we propose a piecewise neural network approach to obtain a large interval numerical solution for initial value problems of differential equations. In this method, we first divide the solution interval, on which the initial problem is to be solved, into several smaller intervals. Neural networks with a unified structure are then employed on each sub-interval to solve the related sub-problems. By assembling these neural network solutions, a piecewise expression of the large interval solution to the problem is constructed, referred to as the piecewise neural network solution. The continuous differentiability of the solution over the entire interval, except for finite points, is proven through theoretical analysis and employing a parameter transfer technique. Additionally, a parameter transfer and multiple rounds of pre-training technique are utilized to enhance the accuracy of the approximation solution. Compared with existing neural network algorithms, this method does not increase the network size and training data scale for training the network on each sub-domain. Finally, several numerical experiments are presented to demonstrate the efficiency of the proposed algorithm.

A piecewise neural network method for solving large interval solution to initial value problem of ordinary differential equations

TL;DR

This paper proposes a piecewise neural network approach to obtain a large interval numerical solution for initial value problems of differential equations, and proves the continuous differentiability of the solution over the entire interval, except for finite points.

Abstract

Various traditional numerical methods for solving initial value problems of differential equations often produce local solutions near the initial value point, despite the problems having larger interval solutions. Even current popular neural network algorithms or deep learning methods cannot guarantee yielding large interval solutions for these problems. In this paper, we propose a piecewise neural network approach to obtain a large interval numerical solution for initial value problems of differential equations. In this method, we first divide the solution interval, on which the initial problem is to be solved, into several smaller intervals. Neural networks with a unified structure are then employed on each sub-interval to solve the related sub-problems. By assembling these neural network solutions, a piecewise expression of the large interval solution to the problem is constructed, referred to as the piecewise neural network solution. The continuous differentiability of the solution over the entire interval, except for finite points, is proven through theoretical analysis and employing a parameter transfer technique. Additionally, a parameter transfer and multiple rounds of pre-training technique are utilized to enhance the accuracy of the approximation solution. Compared with existing neural network algorithms, this method does not increase the network size and training data scale for training the network on each sub-domain. Finally, several numerical experiments are presented to demonstrate the efficiency of the proposed algorithm.
Paper Structure (14 sections, 26 equations, 13 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 26 equations, 13 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A brief recall of PINN
  3. A piecewise ANN method
  4. Method
  5. Theoretical analysis and a parameter transfer method
  6. Approximation of large interval solution
  7. Transfer of network parameters
  8. Implementation of PWNN
  9. Experiment
  10. Example 1
  11. Example 2
  12. Example 3
  13. Example 4
  14. Discussion and Conclusion

Figures (13)

  • Figure 1: Network structure of PINN for solving IVP (\ref{['eq:odes']}).
  • Figure 2: Network structure of PWNN for solving IVP (\ref{['eq:odes']}).
  • Figure 3: PINN results for IVP (\ref{['eq:exam1']}) with final loss function value $6.58*10^{-5}$.
  • Figure 4: Result figures of PWNNs: the first five is $m_k$ with final loss function value on each sub-interval $\Delta_k (k=1, 2, \cdots, 5)$. The last one is their combinations $\hat{y}_1(x)$ and $\hat{y}_2(x)$ .
  • Figure 5: Comparisons of the results of solving IVP (\ref{['eq:exam1']}) using PINN, PWNN and RK4 methods.
  • ...and 8 more figures