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
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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. 该方法首先将求解时域分解为多个连续阶段, 在每个阶段内, 基于当前残差点的损失构建采样概率密度函数, 并依据该函数重新采样部分样本以替换原有残差点, 且每间隔一定训练周期重复上述采样更新过程. 通过引入这一自适应重采样策略, 残差点的空间分布可被动态调整, 从而使样本点自适应聚焦于方程求解的刚性区域, 进而显著加速网络的收敛过程. 考虑到早期阶段的预测精度会直接影响后续阶段的求解结果, STAR-PINNs 引入了因果加权算法, 并提出一种新的因果强度系数自适应更新策略, 使其在训练过程中能够动态调整加权强度. 该设计有效抑制了误差随时间演化的累积效应, 显著提升了长期预测的稳定性与精度. 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 attracted considerable attention as a novel approach for solving partial differential equations (PDEs). Although PINNs offer numerous advantages over traditional numerical methods, effectively ensuring model convergence and accuracy remains a core challenge that demands urgent resolution. To address this, this paper proposes a stage-adaptive resampling physics-informed neural network (stage-Adaptive resampling physics-informed neural networks, STAR-PINNs) for solving evolutionary equations. The method first discretizes the solution time domain into multiple consecutive stages. Within each stage, a sampling probability density function is constructed based on the loss values of the current residual points, and a subset of new sample points is resampled according to this function to replace the original residual points. This resampling and update process is performed repeatedly at fixed training intervals. By incorporating this adaptive resampling strategy, the spatial distribution of residual points is dynamically adjusted, enabling the sample points to adaptively focus on the stiff regions of the equation solution and thereby substantially accelerating the network convergence process. Recognizing that the prediction accuracy of early stages directly impacts the solution results of subsequent stages, STAR-PINNs introduces a causality weighting algorithm and proposes a novel adaptive update strategy for the causality strength coefficient, which enables dynamic adjustment of the weighting intensity during training. This design effectively suppresses the accumulation effect of errors evolving over time, significantly enhancing the stability and accuracy of long-term predictions. To validate its effectiveness, this paper adopts the Allen-Cahn equation—a challenging case for PINNs—as a test case for solution and further compares it with causal training. The results demonstrate that STAR-PINNs significantly reduces training costs while improving accuracy by approximately one order of magnitude, achieving a minimum relative L2 error of 3.11 × 10−5. Further solutions to the reaction equation, reaction-diffusion equation, and wave equation show that the predicted solutions of STAR-PINNs are highly consistent with the reference solutions of the equations.
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