TorchCurves: Differentiable Parametric Curves

Kolmogorov-Arnold Networks (KANs) place learnable B-spline activations on network edges rather than fixed activations on nodes. The standard Cox-de Boor recursion evaluates these activations through $k$ sequential passes for degree-$k$ splines, consuming over 90% of forward-pass time. InKAN replaces this recursion with the truncated power form, a classical result from approximation theory that expresses each uniform cubic B-spline as five $(x)_+^3$ terms at shifted knot positions. The resulting expression computes exact B-spline basis values: the same mathematical function as the Cox-de Boor recursion, evaluated without sequential passes. This paper documents three contributions: (1) an implementation structured for torch . compile fusion, eliminating recursion, span lookup, and scatter-gather operations; (2) a bounded-coordinate evaluation that clamps the normalized input to $[0, k{+}1]$, preventing the growth of cancellation error at large off-support coordinates; and (3) an open-source package (pip install inkan). In the tested configurations, InKAN has 2.8--3.5$\times$ lower forward-pass latency than the Cox-de Boor recursion. Partition-of-unity errors remain below $10^{-5}$ for grid sizes up to 200.

Kolmogorov-Arnold networks (KANs) offer a potent combination of accuracy and interpretability, thanks to their compositions of learnable univariate activation functions. However, the activations of well-fitting KANs tend to exhibit pathologically high-curvature oscillations, making them difficult to interpret, and standard regularization penalties do not prevent this. Here we derive a basis-agnostic curvature penalty and show that penalized models can maintain accuracy while achieving substantially smoother activations. Accounting for how function composition shapes curvature, we prove an upper bound on the full model's curvature relative to the curvature penalty, and use this to motivate richer forms of penalties. Scientific machine learning is increasingly bottlenecked by the trade-off between accuracy and interpretability. Results such as ours that improve interpretability without sacrificing accuracy will further strengthen KANs as a practical tool for both prediction and insight.

KANLab is a browser-based visualization tool designed to interactively teach the core concepts of Kolmogorov-Arnold Networks (KAN), forked from the widely successful TensorFlow Playground. To maximize educational clarity, KANLab omits residual connections and learnable scaling parameters found in modern KAN implementations, ensuring that displayed B-spline activation curves directly correspond to their underlying trainable control points. Users can inspect and modify activation functions, initialization schemes and observe gradient descent correcting manual disruptions. Furthermore, user can explore phenomena such as activation variance, interactive pruning, and re-/training within a compact 2D machine learning setting. KANLab is open-source and available at https://kanlab.app/