CI-PINN：用于求解演化方程的因果积分物理信息神经网络

中文摘要 AI 辅助

物理信息神经网络（PINN）通过将控制物理定律纳入训练损失来求解偏微分方程（PDE）。然而，对于演化方程，其传统的逐点时空表示并未显式编码时间依赖性，这可能阻碍准确预测。为缓解这一局限，本工作提出了一种新颖的神经网络架构，称为因果积分神经网络（CinNet）。CinNet的核心模块是一个Volterra型因果积分项，它聚合历史特征以编码时间依赖性，从而在架构层面而非像许多现有方法那样在训练层面引入时间因果性。基于CinNet，我们进一步开发了用于求解演化方程的因果积分物理信息神经网络（CI-PINN）。在基准演化方程上的大量数值实验表明，所提方法在解精度方面优于各种基线PINN变体，且在稀疏配置场景下具有显著优势。额外的实证分析显示，CI-PINN对超参数选择具有低敏感性，而消融研究证实了所提网络组件的有效性。

英文摘要

Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by incorporating governing physical laws into the training loss. For evolution equations, however, their conventional pointwise space--time representation does not explicitly encode temporal dependence, which can hinder accurate prediction. To mitigate this limitation, this work proposes a novel neural architecture termed a causal integral neural network (CinNet). The core module of CinNet is a Volterra-type causal integral term, which aggregates historical features to encode temporal dependence, thereby incorporating temporal causality at the architectural level rather than through training-level modifications as in many existing methods. Building on CinNet, we further develop a causal integral physics-informed neural network (CI-PINN) for solving evolution equations. Extensive numerical experiments on benchmark evolution equations demonstrate that the presented method outperforms various baseline PINN variants in terms of solution accuracy, with pronounced superiority under sparse-collocation scenarios. Additional empirical analyses show that CI-PINN exhibits low sensitivity to hyperparameter choices, while ablation studies confirm the effectiveness of the proposed network components.

发表机构

• Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）
• Beijing Normal University（北京师范大学）
• Guangzhou Nanfang College（广州南方学院）
• Louisiana State University（路易斯安那州立大学）
• Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）

机构由 AI 辅助整理，请以论文原文为准。