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

arXiv: https://arxiv.org/abs/2409.06649 · HTML: https://mlsanity.com/paper/2409.06649/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 80 references that are in the map.
- First posted on arXiv 10 Sep 2024.

## Builds on (most cited first, 28 of 28)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions](https://mlsanity.com/paper/2406.07456/) (2024)
- [Physics-Informed Neural Networks for AC Optimal Power Flow](https://mlsanity.com/paper/2110.02672/) (2021)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2407.17611/) (2024)
- [rKAN: Rational Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.14495/) (2024)
- [Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems](https://mlsanity.com/paper/2406.19662/) (2024)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Real-time optimal control via Deep Neural Networks: study on landing problems](https://mlsanity.com/paper/1610.08668/) (2016)
- [Solving inverse-PDE problems with physics-aware neural networks](https://mlsanity.com/paper/2001.03608/) (2020)
- [Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN](https://mlsanity.com/paper/2407.18373/) (2024)
- [Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations](https://mlsanity.com/paper/2203.08501/) (2022)
- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [Physics-Informed Kolmogorov-Arnold Networks for Power System Dynamics](https://mlsanity.com/paper/2408.06650/) (2024)
- [AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems](https://mlsanity.com/paper/2302.02076/) (2023)
- [deepFDEnet: A Novel Neural Network Architecture for Solving Fractional Differential Equations](https://mlsanity.com/paper/2309.07684/) (2023)
- [Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving Linear Elasticity Problems](https://mlsanity.com/paper/2407.01088/) (2024)
- [A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints](https://mlsanity.com/paper/2310.02286/) (2023)
- [Physics-informed ConvNet: Learning Physical Field from a Shallow Neural Network](https://mlsanity.com/paper/2201.10967/) (2022)
- [Fractional Calculus In Image Processing: A Review](https://mlsanity.com/paper/1608.03240/) (2016)
- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)
- [Accelerating Fractional PINNs using Operational Matrices of Derivative](https://mlsanity.com/paper/2401.14081/) (2024)
- [PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems](https://mlsanity.com/paper/2409.01899/) (2024)

## Built on by (most cited first, 3 of 3)

- [On the Convergence of (Stochastic) Gradient Descent for Kolmogorov--Arnold Networks](https://mlsanity.com/paper/2410.08041/) (2024)
- [CIKAN: Constraint Informed Kolmogorov-Arnold Networks for Autonomous Spacecraft Rendezvous using Time Shift Governor](https://mlsanity.com/paper/2412.03710/) (2024)
- [High-Fidelity Modeling of Stochastic Chemical Dynamics on Complex Manifolds: A Multi-Scale SIREN-PINN Framework for the Curvature-Perturbed Ginzburg-Landau Equation](https://mlsanity.com/paper/2601.08104/) (2026)

## Abstract (arXiv)

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.