PO-CKAN:Physics Informed Deep Operator Kolmogorov Arnold Networks with Chunk Rational Structure

TL;DR

This work addresses the challenge of learning accurate, physically consistent solution operators for parametric PDEs. It introduces PO-CKAN, a physics-informed DeepONet that replaces MLPs with Chunk-rational Kolmogorov–Arnold Networks (CKAN) featuring Enhanced Rational Unit activations and chunk-wise parameter sharing to curb quadratic growth. The framework enforces physics via a composite loss including data, initial/boundary, and PDE residual terms, and demonstrates substantial accuracy and efficiency gains over PI-DeepONet across Burgers', Eikonal, fractional, and diffusion–reaction PDEs, achieving up to an ~80% reduction in mean relative $L^2$ error in some cases. Overall, PO-CKAN provides a scalable, interpretable operator-learning paradigm that can preserve physical laws while delivering train-once, predict-many capability for complex, nonlocal, and stiff PDEs, with broad potential for multi-physics and uncertainty-quantified extensions.

Abstract

We propose PO-CKAN, a physics-informed deep operator framework based on Chunkwise Rational Kolmogorov--Arnold Networks (KANs), for approximating the solution operators of partial differential equations. This framework leverages a Deep Operator Network (DeepONet) architecture that incorporates Chunkwise Rational Kolmogorov-Arnold Network (CKAN) sub-networks for enhanced function approximation. The principles of Physics-Informed Neural Networks (PINNs) are integrated into the operator learning framework to enforce physical consistency. This design enables the efficient learning of physically consistent spatio-temporal solution operators and allows for rapid prediction for parametric time-dependent PDEs with varying inputs (e.g., parameters, initial/boundary conditions) after training. Validated on challenging benchmark problems, PO-CKAN demonstrates accurate operator learning with results closely matching high-fidelity solutions. PO-CKAN adopts a DeepONet-style branch--trunk architecture with its sub-networks instantiated as rational KAN modules, and enforces physical consistency via a PDE residual (PINN-style) loss. On Burgers' equation with $ν=0.01$, PO-CKAN reduces the mean relative $L^2$ error by approximately 48\% compared to PI-DeepONet, and achieves competitive accuracy on the Eikonal and diffusion--reaction benchmarks.

Figures (10)

• Figure 1: This figure illustrates the PO-CKAN framework. The model adopts the DeepONet architecture to learn the solution operator mapping input functions to solution functions. The branch and trunk nets are both constructed from our novel CKAN layers. Base Function (Top Panel): The Enhanced Rational Unit (ERU) is used as the computationally efficient and numerically stable base function for all CKAN layers. CKAN Layer (Middle Panel): The CKAN layer is the core innovation. It reduces parameters through a chunk-wise sharing mechanism. Chunks of edges share a single base ERU but each edge retains an individual scalar weight. Training Objective (Bottom Panel): The loss is computed via automatic differentiation and comprises data ($\mathcal{L}_{\text{data}}$), initial condition ($\mathcal{L}_{\text{ic}}$), boundary condition ($\mathcal{L}_{\text{bc}}$), and PDE residual ($\mathcal{L}_{r}$) terms.
• Figure 2: A comparison between our CKAN (configured with a 2×2 chunk) and standard KAN and MLP models. Unlike the conventional KAN, which assigns a distinct function to every input–output connection, CKAN employs one shared base function for each chunk of edges.
• Figure 3: Modified network architecture
• Figure 4: PO-CKAN vs. PI-DeepONet:loss at different viscosity coefficients $\nu$. ($\nu$ = 0.01, 0.03, 0.05)
• Figure 5: Visualization of PI-DeepONet and PO-CKAN outputs for Burgers' equation ($\nu=0.05$). Shown, from left to right, are the exact solution, the network approximation, and the absolute error.