Physics-Informed Kolmogorov-Arnold Networks (PI-KAN): Embedding Physical Laws into Neural Networks

Forum topic · 小凯 · 2026-05-03

Summary

This forum post discusses Physics-Informed Kolmogorov-Arnold Networks (PI-KAN), a neural architecture for solving partial differential equations (PDEs) with physics embedded directly into the model. The author contrasts traditional MLPs, which approximate physical curves with piecewise-linear activations like ReLU and can violate conservation laws when extrapolating, with KAN's approach of placing learnable activations (B-splines) on edges rather than fixed nodes. PI-KAN adds physics-informed loss terms based on equations such as Navier-Stokes, penalizing predictions that violate physical laws. The post claims PI-KAN achieves high accuracy with tens of times fewer parameters than traditional PINNs, converges faster, and resists catastrophic forgetting due to the local nature of splines. The takeaway: rather than stacking layers in a black box, upgrading the basis functions and aligning network structure with physical equations yields efficient, transparent scientific computing.

This post from zhichai.net offers an accessible take on Physics-Informed Kolmogorov-Arnold Networks (PI-KAN), arguing that they bring a true "physics scalpel" to the field of PDE solving.

Why traditional neural networks struggle with physics

The author compares a conventional MLP to a "student who only draws straight lines":

The pain point: when simulating fluid dynamics or heat conduction, MLPs approximate smooth physical curves with thousands of piecewise-linear segments (ReLU-style activations). With no built-in formulas, predictions can fly wildly out of range once inputs leave the training distribution — described as the collapse of physical conservation laws during fitting.

How PI-KAN works

Learnable activations: KAN moves activation functions from nodes to edges. Instead of rigid polyline segments, PI-KAN traces physical laws with flexible, self-adaptive B-splines.
Physics-informed training: physical laws such as Newton's laws and the Navier-Stokes equations are added directly as penalty terms in the loss function. Curves that violate energy conservation get "severely penalized" —压制 probabilistic guessing with high-dimensional physical rules.
Efficiency and stability: experiments cited in the referenced paper show PI-KAN solves difficult physical equations with high accuracy using far fewer parameters (tens of times fewer than traditional PINNs), and the locality of splines makes it naturally resistant to catastrophic forgetting.

The Feynman-style takeaway

> "Doing science with AI" is not about feeding data into a giant black box. It is about whether you can replicate the mathematical logic of the universe in the underlying topology of the neural network.

When the algorithm's structure is highly isomorphic to the structure of the physical equations, the author argues, you don't need AI to "guess" how gravity works — you weave it into the network's instincts via KAN's elegant mathematical edges.

Practical advice from the post: when tackling scientific computing problems, stop blindly adding network 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.