Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Orthogonal Polynomial Kernel-Based Machine Learning Model for Differential-Algebraic Equations

Tayebeh Taheri, Alireza Afzal Aghaei, Kourosh Parand

TL;DR

The paper tackles the challenge of solving complex differential-algebraic equations (DAEs), including fractional, integro-differential, and partial forms, by developing a collocation-weighted residual LS-SVR framework with a Legendre orthogonal kernel. Solutions are expressed as a Legendre-basis expansion and solved via a dual quadratic programming problem, yielding exponential convergence and a symmetric positive definite system. The method is validated across multiple DAE classes with high accuracy and is shown to offer regularization benefits and scalability relative to existing approaches. These results suggest a practical, kernel-based ML approach for robustly solving diverse DAEs with potential extensions to other polynomial kernels and problem types.

Abstract

The recent introduction of the Least-Squares Support Vector Regression (LS-SVR) algorithm for solving differential and integral equations has sparked interest. In this study, we expand the application of this algorithm to address systems of differential-algebraic equations (DAEs). Our work presents a novel approach to solving general DAEs in an operator format by establishing connections between the LS-SVR machine learning model, weighted residual methods, and Legendre orthogonal polynomials. To assess the effectiveness of our proposed method, we conduct simulations involving various DAE scenarios, such as nonlinear systems, fractional-order derivatives, integro-differential, and partial DAEs. Finally, we carry out comparisons between our proposed method and currently established state-of-the-art approaches, demonstrating its reliability and effectiveness.

An Orthogonal Polynomial Kernel-Based Machine Learning Model for Differential-Algebraic Equations

TL;DR

The paper tackles the challenge of solving complex differential-algebraic equations (DAEs), including fractional, integro-differential, and partial forms, by developing a collocation-weighted residual LS-SVR framework with a Legendre orthogonal kernel. Solutions are expressed as a Legendre-basis expansion and solved via a dual quadratic programming problem, yielding exponential convergence and a symmetric positive definite system. The method is validated across multiple DAE classes with high accuracy and is shown to offer regularization benefits and scalability relative to existing approaches. These results suggest a practical, kernel-based ML approach for robustly solving diverse DAEs with potential extensions to other polynomial kernels and problem types.

Abstract

The recent introduction of the Least-Squares Support Vector Regression (LS-SVR) algorithm for solving differential and integral equations has sparked interest. In this study, we expand the application of this algorithm to address systems of differential-algebraic equations (DAEs). Our work presents a novel approach to solving general DAEs in an operator format by establishing connections between the LS-SVR machine learning model, weighted residual methods, and Legendre orthogonal polynomials. To assess the effectiveness of our proposed method, we conduct simulations involving various DAE scenarios, such as nonlinear systems, fractional-order derivatives, integro-differential, and partial DAEs. Finally, we carry out comparisons between our proposed method and currently established state-of-the-art approaches, demonstrating its reliability and effectiveness.
Paper Structure (13 sections, 2 theorems, 44 equations, 5 figures, 9 tables)

This paper contains 13 sections, 2 theorems, 44 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A system of differential-algebraic equations
  4. Integro-differential algebraic equations
  5. Partial differential-algebraic equations
  6. Fractional differential-algebraic equations
  7. Methodology
  8. Legendre Orthogonal Kernel
  9. LS-SVR for solving differential-algebraic equations
  10. Numerical results
  11. Conclusion
  12. Declarations
  13. Data availability statement

Key Result

Theorem 1

Let $0 < \alpha < 1$ and the interval $[x_0, x_m]$ is discretized to $m$ points, $0 = x_0 < x_1 < \dots < x_m$. By subtracting an integral part in Caputo’s definition to $m$ sub-interval, Caputo’s fractional derivative is defined as: And since the first derivative of the function $h(x)$ is estimated as: that $h_k=h(f_k)$, then the formula (form_des) can be expressed as: with $g_k$ defined as:

Figures (5)

  • Figure 1: (a) $|u_1(t)-\tilde{u}_1(t)|$, (b) $|u_2(t)-\tilde{u}_2(t)|$, and (c) $|u_3(t)-\tilde{u}_3(t)|$ for example \ref{['ex1']} with $m=4$.
  • Figure 2: (a) $|u_1(t)-\tilde{u}_1(t)|$ and (b) $|u_2(t)-\tilde{u}_2(t)|$ for example \ref{['ex2']} with $m=14$.
  • Figure 3: (a) $|u_1(t)-\tilde{u}_1(t)|$, (b) $|u_2(t)-\tilde{u}_2(t)|$, and (c) $|u_3(t)-\tilde{u}_3(t)|$ for example \ref{['ex3']} with $m=10$.
  • Figure 4: (a) $|u_1(t)-\tilde{u}_1(t)|$, (b) $|u_2(t)-\tilde{u}_2(t)|$, and (c) $|u_3(t)-\tilde{u}_3(t)|$ for example \ref{['ex4']} with $m=10$.
  • Figure 5: (a) $|u_1(x,t)-\tilde{u}_1(x,t)|$, (b) $|u_2(x,t)-\tilde{u}_2(x,t)|$, and (c) $|u_3(x,t)-\tilde{u}_3(x,t)|$ for example \ref{['ex5']} with m=6.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Definition 3
  • Definition 4
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • ...and 6 more