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<p>Training neural networks to solve partial differential equations (PDEs) can be formulated as a semi-supervised learning problem. Exact values are available only at initial and boundary points, while collocation points across the spatiotemporal domain lack explicit labels. This creates significant challenges during training, making the choice of collocation points and sampling strategies critical for efficiency and accuracy. In this paper, we introduce a novel probability distribution for adaptive sampling, which dynamically concentrates computational effort in regions where the solution exhibits complex behavior and requires higher information density. Integrated with the Deep Galerkin Method (DGM), our approach achieves higher accuracy and faster convergence without additional computational cost. Numerical experiments on benchmark problems including cases with sharply localized peaks, irregular solutions, time-dependent problem, nonlinear problem and high-dimensional domains, demonstrate the robustness and versatility of the method. These results highlight the potential of adaptive sampling techniques to enhance deep learning approaches for PDEs, providing improved performance and computational efficiency applicable to a wide range of real-world problems.</p>

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Prioritized sampling for scalable neural network solutions to complex partial differential equations

  • 期刊论文

  • 发表日期：

    2026年4月23日

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  • Idriss Barbara,
  • Elmehdi Amhraoui,
  • Tawfik Masrour,
  • Mohammed Hadda

摘要

Training neural networks to solve partial differential equations (PDEs) can be formulated as a semi-supervised learning problem. Exact values are available only at initial and boundary points, while collocation points across the spatiotemporal domain lack explicit labels. This creates significant challenges during training, making the choice of collocation points and sampling strategies critical for efficiency and accuracy. In this paper, we introduce a novel probability distribution for adaptive sampling, which dynamically concentrates computational effort in regions where the solution exhibits complex behavior and requires higher information density. Integrated with the Deep Galerkin Method (DGM), our approach achieves higher accuracy and faster convergence without additional computational cost. Numerical experiments on benchmark problems including cases with sharply localized peaks, irregular solutions, time-dependent problem, nonlinear problem and high-dimensional domains, demonstrate the robustness and versatility of the method. These results highlight the potential of adaptive sampling techniques to enhance deep learning approaches for PDEs, providing improved performance and computational efficiency applicable to a wide range of real-world problems.