Efficiently Solving Complex PDEs with DRM-QMC Algorithm

doi:10.29350/qjps.2025.190627

	Efficiently Solving Complex PDEs with DRM-QMC Algorithm
Al-Qadisiyah Journal of Pure Science
Volume 30, Issue 2, December 2025, Pages 148-164
DOI: 10.29350/qjps.2025.190627
Abstract
The Deep Ritz Method - Quasi-Monte Carlo (DRM-QMC) algorithm represents a novel approach that integrates deep learning and Quasi-Monte Carlo sampling for efficiently solving complex partial differential equations (PDEs). By leveraging neural networks for solution approximation and QMC sampling for point generation, the algorithm aims to provide accurate solutions with improved convergence rates for high-dimensional PDEs. The DRM-QMC algorithm demonstrates promising potential in accurately solving complex PDEs with enhanced convergence properties. Through numerical experiments and convergence analysis, the algorithm showcases improved accuracy and convergence rates, highlighting the effectiveness of deep learning methods in tackling high-dimensional PDEs. The comparison of total errors provides valuable insights into the algorithm's performance and reliability in solving intricate PDEs. Regularization techniques and QMC sampling are identified as key factors influencing the convergence properties of the DRM-QMC algorithm. By optimizing these components, researchers can enhance the algorithm's efficiency and reliability in approximating solutions to high-dimensional PDEs. The equation 𝐸(𝜃) ≤ 𝐶(log 𝑓(𝑁))^𝛼𝑁^−𝛽 serves as a quantitative measure for assessing the algorithm's convergence properties and guiding future advancements in deep learning methods for PDEs. Overall, the DRM-QMC algorithm represents a significant advancement in computational mathematics, offering a promising approach for efficiently solving complex PDEs with improved accuracy and convergence rates.
Keywords
DRM; QMC; PDEs; Convergence Analysis; and Deep Learning

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