KAN 2.0: Kolmogorov-Arnold Networks Meet Science

TL;DR

The paper tackles the AI/Science gap by extending Kolmogorov-Arnold Networks (KANs) to support curiosity-driven scientific discovery. It introduces MultKAN, a kanpiler to translate symbolic formulas into KANs, and a tree converter to visualize parse graphs, enabling bidirectional knowledge flow between science and KANs. By incorporating inductive biases such as important features, modularity, and symbolic formulas, and by developing methods for feature attribution, modularity discovery, and symbolic regression, the framework aims to extract interpretable scientific insights from data. Across applications like conserved quantities, Lagrangians, hidden symmetry, and constitutive laws, the approach demonstrates interpretable representations and accurate symbolic recoveries, highlighting a practical path for interpretable AI-assisted scientific discovery.

Abstract

A major challenge of AI + Science lies in their inherent incompatibility: today's AI is primarily based on connectionism, while science depends on symbolism. To bridge the two worlds, we propose a framework to seamlessly synergize Kolmogorov-Arnold Networks (KANs) and science. The framework highlights KANs' usage for three aspects of scientific discovery: identifying relevant features, revealing modular structures, and discovering symbolic formulas. The synergy is bidirectional: science to KAN (incorporating scientific knowledge into KANs), and KAN to science (extracting scientific insights from KANs). We highlight major new functionalities in the pykan package: (1) MultKAN: KANs with multiplication nodes. (2) kanpiler: a KAN compiler that compiles symbolic formulas into KANs. (3) tree converter: convert KANs (or any neural networks) to tree graphs. Based on these tools, we demonstrate KANs' capability to discover various types of physical laws, including conserved quantities, Lagrangians, symmetries, and constitutive laws.

Figures (14)

• Figure 1: Synergizing science and the Kolmogorov-Arnold Network (KAN).
• Figure 2: Top: comparing KAN and MultKAN diagrams. MultKAN has extra multiplication layers $\mathbf{M}$. Bottom: After training on $f(x,y)=xy$, KAN learns an algorithm requiring two addition nodes, while MultKAN requires only one multiplication node.
• Figure 3: Adding auxiliary variables to inputs enhances interpretability. For the relativistic mass equation, $m=m_0/\sqrt{1-v^2/c^2}$, (a) a two-layer KAN is needed if only $(m_0, v, c)$ are used as inputs. (b) If we add $\beta\equiv v/c$ and $\gamma\equiv1/\sqrt{1-\beta^2}$ as auxiliary variables to KANs, a one-layer KAN suffices (seed 0). (c) seed 1 finds a different solution, which is sub-optimal and can be avoided through hypothesis testing (Section 4.3).
• Figure 4: Building modular structures to KANs: (a) multiplicative separability;(b) symmetries.
• Figure 5: KAN compiler (kanpiler) converts symbolic expressions to KANs. (a) how kanpiler works: the symbolic formula is first parsed to an expression tree, which is then converted to a KAN. (b) Applying KANs to 10 equations (selected from the Feynman dataset). (c) Expand a compiled KAN to increase its expressive power.