Physics-Informed Kolmogorov-Arnold Networks (PI-KAN): Writing Newton's Laws Directly into Neural Networks

Forum topic · 小凯 · 2026-05-03

Summary

This post from zhichai.net explains Physics-Informed Kolmogorov-Arnold Networks (PI-KAN), an architecture for solving partial differential equations (PDEs) with stronger physics consistency than conventional neural networks. It argues that standard MLPs approximate physical curves with fixed activations like ReLU, causing conservation-law violations and poor extrapolation outside training data. PI-KAN combines the KAN architecture—where learnable B-spline activation functions sit on edges rather than nodes—with physics-informed training that embeds governing equations such as Newton's laws and Navier-Stokes as penalty terms in the loss function. The author reports that PI-KAN achieves high PDE-solving accuracy with tens of times fewer parameters than traditional physics-informed neural networks (PINNs), converges faster, and resists catastrophic forgetting thanks to the locality of splines. The post closes with a Feynman-inspired takeaway: scientific AI should mirror the mathematical structure of physical laws rather than guess from data in a black box; upgrading basis functions is more productive than stacking layers.

Translated and adapted from a Chinese forum post on zhichai.net.

Why ordinary neural networks struggle with physics

Traditional MLPs are compared to a student who can only draw straight lines: when simulating fluid dynamics or heat conduction, they approximate elegant physical curves with thousands of rigid "folded lines" (fixed activations such as ReLU). Because no physical formulas are built in, once inputs go beyond the training range, predictions fly off — what the author calls the collapse of physical conservation laws during fitting.

What PI-KAN does differently

Learnable activations on edges (KAN): KAN moves activation functions off the nodes and onto the edges of the network. Instead of stacking dead fold-lines, PI-KAN traces physical laws with flexible, self-adaptive B-spline curves.
Physics takes over the loss: Physics-informed training embeds Newton's laws and the Navier-Stokes equations directly as penalty terms in the loss. If the network's solution violates energy conservation, it is penalized — "using high-dimensional physical laws to suppress low-dimensional probabilistic guessing."
Fewer parameters, faster convergence, no forgetting: Experiments cited in the paper show PI-KAN solves difficult PDEs with far fewer parameters (dozens of times fewer than traditional PINNs) at high accuracy. The locality of spline basis functions also makes it naturally resistant to catastrophic forgetting.

The Feynman-style judgment: intelligence comes from structural isomorphism

The author argues that "AI for science" should not mean feeding data into a giant black box. The real goal is to replicate the mathematical logic of the universe in the underlying topology of the network. When the algorithm's structure is highly isomorphic to the structure of the physical equations, AI no longer has to guess how gravity works — gravity is woven directly into its instincts.

Takeaway

For scientific computing, stop blindly adding layers. Upgrade your basis functions instead. If your algorithm grows along the texture of physical law, you can find truth with minimal compute — and read nature's hidden equations out of a transparent structure.

*Source tags: PIKAN, KolmogorovArnoldNetworks, PhysicsInformed, AI4Science, PDE.*