Softly Symbolifying Kolmogorov-Arnold Networks
S2KAN integrates symbolic primitives with differentiable sparsity, achieving compact, interpretable models with high accuracy.
Key Findings
Methodology
S2KAN combines symbolic primitives, sparse bases, and dense representations with differentiable gating mechanisms. It optimizes a Minimum Description Length (MDL) objective to balance accuracy and complexity, dynamically selecting between symbolic and dense representations during training.
Key Results
- On the Nguyen benchmarks, S2KAN achieved 100% symbolic terms for F1-F9 problems with test R² near 1.0, outperforming post-hoc symbolification in traditional KAN.
- In the Ikeda chaotic system, S2KAN reduced parameters by nearly 6x, achieving 1-step prediction RMSE of 0.0196, comparable to the baseline.
- In the ecosystem model, unregularized S2KAN self-sparsified, reducing parameters by 87% while lowering multi-step prediction error to 0.1108.
Significance
S2KAN addresses key limitations in traditional KANs by integrating symbolic primitives directly into training. This enables compact, interpretable models suitable for scientific machine learning tasks like chaotic system modeling and real-world predictions, advancing both academic and industrial applications.
Technical Contribution
Key contributions include: 1) dynamic integration of symbolic and dense representations; 2) differentiable sparsity for end-to-end optimization; 3) MDL-based loss balancing accuracy and complexity; 4) superior performance on benchmarks and real-world tasks.
Novelty
S2KAN is the first to integrate symbolic primitives directly into training with differentiable sparsity, replacing post-hoc symbolification. This proactive approach ensures higher symbolic fidelity and compactness compared to traditional KANs.
Limitations
- Limited expressiveness of symbolic primitives for complex physical systems like superconductivity prediction.
- Sensitivity to hyperparameter β, which affects sparsity and performance.
- Over-sparsification may degrade multi-step prediction performance.
Future Work
Future work could explore richer symbolic libraries, adaptive sparsity mechanisms, and dynamic regularization to balance sparsity and accuracy across diverse tasks.
AI Executive Summary
Kolmogorov-Arnold Networks (KANs) are valued for their interpretability and performance in scientific machine learning. However, traditional KANs often fail to produce interpretable symbolic forms, relying on dense representations that lack symbolic fidelity.
S2KAN introduces symbolic primitives and sparse bases into training, leveraging differentiable gating mechanisms for dynamic selection. Its MDL-based objective balances model complexity and accuracy. Experiments show S2KAN achieves 100% symbolic terms on Nguyen benchmarks and significantly reduces parameters in chaotic system modeling while maintaining high accuracy.
While dense representations remain necessary for some tasks, S2KAN offers a transformative approach to symbolic regression and scientific modeling. Future research could expand symbolic libraries and refine sparsity mechanisms for broader applications.
Deep Analysis
Background
KANs leverage the Kolmogorov representation theorem to model complex data with learnable activation functions. However, traditional KANs rely on dense representations like B-splines, which hinder symbolic interpretability.
Core Problem
Post-hoc symbolification in KANs faces challenges: 1) dense activations may not map to symbolic forms; 2) symbolic forms are explored independently of network structure; 3) symbolic fidelity is not optimized during training.
Innovation
S2KAN's innovations include: 1) integrating symbolic primitives and sparse bases with dense representations; 2) differentiable gating for sparsity; 3) MDL-based loss to balance accuracy and complexity.
Methodology
- �� Activation dictionary: combines symbolic primitives (e.g., sin, log), sparse bases (e.g., Chebyshev polynomials), and dense representations (e.g., B-splines).
- �� Differentiable sparsity: uses Hard Concrete distribution for binary gating.
- �� MDL objective: balances model complexity and residual error for optimal performance.
Experiments
Experiments include Nguyen benchmarks, chaotic system modeling (Ikeda map, ecosystem), and real-world tasks (concrete strength, superconductivity). Metrics include R², RMSE, and sparsity levels. Ablations study β's impact on performance.
Results
S2KAN achieved 100% symbolic terms on F1-F9 Nguyen benchmarks. In the Ikeda map, it reduced parameters by 6x with RMSE of 0.0196. In the ecosystem, unregularized S2KAN self-sparsified, reducing parameters by 87%.
Applications
S2KAN is applicable to symbolic regression, dynamic system modeling, and scientific discovery. Examples include predicting superconducting properties and optimizing concrete strength.
Limitations & Outlook
S2KAN relies on dense representations for complex systems, is sensitive to regularization hyperparameters, and may over-sparsify, impacting multi-step predictions.
Plain Language Accessible to non-experts
Imagine a chef preparing a meal. Traditional KAN is like a chef with only a knife, chopping everything into pieces. S2KAN, however, is like a smart chef who uses the right tool for each task—chopping, blending, or peeling—depending on the ingredient. It even decides which tools to skip, saving time and making the dish more delicious!
ELI14 Explained like you're 14
Imagine playing Minecraft. Regular KAN is like building a house with just blocks—functional but boring. S2KAN is like a mod that helps you build a castle using different materials and tools. It even removes extra blocks to make your castle look perfect! Cool, right?
Glossary
Kolmogorov-Arnold Network (KAN)
A neural network with learnable activation functions for modeling complex data.
Used for dynamic system modeling and symbolic regression.
Symbolic Primitives
Basic mathematical functions like sin and log.
Used in S2KAN to create interpretable activation functions.
Sparsity
Reducing unnecessary components in a model.
Achieved in S2KAN via differentiable gating.
Minimum Description Length (MDL)
An objective balancing model complexity and data fit.
Guides S2KAN's optimization.
Hard Concrete Distribution
A differentiable relaxation of binary gating for optimization.
Enables sparsity in S2KAN.
Open Questions Unanswered questions from this research
- 1 How can symbolic libraries be expanded for complex physical systems?
- 2 Can sparsity mechanisms adapt dynamically to different tasks?
Applications
Immediate Applications
Dynamic System Modeling
Predict chaotic systems like climate or ecosystems.
Engineering Optimization
Improve concrete strength prediction for efficient material design.
Long-term Vision
Automated Scientific Discovery
Discover physical laws via symbolic models, accelerating research.
Abstract
Kolmogorov-Arnold Networks (KANs) offer a promising path toward interpretable machine learning: their learnable activations can be studied individually, while collectively fitting complex data accurately. In practice, however, trained activations often lack symbolic fidelity, learning pathological decompositions with no meaningful correspondence to interpretable forms. We propose Softly Symbolified Kolmogorov-Arnold Networks (S2KAN), which integrate symbolic primitives directly into training. Each activation draws from a dictionary of symbolic and dense terms, with learnable gates that sparsify the representation. Crucially, this sparsification is differentiable, enabling end-to-end optimization, and is guided by a principled Minimum Description Length objective. When symbolic terms suffice, S2KAN discovers interpretable forms; when they do not, it gracefully degrades to dense splines. We demonstrate competitive or superior accuracy with substantially smaller models across symbolic benchmarks, dynamical systems forecasting, and real-world prediction tasks, and observe evidence of emergent self-sparsification even without regularization pressure.