Personalized Control for Lower Limb Prosthesis Using Kolmogorov-Arnold Networks

Objective: This paper investigates the potential of learnable activation functions in Kolmogorov-Arnold Networks (KANs) for personalized control in a lower-limb prosthesis. In addition, user-specific vs. pooled training data is evaluated to improve machine learning (ML) and Deep Learning (DL) performance for turn intent prediction. Method: Inertial measurement unit (IMU) data from the shank were collected from five individuals with lower-limb amputation performing turning tasks in a laboratory setting. Ability to classify an upcoming turn was evaluated for Multilayer Perceptron (MLP), Kolmogorov-Arnold Network (KAN), convolutional neural network (CNN), and fractional Kolmogorov-Arnold Networks (FKAN). The comparison of MLP and KAN (for ML models) and FKAN and CNN (for DL models) assessed the effectiveness of learnable activation functions. Models were trained separately on user-specific and pooled data to evaluate the impact of training data on their performance. Results: Learnable activation functions in KAN and FKAN did not yield significant improvement compared to MLP and CNN, respectively. Training on user-specific data yielded superior results compared to pooled data for ML models ($p < 0.05$). In contrast, no significant difference was observed between user-specific and pooled training for DL models. Significance: These findings suggest that learnable activation functions may demonstrate distinct advantages in datasets involving more complex tasks and larger volumes. In addition, pooled training showed comparable performance to user-specific training in DL models, indicating that model training for prosthesis control can utilize data from multiple participants.

TabKAN: Advancing Tabular Data Analysis using Kolmograv-Arnold Network

Tabular data analysis presents unique challenges due to its heterogeneous feature types, missing values, and complex interactions. While traditional machine learning methods, such as gradient boosting, often outperform deep learning approaches, recent advancements in neural architectures offer promising alternatives. This paper introduces TabKAN, a novel framework that advances tabular data modeling using Kolmogorov-Arnold Networks (KANs). Unlike conventional deep learning models, KANs leverage learnable activation functions on edges, enhancing both interpretability and training efficiency. Our contributions include: (1) the introduction of modular KAN-based architectures tailored for tabular data analysis, (2) the development of a transfer learning framework for KAN models, enabling effective knowledge transfer between domains, (3) the development of model-specific interpretability for tabular data learning, reducing reliance on post hoc and model-agnostic analysis, and (4) comprehensive evaluation of vanilla supervised learning across binary and multi-class classification tasks. Through extensive benchmarking on diverse public datasets, TabKAN demonstrates superior performance in supervised learning while significantly outperforming classical and Transformer-based models in transfer learning scenarios. Our findings highlight the advantage of KAN-based architectures in efficiently transferring knowledge across domains, bridging the gap between traditional machine learning and deep learning for structured data.

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

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.

PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems

This paper introduces an efficient tensor-vector product technique for the rapid and accurate approximation of integral operators within physics-informed deep learning frameworks. Our approach leverages neural network architectures to evaluate problem dynamics at specific points, while employing Gaussian quadrature formulas to approximate the integral components, even in the presence of infinite domains or singularities. We demonstrate the applicability of this method to both Fredholm and Volterra integral operators, as well as to optimal control problems involving continuous time. Additionally, we outline how this approach can be extended to approximate fractional derivatives and integrals and propose a fast matrix-vector product algorithm for efficiently computing the fractional Caputo derivative. In the numerical section, we conduct comprehensive experiments on forward and inverse problems. For forward problems, we evaluate the performance of our method on over 50 diverse mathematical problems, including multi-dimensional integral equations, systems of integral equations, partial and fractional integro-differential equations, and various optimal control problems in delay, fractional, multi-dimensional, and nonlinear configurations. For inverse problems, we test our approach on several integral equations and fractional integro-differential problems. Finally, we introduce the pinnies Python package to facilitate the implementation and usability of the proposed method.

rKAN: Rational Kolmogorov-Arnold Networks

The development of Kolmogorov-Arnold networks (KANs) marks a significant shift from traditional multi-layer perceptrons in deep learning. Initially, KANs employed B-spline curves as their primary basis function, but their inherent complexity posed implementation challenges. Consequently, researchers have explored alternative basis functions such as Wavelets, Polynomials, and Fractional functions. In this research, we explore the use of rational functions as a novel basis function for KANs. We propose two different approaches based on Pade approximation and rational Jacobi functions as trainable basis functions, establishing the rational KAN (rKAN). We then evaluate rKAN's performance in various deep learning and physics-informed tasks to demonstrate its practicality and effectiveness in function approximation.

fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions

Recent advancements in neural network design have given rise to the development of Kolmogorov-Arnold Networks (KANs), which enhance speed, interpretability, and precision. This paper presents the Fractional Kolmogorov-Arnold Network (fKAN), a novel neural network architecture that incorporates the distinctive attributes of KANs with a trainable adaptive fractional-orthogonal Jacobi function as its basis function. By leveraging the unique mathematical properties of fractional Jacobi functions, including simple derivative formulas, non-polynomial behavior, and activity for both positive and negative input values, this approach ensures efficient learning and enhanced accuracy. The proposed architecture is evaluated across a range of tasks in deep learning and physics-informed deep learning. Precision is tested on synthetic regression data, image classification, image denoising, and sentiment analysis. Additionally, the performance is measured on various differential equations, including ordinary, partial, and fractional delay differential equations. The results demonstrate that integrating fractional Jacobi functions into KANs significantly improves training speed and performance across diverse fields and applications.

Accelerating Fractional PINNs using Operational Matrices of Derivative

This paper presents a novel operational matrix method to accelerate the training of fractional Physics-Informed Neural Networks (fPINNs). Our approach involves a non-uniform discretization of the fractional Caputo operator, facilitating swift computation of fractional derivatives within Caputo-type fractional differential problems with $0<\alpha<1$. In this methodology, the operational matrix is precomputed, and during the training phase, automatic differentiation is replaced with a matrix-vector product. While our methodology is compatible with any network, we particularly highlight its successful implementation in PINNs, emphasizing the enhanced accuracy achieved when utilizing the Legendre Neural Block (LNB) architecture. LNB incorporates Legendre polynomials into the PINN structure, providing a significant boost in accuracy. The effectiveness of our proposed method is validated across diverse differential equations, including Delay Differential Equations (DDEs) and Systems of Differential Algebraic Equations (DAEs). To demonstrate its versatility, we extend the application of the method to systems of differential equations, specifically addressing nonlinear Pantograph fractional-order DDEs/DAEs. The results are supported by a comprehensive analysis of numerical outcomes.

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

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.

deepFDEnet: A Novel Neural Network Architecture for Solving Fractional Differential Equations

The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the proposed design. In each equation, a deep neural network is used to approximate the unknown function. Three forms of fractional differential equations have been examined to highlight the method's versatility: a fractional ordinary differential equation, a fractional order integrodifferential equation, and a fractional order partial differential equation. The results show that the proposed architecture solves different forms of fractional differential equations with excellent precision.

Solving Falkner-Skan type equations via Legendre and Chebyshev Neural Blocks

In this paper, a new deep-learning architecture for solving the non-linear Falkner-Skan equation is proposed. Using Legendre and Chebyshev neural blocks, this approach shows how orthogonal polynomials can be used in neural networks to increase the approximation capability of artificial neural networks. In addition, utilizing the mathematical properties of these functions, we overcome the computational complexity of the backpropagation algorithm by using the operational matrices of the derivative. The efficiency of the proposed method is carried out by simulating various configurations of the Falkner-Skan equation.