Fractional Hermite networks improve learning and physics modeling accuracy
FHKAN: Fractional Hermite Kolmogorov-Arnold Networks for Data-Driven and Physics-Informed Learning
Machine Learning
Summary
Many neural networks use fixed shapes for their calculations, which can limit how well they learn. The authors created a new type of network called FHKAN that uses flexible shapes based on math functions called fractional Hermite functions. These shapes work well on data that stretches out infinitely and avoid squashing parts of the network that can lose detail. Tests showed that FHKAN works better than many other methods on problems like predicting data, recognizing images, cleaning up noisy pictures, understanding text feelings, and solving physics equations.
What this means in practice
- •For machine learning engineers: Build neural networks that adapt better to continuous range data without distortion, improving accuracy in classification and regression tasks.
- •For computational physicists: Use improved neural architectures to solve complex physics equations with higher precision and lower error rates.
- •For image processing teams: Develop image denoising tools that produce clearer results by incorporating fractional Hermite activation blocks.$Commercial implications: Enables creation of advanced image enhancement products with superior noise reduction powered by new neural components.
Authors
Arsha Farnoush, Amirhossein Sadr, Aida Pakniyat, Dara Rahmati, Saeid Gorgin, Kourosh Parand
Abstract
Kolmogorov-Arnold Networks (KANs) replace fixed scalar weights by learnable univariate maps, yet most spectral KANs still rely on Chebyshev polynomials that live on a bounded interval and therefore require a saturating squashing of the pre-activation. We introduce the Fractional Hermite Kolmogorov-Arnold Network (FHKAN), a neural architecture that equips KANs with an adaptive, trainable fractional-order generalized Hermite basis. With a nonnegative shape parameter, the underlying generalized Hermite functions are orthonormal on the entire real line and uniformly bounded under a Gaussian envelope, so they act directly on unbounded pre-activations without any saturating map. We derive closed-form derivatives of the generalized, fractional, and scaled basis away from the origin and determine its order of differentiability at the origin, where the fractional power map creates a singular point. On synthetic regression, FHKAN attains the lowest test error at every network depth among nine fixed activations and the spline-based KAN. On image classification, image denoising, and sentiment analysis, the best-performing fractional Hermite block yields the highest accuracy or reconstruction quality. On physics-informed problems, FHKAN reaches mean absolute errors of order 10^-6 to 10^-5 on the Lane-Emden equation, residuals of order 10^-4 on the Burgers equation, and absolute errors below 10^-4 on a fractional delay differential equation.