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  • Open Access

Quantum physics-informed neural network with residual-based adaptive refinement for solving partial differential equations

Le Li1, Junkai Yang1, Qingle Wang2, and Zhichao Zhang1,*

  • *Contact author: zhichao@ustb.edu.cn
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Phys. Rev. Research 8, 033042 – Published 10 July, 2026

DOI: https://doi.org/10.1103/lb19-wv6f

Abstract

In this paper, we propose a quantum physics-informed neural network (QPINN) to solve partial differential equations (PDEs) by using residual-based adaptive refinement (RAR). The method incorporates a residual-driven adaptive sampling mechanism into the training of QPINN, which aims to enhance convergence and accuracy by enforcing physical constraints in regions with high residuals. Then, we conduct numerical experiments on the one-dimensional viscous Burgers’ equation and diffusion equation, which have different solution structures, as well as the three-dimensional heat equation. The results show that QPINN with RAR reduces the number of parameters by about 44%, and achieves reductions of 31.65%–39.64%, 15.77%–20%, and 3.31%–23.40% over physics-informed neural network (PINN), PINN with RAR, and QPINN in mean relative L2 error, respectively. In summary, it improves model convergence and solution accuracy even with limited sampling points and fewer parameters. These advantages further illustrate that QPINN with RAR is an efficient and stable pathway to solve complex and high-dimensional PDEs.

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Outline

Information

Phys. Rev. Research 8, 033042– Published 10 July, 2026
  • Received 23 April 2026
  • Accepted 22 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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