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Model Comparisons: XNet Outperforms KAN

Xin Li, Zhihong Jeff Xia, Xiaotao Zheng

TL;DR

The study introduces XNet, a complex-domain network leveraging a Cauchy activation $\phi_a(x)=\frac{\lambda_1 x}{x^2+d^2}+\frac{\lambda_2}{x^2+d^2}$, and demonstrates its superior function-approximation capabilities relative to KAN and MLP baselines, especially for discontinuous and high-dimensional functions. In PINN contexts, XNet achieves smaller $L^2$ errors and faster runtimes than both PINN and KAN, suggesting strong potential for PDE-model reduction. By integrating XNet into LSTM (forming XLSTM), the approach yields substantial gains in time-series forecasting accuracy, including noisy and real-world data like stock prices. Collectively, the results indicate that XNet provides a more accurate, efficient representation for complex mappings, with significant implications for PDE solvers, high-dimensional approximation, and sequential modeling across engineering and scientific domains.

Abstract

In the fields of computational mathematics and artificial intelligence, the need for precise data modeling is crucial, especially for predictive machine learning tasks. This paper explores further XNet, a novel algorithm that employs the complex-valued Cauchy integral formula, offering a superior network architecture that surpasses traditional Multi-Layer Perceptrons (MLPs) and Kolmogorov-Arnold Networks (KANs). XNet significant improves speed and accuracy across various tasks in both low and high-dimensional spaces, redefining the scope of data-driven model development and providing substantial improvements over established time series models like LSTMs.

Model Comparisons: XNet Outperforms KAN

TL;DR

The study introduces XNet, a complex-domain network leveraging a Cauchy activation , and demonstrates its superior function-approximation capabilities relative to KAN and MLP baselines, especially for discontinuous and high-dimensional functions. In PINN contexts, XNet achieves smaller errors and faster runtimes than both PINN and KAN, suggesting strong potential for PDE-model reduction. By integrating XNet into LSTM (forming XLSTM), the approach yields substantial gains in time-series forecasting accuracy, including noisy and real-world data like stock prices. Collectively, the results indicate that XNet provides a more accurate, efficient representation for complex mappings, with significant implications for PDE solvers, high-dimensional approximation, and sequential modeling across engineering and scientific domains.

Abstract

In the fields of computational mathematics and artificial intelligence, the need for precise data modeling is crucial, especially for predictive machine learning tasks. This paper explores further XNet, a novel algorithm that employs the complex-valued Cauchy integral formula, offering a superior network architecture that surpasses traditional Multi-Layer Perceptrons (MLPs) and Kolmogorov-Arnold Networks (KANs). XNet significant improves speed and accuracy across various tasks in both low and high-dimensional spaces, redefining the scope of data-driven model development and providing substantial improvements over established time series models like LSTMs.
Paper Structure (16 sections, 8 equations, 38 figures, 12 tables)

This paper contains 16 sections, 8 equations, 38 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Experimental Setup
  3. RESULTS
  4. Heaviside step function apprxiamtion
  5. Function Approximation with $\exp(\sin(\pi x) + y^2)$ and $xy$
  6. Approximation with high-dimensional functions
  7. Possion function
  8. XNet enhance the LSTM
  9. Summary and Outlook
  10. Appendix
  11. ADDITIONAL EXPERIMENT DETAILS
  12. A.1 FUNCTION APPROXIMATION
  13. A.2 Time sereis
  14. Time series
  15. high-dimensional function 1
  16. ...and 1 more sections

Figures (38)

  • Figure 1: Comparing the MSE and training time for: PINN, XNet(20), KAN, and XNet(200). The MSE values are displayed on a logarithmic scale to better visualize the differences among the models.
  • Figure 2: Solution of the Poisson equation
  • Figure 3: Apple's stock price: 7/1/2016 - 7/1/2017
  • Figure 4: XNet approximation, with 64 basis functions
  • Figure 5: [1,1] KAN approximation, with k=3, grid =200
  • ...and 33 more figures