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物理信息神经网络(PINNs)作为人工智能助力科学研究(AI for Science)求解偏微分方程(PDEs)的一种无网格化求解框架, 近年来受到广泛关注. 然而, 传统PINNs存在局限性: 一方面, PINNs网络结构使用单向信息传递的多层感知机(MLPs), 难以有效聚焦序列数据中蕴含的关键特征, 信息表征能力弱; 另一方面, PINNs的损失函数为嵌入物理约束的二次罚函数, 其未受约束而无限膨胀的惩罚因子影响模型训练寻优效率. 为应对上述挑战, 本文提出一种基于信息表征-损失优化改进的PINNs—allaPINNs, 旨在增强模型关键特征提取和训练寻优能力, 提升其求解PDEs数值解的准确性和泛化能力. 在信息表征方面, allaPINNs引入高效线性注意力(LA)增强模型关键特征识别能力, 同时降低权重动态加权的计算复杂度. 在损失优化方面, allaPINNs通过引入增广拉格朗日(AL)函数重构目标损失函数, 利用可学习的拉格朗日乘子和惩罚因子有效调控各损失残差项的相互作用. 由Helmholtz、Black-Scholes、Burgers和非线性Schrödinger四个基准方程验证allaPINNs的有效性. 结果表明, allaPINNs能够有效求解不同类型PDEs, 并展现出卓越的数值解预测精度与泛化能力. 相较于当前先进PINNs其预测精度提升一至两个数量级.Physics-informed neural networks (PINNs) have recently garnered significant attention as a meshless solution framework for solving partial differential equations (PDEs) in the context of AI-assisted scientific research (AI for Science). However, traditional PINNs exhibit certain limitations. On one hand, their network architecture, typically multilayer perceptrons (MLPs) with unidirectional information transfer, struggles to effectively capture key features embedded in sequential data, resulting in weak information characterization. On the other hand, the loss function of PINNs, a quadratic penalty function embedded with physical constraints, has an unconstrained and infinitely inflated penalty factor that affects the efficiency of the model's training optimization search. To address these challenges, this paper proposes an improved PINN based on information representation and loss optimization, termed allaPINNs, which aims to enhance the model’s key feature extraction capability and training optimization search ability, thereby improving its accuracy and generalization for solving numerical solutions of PDEs. In terms of information characterization, allaPINNs introduces efficient linear attention (LA) to enhance the model’s ability to identify key features while reducing the computational complexity of dynamic weighting. In terms of loss optimization, allaPINNs reconstructs the objective loss function by introducing the augmented Lagrangian (AL) function, utilizing learnable Lagrangian multipliers and penalty factors to efficiently regulate the interaction of each loss residual term. The feasibility of allaPINNs is validated through four benchmark equations: Helmholtz, Black-Scholes, Burgers, and nonlinear Schrödinger. The results demonstrate that allaPINNs can effectively solve various PDEs of different complexities and exhibit excellent numerical solution prediction accuracy and generalization ability. Compared to the current state-of-the-art PINNs, the predictive accuracy is improved by one to two orders of magnitude.
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Keywords:
- physics-informed neural networks /
- linear attention /
- augmented Lagrangian functions /
- solving partial differential equations
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图 2 使用allaPINNs求解基准方程的预测解和解析解对比结果
Fig. 2. Comparison of predicted and analytical solutions for solving the benchmark equations using allaPINNs.
图 3 使用allaPINNs在$ t = 0,\; 0.3,\; 0.66,\; 0.88 $四个时刻求解非线性Schrödinger方程的振幅快照
Fig. 3. Snapshots of amplitudes for solving nonlinear Schrödinger equation at four moments $ t = 0,\; 0.3,\; 0.66,\; 0.88 $ using allaPINNs
图 4 使用allaPINNs求解基准方程的逐点绝对误差分布
Fig. 4. Pointwise absolute error distributions for solving benchmark equations using allaPINNs.
图 5 最优拉格朗日乘子$ \lambda_{*} $的频率直方图, 其中, Helmholtz方程包含$ \lambda_{1}^{*} $, Black-Scholes方程包含$ \lambda_{1}^{*}{\text{—}}\lambda_{3}^{*} $, Burgers方程包含$ \lambda_{1}^{*} $和$ \lambda_{2}^{*} $, 非线性Schrödinger方程包含$ \lambda_{1}^{*}{\text{—}}\lambda_{6}^{*} $
Fig. 5. The frequency histogram show the optimal Lagrangian multipliers $ \lambda^{*} $, where the Helmholtz equation contains $ \lambda_{1}^{*} $, the Black-Scholes equation contains $ \lambda_{1}^{*}-\lambda_{3}^{*} $, the Burgers equation contains $ \lambda_{1}^{*}, \lambda_{2}^{*} $, and nonlinear Schrödinger equation contains $ \lambda_{1}^{*}-\lambda_{6}^{*} $.
图 6 惩罚因子σ的学习过程, 其中(a) (b) (c) (d)中的$ \sigma^{*} $分别位于第48665、49420、43467和48933轮迭代, 对应的数值分别为2517.21、35.45、5.02和52.08
Fig. 6. The learning process of the penalty factor σ, where $ \sigma^{*} $ for (a) (b) (c) (d) are located at rounds 48665, 49420, 43467, and 48933, corresponding to values of 2517.21, 35.45, 5.02, and 52.08, respectively.
图 7 allaPINNs在训练过程中的$ L^{2} $相对泛化误差
Fig. 7. The $ L^{2} $ relative generalization error of allaPINNs during training.
表 1 allaPINNs模型和当前先进PINNs模型求解四个基准方程实验的$ L^{2} $相对误差对比结果
Table 1. Comparison of $ L^{2} $ relative errors between allaPINNs model and current state-of-the-art PINNs model for solving the four benchmark equations.
物理信息神经求解器 求解方程 PINNs[23] AL-PINNs[61] f-PICNN[30] KINN[31] allaPINNs (本文) Helmholtz 5.63 × 10–2 1.82 × 10–3 2.51 × 10–3 1.08 × 10–3 8.06 × 10–4 Black-Scholes 7.18 × 10–2 7.41 × 10–3 5.24 × 10–3 4.35 × 10–3 3.48 × 10–4 Burgers 7.04 × 10–2 3.39 × 10–3 2.49 × 10–3 3.02 × 10–3 8.31 × 10–4 非线性Schrödinger 2.09 × 10–2 1.55 × 10–3 4.18 × 10–3 1.37 × 10–3 6.71 × 10–4 
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表 2 allaPINNs模型在不同网络结构和损失函数条件下的平均$ L^{2} $相对误差对比结果
Table 2. Comparison of mean $ L^{2} $ relative errors of allaPINNs model with different network structure and loss function conditions.
网络结构 基准方程平均$ L^{2} $
相对误差损失函数 基准方程平均$ L^{2} $
相对误差MLPs 7.26 × 10–2 罚函数 8.55 × 10–3 CNN 3.09 × 10–3 罚函数
(高斯噪声)1.63 × 10–3 LA 6.61 × 10–4 AL 6.61 × 10–4
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表 B1 各对比模型中超参数配置和求解基准方程时间消耗
Table B1. Hyperparameter configuration and time consumption for solving the benchmark equations in each comparison model.
物理信息神经求解器 对比指标 PINNs[23] AL-PINNs[61] f-PICNN[30] KINN[31] allaPINNs (本文) 网络层数 8 8 6 5 5 神经元数量 200 256 128 80 64 参数学习率 1 × 10–3 1 × 10–4 1 × 10–4 1 × 10–4 1 × 10–3 求解Helmholtz时间消耗/s 1476 1520 1529 1606 1513 求解Black-Scholes时间消耗/s 1538 1585 1604 1714 1624 求解Burgers时间消耗/s 1519 1574 1593 1636 1588 求解非线性Schrödinger时间消耗/s 1545 1628 1650 1745 1661
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