Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs

Research Approach

Traditional Physics-Informed Neural Networks (PINNs) typically rely on Multi-Layer Perceptrons (MLPs), which often suffer from spectral bias—a tendency to learn low-frequency components of a solution while struggling with high-frequency details or sharp gradients. To address this, the authors propose QCPIKAN, a hybrid framework that replaces the standard MLP architecture with Kolmogorov-Arnold Networks (KANs) integrated into a quantum-classical pipeline. The model uses Chebyshev polynomials as learnable basis functions within the KAN layers to enhance function representation, followed by a parameterized quantum circuit (the quantum core) to map features into a high-dimensional Hilbert space.

Key Contributions

The primary innovation of QCPIKAN is its ability to leverage the spectral-approximation properties of Chebyshev polynomials. By replacing fixed activation functions with learnable univariate functions, the network can adaptively represent complex nonlinear mappings. Theoretical analysis in the paper demonstrates that this design shifts the error convergence rate for high-frequency components from algebraic to exponential. This is particularly significant for simulating fluid flow in porous media, where steep gradients and displacement fronts often lead to numerical dispersion in standard PINN models.

Performance and Validation

The authors validated QCPIKAN across three canonical seepage scenarios: single-phase flow, component transport, and two-phase flow. In these tests, QCPIKAN consistently outperformed traditional quantum-classical PINNs (QCPINNs). Specifically, the model showed superior global prediction accuracy, more precise tracking of dynamic evolution, and better localization of displacement fronts. By combining the parameter efficiency of KANs with the high-dimensional feature-mapping capabilities of quantum circuits, the framework provides a robust alternative for solving complex, multiscale PDEs.