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一种基于阶段自适应重采样的物理信息神经网络

A STAGE-ADAPTIVE RESAMPLING PHYSICS-INFORMED NEURAL NETWORK

  • 摘要: 近年来,物理信息神经网络 (Physics-Informed Neural Networks, PINNs) 作为求解偏微分方程的新方法受到广泛关注。尽管PINNs相较于传统数值方法具有诸多优势,但如何有效保证模型的收敛性及其精度仍是当前亟待解决的核心问题。为此,文章提出了一种基于阶段自适应重采样的物理信息神经网络 (Stage-Adaptive Resampling Physics-Informed Neural Networks, STAR-PINNs) 用于求解发展方程。该方法通过将时域分解为多个阶段,并在每个阶段内采用一种自适应重采样策略动态调整残差点分布,使得残差点聚焦于难以拟合的刚性区域,从而加速网络收敛。鉴于早期阶段的预测精度直接影响后期结果,STAR-PINNs 在各阶段损失函数中设计并引入一种因果加权策略,确保训练过程严格满足物理因果关系,降低了随时间推移产生的误差累积。为验证效果,文章以PINNs难以求解的Allen-Cahn方程作为测试案例进行求解,并进一步与时间因果算法进行了对比,结果表明STAR-PINNs 显著降低了训练成本且精度提升约一个数量级,相对L2误差最低达到了3.11×10-5。进一步对反应方程、反应扩散方程及波动方程进行了求解,结果表明STAR-PINNs的预测解与方程的参考解保持高度一致。

     

    Abstract: In recent years, Physics-Informed Neural Networks (PINNs) have gained significant attention as a novel approach for solving partial differential equations. Despite their advantages over traditional numerical methods, effectively ensuring model convergence and accuracy remains a critical challenge. To address this, we propose Stage-Adaptive Resampling Physics-Informed Neural Networks (STAR-PINNs) for solving evolution equations. This method decomposes the temporal domain into multiple stages and employs an adaptive resampling strategy within each stage to dynamically adjust the distribution of collocation points, concentrating them on stiff regions difficult to fit, thereby accelerating network convergence. Given that prediction accuracy in early stages directly impacts later results, STAR-PINNs incorporates a causality weighting strategy into the stage-wise loss functions, ensuring strict adherence to physical causality during training and reducing error accumulation over time. To validate efficacy, the notoriously challenging Allen-Cahn equation—known for its difficulty in conventional PINN solutions—serves as a test case. Comparative results with temporally causal algorithms demonstrate that STAR-PINNs significantly reduce training costs while improving accuracy by approximately an order of magnitude, achieving a minimal relative L2 error of 3.11×10-5. Further validation on reaction equations, reaction-diffusion equations, and wave equations confirms that STAR-PINNs' predictions maintain high consistency with reference solutions across all cases.

     

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