Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems

Alireza Afzal Aghaei

TL;DR

KANtrol is introduced, a Kolmogorov–Arnold Network (KAN)-based framework for solving optimal control problems in continuous-time systems that leverages Gaussian quadrature to approximate integral components, including cost function and integro-differential state equations.

Abstract

In this paper, we introduce the KANtrol framework, which utilizes Kolmogorov-Arnold Networks (KANs) to solve optimal control problems involving continuous time variables. We explain how Gaussian quadrature can be employed to approximate the integral parts within the problem, particularly for integro-differential state equations. We also demonstrate how automatic differentiation is utilized to compute exact derivatives for integer-order dynamics, while for fractional derivatives of non-integer order, we employ matrix-vector product discretization within the KAN framework. We tackle multi-dimensional problems, including the optimal control of a 2D heat partial differential equation. The results of our simulations, which cover both forward and parameter identification problems, show that the KANtrol framework outperforms classical MLPs in terms of accuracy and efficiency.

KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems

TL;DR

KANtrol is introduced, a Kolmogorov–Arnold Network (KAN)-based framework for solving optimal control problems in continuous-time systems that leverages Gaussian quadrature to approximate integral components, including cost function and integro-differential state equations.

Abstract

In this paper, we introduce the KANtrol framework, which utilizes Kolmogorov-Arnold Networks (KANs) to solve optimal control problems involving continuous time variables. We explain how Gaussian quadrature can be employed to approximate the integral parts within the problem, particularly for integro-differential state equations. We also demonstrate how automatic differentiation is utilized to compute exact derivatives for integer-order dynamics, while for fractional derivatives of non-integer order, we employ matrix-vector product discretization within the KAN framework. We tackle multi-dimensional problems, including the optimal control of a 2D heat partial differential equation. The results of our simulations, which cover both forward and parameter identification problems, show that the KANtrol framework outperforms classical MLPs in terms of accuracy and efficiency.
Paper Structure (7 sections, 1 theorem, 21 equations, 4 figures, 4 tables)

This paper contains 7 sections, 1 theorem, 21 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Kolmogorov-Arnold Network
  3. Methodology
  4. Fractional Order Derivatives
  5. Integro-Differential Equations
  6. Numerical Experiments
  7. Conclusion

Key Result

Theorem 1

The Caputo fractional derivative can be approximated using the matrix-vector product: where $\tilde{\xi}_j = \xi(\tau_j)$, $\tau_j$ for $j=1,\ldots, M$ are some points in the problem domain, and the lower triangular matrix $\mathcal{D}^\alpha$ is defined as: where $\mu_k^{(i)} = {\left[\lambda_k^{(i)} - \lambda_{k-1}^{(i)}\right]}/{\Gamma(2-\alpha)}$ and: for any $0\le k < M$.

Figures (4)

  • Figure 1: Overall Structure of KANtrol framework for solving optimal control problems using Kolmogorov-Arnold Networks.
  • Figure 2: Parameter identification of optimal control problems using the KANtrol framework, for Example \ref{['ex:frac-inverse']}.
  • Figure 3: The training loss and convergence of the KANtrol framework towards the exact $\kappa$ value during the training phase are depicted. The y-axis of the plots is on a logarithmic scale.
  • Figure 4: Simulation results for the optimal control of a 2D heat PDE over the domain $[0, \pi]^2$ at different time values up to one. The top row displays the control variable, while the bottom row illustrates the evolution of heat as the state variable.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5