Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

HANN: Homotopy auxiliary neural network for solving nonlinear algebraic equations

Ling-Zhe Zai, Lei-Lei Guo, Zhi-Yong Zhang

TL;DR

The paper tackles the challenge of solving nonlinear algebraic equations by integrating the homotopy continuation method with physics-informed neural networks to create the Homotopy Auxiliary Neural Network (HANN). It introduces HANN-1, which treats the homotopy parameter $t$ as the neural network input to trace solutions from an easy starting system to the target system, and HANN-2, which iteratively refines these solutions for higher accuracy. Through extensive benchmarks—including a single equation, transcendental systems, high-dimensional ill-conditioned problems, and time-varying equations—HANN demonstrates strong capability to locate multiple isolated solutions and achieve high precision, often outperforming Python’s Fsolve and certain evolutionary methods. The results indicate that HANN offers a flexible, low-cost approach for challenging nonlinear algebraic problems, with a recommended workflow of using HANN-1 to identify candidate solutions followed by HANN-2 or Fsolve for refinement.

Abstract

Solving nonlinear algebraic equations is a fundamental but challenging problem in scientific computations and also has many applications in system engineering. Though traditional iterative methods and modern optimization algorithms have exerted effective roles in addressing certain specific problems, there still exist certain weaknesses such as the initial value sensitivity, limited accuracy and slow convergence rate, particulary without flexible input for the neural network methods. In this paper, we propose a homotopy auxiliary neural network (HANN) for solving nonlinear algebraic equations which integrates the classical homotopy continuation method and popular physics-informed neural network. Consequently, the HANN-1 has strong learning ability and can rapidly give an acceptable solution for the problem which outperforms some known methods, while the HANN-2 can further improve its accuracy. Numerical results on the benchmark problems confirm that the HANN method can effectively solve the problems of determining the total number of solutions of a single equation, finding solutions of transcendental systems involving the absolute value function or trigonometric function, ill-conditioned and normal high-dimensional nonlinear systems and time-varying nonlinear problems, for which the Python's built-in Fsolve function exhibits significant limitations, even fails to work.

HANN: Homotopy auxiliary neural network for solving nonlinear algebraic equations

TL;DR

The paper tackles the challenge of solving nonlinear algebraic equations by integrating the homotopy continuation method with physics-informed neural networks to create the Homotopy Auxiliary Neural Network (HANN). It introduces HANN-1, which treats the homotopy parameter as the neural network input to trace solutions from an easy starting system to the target system, and HANN-2, which iteratively refines these solutions for higher accuracy. Through extensive benchmarks—including a single equation, transcendental systems, high-dimensional ill-conditioned problems, and time-varying equations—HANN demonstrates strong capability to locate multiple isolated solutions and achieve high precision, often outperforming Python’s Fsolve and certain evolutionary methods. The results indicate that HANN offers a flexible, low-cost approach for challenging nonlinear algebraic problems, with a recommended workflow of using HANN-1 to identify candidate solutions followed by HANN-2 or Fsolve for refinement.

Abstract

Solving nonlinear algebraic equations is a fundamental but challenging problem in scientific computations and also has many applications in system engineering. Though traditional iterative methods and modern optimization algorithms have exerted effective roles in addressing certain specific problems, there still exist certain weaknesses such as the initial value sensitivity, limited accuracy and slow convergence rate, particulary without flexible input for the neural network methods. In this paper, we propose a homotopy auxiliary neural network (HANN) for solving nonlinear algebraic equations which integrates the classical homotopy continuation method and popular physics-informed neural network. Consequently, the HANN-1 has strong learning ability and can rapidly give an acceptable solution for the problem which outperforms some known methods, while the HANN-2 can further improve its accuracy. Numerical results on the benchmark problems confirm that the HANN method can effectively solve the problems of determining the total number of solutions of a single equation, finding solutions of transcendental systems involving the absolute value function or trigonometric function, ill-conditioned and normal high-dimensional nonlinear systems and time-varying nonlinear problems, for which the Python's built-in Fsolve function exhibits significant limitations, even fails to work.

Paper Structure

This paper contains 14 sections, 1 theorem, 19 equations, 6 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methods
  3. Numerical results
  4. A single equation
  5. Transcendental systems containing two equations
  6. System containing absolute value function
  7. System containing trigonometric functions
  8. High-dimensional systems containing ten equations
  9. Interval arithmetic benchmark
  10. Combustion problem
  11. Multivariate time-varying equations
  12. Further discussions
  13. Systematic investigations on the six equations
  14. Conclusion

Key Result

Theorem 1

Let $U$ be an open subset of $\mathbb{R}^n$ and $\textbf{x}\in U$. Suppose that $H(\textbf{x},t):=(H_1(\textbf{x},t),$$H_2(\textbf{x},t),\dots,H_n(\textbf{x},t))$ is continuously differentiable in an open set containing $U \times[0,1]$, that the function $H(\textbf{x},0)$ has a zero $\textbf{x}_0$ i If $D_x H$ is nonsingular and $\left| (D_x H)^{-1} D_t H \right| < d$ holds, then there exists a co

Figures (6)

  • Figure 1: (Color online) Equation (\ref{['exa-1']}): (A). Graph of $y = 1/x - \sin x + 1$. (B). Distribution of predicted solutions by HANN-1 under different numbers of sub-intervals. (C). Equation residual distribution of different predicted solutions in different sub-intervals, where the color of points correspond to the one of solutions in (B). (D). Loss history of HANN-2 with a random initial value -20.625. (E). Comparison of equation residuals of the learned 32 solutions by HANN-1 and HANN-2.
  • Figure 2: (Color online) System (\ref{['example-2']}): (A). The mappings by HANN-1 from the 49 initial values to two predicted solutions where the two solid red points are the predicted solutions under the threshold $3.54\times 10^{-2}$; (B). Comparisons of equation residuals of the 49 predicted solutions with HANN-1 and HANN-2. (C) Loss decent tendency by HANN-2 over iterations. (D) Homotopy curve $x_1(t)$. (E) Homotopy curve $x_2(t)$.
  • Figure 3: (Color online) System (\ref{['example-3']}): (A). The mappings by HANN-1 with the 100 initial values where the eight accumulated solid red points are the predicted solutions; (B). Equation residuals of the predicted solutions by HANN-1 with the 100 initial values.
  • Figure 4: (Color online) System (\ref{['example-6']}): Comparisons of equation residuals among the EVO and HANN-1 and HANN-2 methods: (A) EVO and HANN-1 (B) HANN-1 and HANN-2 for the seven solutions than cannot be improved the Fsolve function.
  • Figure 5: (Color online) System (\ref{['example-7']}): Comparisons among the EVO, HANN-1 and HANN-2 methods. (A) EVO and HANN-1 (B) HANN-1 and HANN-2.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1: 1