🤖 AI Summary
This work addresses persistent challenges in solving partial differential equations (PDEs)—notably slow convergence of high-frequency errors, severe numerical dispersion, and insufficient physical consistency—by introducing QCPIKAN, a hybrid quantum-classical architecture that integrates Chebyshev polynomial-based Kolmogorov–Arnold Network (KAN) layers with parameterized quantum circuits. To the best of our knowledge, this is the first framework to combine KANs with quantum computing for PDE solving, embedding physical constraints directly into the loss function. Theoretical analysis demonstrates exponential convergence of high-frequency errors, substantially mitigating numerical dispersion. Evaluated on three representative scenarios of flow through porous media, QCPIKAN consistently outperforms existing methods in global accuracy, local error control, dynamic evolution tracking, and front localization.
📝 Abstract
We develop QCPIKAN, the first quantum-classical physics-informed Kolmogorov-Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev-polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high-frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single-phase flow, component transport and two-phase flow. Compared with existing quantum-classical physics-informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.