# ExSpliNet: An interpretable and expressive spline-based neural network

arXiv: https://arxiv.org/abs/2205.01510 · HTML: https://mlsanity.com/paper/2205.01510/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 69 references that are in the map.
- First posted on arXiv 3 May 2022.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels](https://mlsanity.com/paper/1711.08920/) (2017)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [On the Expressive Power of Deep Learning: A Tensor Analysis](https://mlsanity.com/paper/1509.05009/) (2015)
- [Deep Neural Decision Trees](https://mlsanity.com/paper/1806.06988/) (2018)
- [Mad Max: Affine Spline Insights into Deep Learning](https://mlsanity.com/paper/1805.06576/) (2018)
- [Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem](https://mlsanity.com/paper/1906.11945/) (2019)
- [An Ensemble of Simple Convolutional Neural Network Models for MNIST Digit Recognition](https://mlsanity.com/paper/2008.10400/) (2020)
- [Neural Random Forests](https://mlsanity.com/paper/1604.07143/) (2016)
- [Fine-Tuning DARTS for Image Classification](https://mlsanity.com/paper/2006.09042/) (2020)
- [Neural Decision Trees](https://mlsanity.com/paper/1702.07360/) (2017)
- [Learning activation functions from data using cubic spline interpolation](https://mlsanity.com/paper/1605.05509/) (2016)
- [Nonlinear system identification with regularized Tensor Network B-splines](https://mlsanity.com/paper/2003.07594/) (2020)

## Built on by (most cited first, 30 of 33)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Wav-KAN: Wavelet Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.12832/) (2024)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions](https://mlsanity.com/paper/2406.07456/) (2024)
- [Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN](https://mlsanity.com/paper/2407.18373/) (2024)
- [AuroRA: Breaking Low-Rank Bottleneck of LoRA with Nonlinear Mapping](https://mlsanity.com/paper/2505.18738/) (2025)
- [FC-KAN: Function Combinations in Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2409.01763/) (2024)
- [A Practitioner's Guide to Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2510.25781/) (2025)
- [Leveraging KANs For Enhanced Deep Koopman Operator Discovery](https://mlsanity.com/paper/2406.02875/) (2024)
- [Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs](https://mlsanity.com/paper/2506.07958/) (2025)
- [Fast and Accurate Gigapixel Pathological Image Classification with Hierarchical Distillation Multi-Instance Learning](https://mlsanity.com/paper/2502.21130/) (2025)
- [KAN we improve on HEP classification tasks? Kolmogorov-Arnold Networks applied to an LHC physics example](https://mlsanity.com/paper/2408.02743/) (2024)
- [Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning](https://mlsanity.com/paper/2603.23854/) (2026)
- [Nonparametric Automatic Differentiation Variational Inference with Spline Approximation](https://mlsanity.com/paper/2403.06302/) (2024)
- [Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertainty](https://mlsanity.com/paper/2501.04757/) (2025)
- [SechKAN: Kolmogorov-Arnold Networks with Hyperbolic Secant Functions](https://mlsanity.com/paper/2607.18290/) (2026)
- [InKAN: B-Spline KANs via Truncated Power Form](https://mlsanity.com/paper/2609.01956/) (2026)
- [Learning Expressive Random Feature Models via Parametrized Activations](https://mlsanity.com/paper/2411.19468/) (2024)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [A Kolmogorov-Arnold Neural Model for Cascading Extremes](https://mlsanity.com/paper/2505.13370/) (2025)
- [BEKAN: Boundary condition-guaranteed evolutionary Kolmogorov-Arnold networks with radial basis functions for solving PDE problems](https://mlsanity.com/paper/2510.03576/) (2025)
- [On the Generalization Properties of Learning the Random Feature Models with Learnable Activation Functions](https://mlsanity.com/paper/2510.15327/) (2025)
- [Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2602.18515/) (2026)
- [SL2A-INR: Single-Layer Learnable Activation for Implicit Neural Representation](https://mlsanity.com/paper/2409.10836/) (2024)
- [Model Comparisons: XNet Outperforms KAN](https://mlsanity.com/paper/2410.02033/) (2024)
- [Hybrid deep additive neural networks](https://mlsanity.com/paper/2411.09175/) (2024)
- [Local Control Networks (LCNs): Optimizing Flexibility in Neural Network Data Pattern Capture](https://mlsanity.com/paper/2501.14000/) (2025)
- [Enhancing Neural Function Approximation: The XNet Outperforming KAN](https://mlsanity.com/paper/2501.18959/) (2025)
- [Polynomial, trigonometric, and tropical activations](https://mlsanity.com/paper/2502.01247/) (2025)
- [Fully Kolmogorov-Arnold Deep Model in Medical Image Segmentation](https://mlsanity.com/paper/2602.03156/) (2026)

## Abstract (arXiv)

In this paper we present ExSpliNet, an interpretable and expressive neural network model. The model combines ideas of Kolmogorov neural networks, ensembles of probabilistic trees, and multivariate B-spline representations. We give a probabilistic interpretation of the model and show its universal approximation properties. We also discuss how it can be efficiently encoded by exploiting B-spline properties. Finally, we test the effectiveness of the proposed model on synthetic approximation problems and classical machine learning benchmark datasets.