Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learning

Introduction

Kolmogorov-Arnold Networks (KANs) have recently emerged as a promising, interpretable alternative to traditional Multi-Layer Perceptrons (MLPs), particularly in scientific computing. However, their adoption in high-stakes scientific machine learning (SciML) is hindered by a lack of reliable uncertainty quantification (UQ). This paper introduces Conformalized-KANs, a framework that integrates Split Conformal Prediction with KAN ensembles to provide rigorous, distribution-free prediction intervals with mathematical coverage guarantees.

The Challenge of Reliability in SciML

In SciML, models often predict physical phenomena where errors can have significant consequences. While KANs are efficient at decomposing multivariate functions, they typically produce "point predictions." Existing UQ methods, such as standard ensembles, often rely on heuristic measures or assume Gaussian error distributions—assumptions that frequently fail in complex, data-sparse scientific environments. There is a critical need for a method that ensures a "95% confidence interval" truly contains the ground truth 95% of the time.

Methodology: From Ensembles to Guarantees

The Conformalized-KANs framework employs a robust two-step process to transform heuristic uncertainty into formal guarantees.

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1. Ensemble Construction: The process begins by training an ensemble of $M$ KAN models with varying initializations. This ensemble provides a mean prediction ($\mu_M$) and a standard deviation ($\sigma_M$) representing the model's internal uncertainty.
2. Split Conformal Prediction: A held-out "calibration dataset" is used to calculate nonconformity scores. These scores quantify the discrepancy between the ground truth and the ensemble mean, normalized by the ensemble's predicted standard deviation.
3. Calibration and Inference: By determining the specific quantile ($\hat{q}_\alpha$) of these scores, the framework "calibrates" the prediction intervals. For any new test point, the final prediction interval is defined as $\mu_M \pm \hat{q}_\alpha \cdot \sigma_M$, ensuring the desired coverage level (e.g., 95%).

This framework is versatile and was extended to advanced variants like Finite Basis KANs (FBKANs) for multiscale problems and Multi-Fidelity KANs (MFKANs) for data-scarce scenarios.

Key Insights: Correcting Overconfidence and the FBKAN Advantage

A primary finding of this research is the tendency of standard KAN ensembles to be "overconfident." In physics-informed tasks (PIKANs), standard ensembles achieved only ~25-28% coverage for a target 95% interval. Conformalization successfully "stretched" these intervals to meet the target.

Furthermore, the study highlights the FBKAN Advantage. By using domain decomposition, FBKANs produce significantly "sharper" (narrower) intervals than standard KANs while maintaining the same coverage guarantee. This suggests that localizing learning through subdomains effectively reduces epistemic uncertainty.

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Experimental Results

The framework was validated across several tasks, including synthetic nonlinear functions, multi-fidelity jump functions, and the wave equation (PDE).

• Guaranteed Coverage: Conformalized-KANs consistently met target coverage levels (e.g., ~95-96% for a 95% target), whereas non-conformalized versions often missed targets by over 60%.
• Interval Efficiency: In 1-D tests, Conformalized-FBKANs achieved target coverage with an average interval width of 0.41, compared to 1.06 for standard Conformalized-KANs, demonstrating superior precision.
• Robustness in Data Scarcity: The Multi-Fidelity experiments proved that even with as few as 5 high-fidelity training points, the framework provides reliable, calibrated intervals.

Limitations and Future Directions

While powerful, the split conformal prediction method requires a separate calibration dataset, which can be a challenge in extremely data-limited settings. Future work may explore adaptive subdomain partitioning for FBKANs to further sharpen intervals by adjusting boundaries based on local uncertainty. Additionally, while KANs are efficient, the computational cost of training ensembles remains a factor for large-scale applications.

@misc{lin2025conformalized,
  author = {Guang Lin and Amirhossein Mollaali and Christian B. Moya and Amanda A. Howard and Alexander Heinlein and Panos Stinis and Guang Lin},
  title = {{Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learning}},
  year = {2025},
  eprint = {2504.15240},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2504.15240},
}