标题： BO-SA-PINNs：基于贝叶斯优化的自适应物理信息神经网络用于自动设计偏微分方程求解器 显示英文标题

标题： BO-SA-PINNs: Self-adaptive physics-informed neural networks based on Bayesian optimization for automatically designing PDE solvers

摘要： 物理信息神经网络（PINNs）正成为求解偏微分方程（PDEs）的一种流行替代方法。然而，它们需要对不同PDE的网络超参数、采样方法以及损失函数权重进行专门的手动修改，这降低了求解器的效率。本文提出了一种通用的多阶段框架，即BO-SA-PINNs，以缓解这一问题。在第一阶段，使用贝叶斯优化（BO）选择训练过程的超参数，并基于预训练结果自动确定适合PDE的网络架构、学习率、采样点分布和损失函数权重。所提出的基于实验结果的超参数搜索空间可以提高BO识别最优超参数的效率。在选择了适当的超参数后，我们在第二阶段引入了全局自适应（SA）机制。利用第二阶段的预训练模型和损失信息，采用指数移动平均（EMA）方法优化损失函数权重，并通过基于残差的自适应细化与分布（RAR-D）优化采样点分布。在第三阶段，使用L-BFGS进行稳定训练。此外，我们引入了一种新的激活函数，使BO-SA-PINNs能够实现更高的精度。在数值实验中，我们进行了对比和消融实验，验证了该模型在Helmholtz、Maxwell、Burgers和高维Poisson方程上的性能。对比实验结果显示，我们的模型在测试用例中可以实现更高的精度和更少的迭代次数，而消融实验表明了每项改进的积极影响。 显示英文摘要

摘要： Physics-informed neural networks (PINNs) is becoming a popular alternative method for solving partial differential equations (PDEs). However, they require dedicated manual modifications to the hyperparameters of the network, the sampling methods and loss function weights for different PDEs, which reduces the efficiency of the solvers. In this paper, we pro- pose a general multi-stage framework, i.e. BO-SA-PINNs to alleviate this issue. In the first stage, Bayesian optimization (BO) is used to select hyperparameters for the training process, and based on the results of the pre-training, the network architecture, learning rate, sampling points distribution and loss function weights suitable for the PDEs are automatically determined. The proposed hyperparameters search space based on experimental results can enhance the efficiency of BO in identifying optimal hyperparameters. After selecting the appropriate hyperparameters, we incorporate a global self-adaptive (SA) mechanism the second stage. Using the pre-trained model and loss information in the second-stage training, the exponential moving average (EMA) method is employed to optimize the loss function weights, and residual-based adaptive refinement with distribution (RAR-D) is used to optimize the sampling points distribution. In the third stage, L-BFGS is used for stable training. In addition, we introduce a new activation function that enables BO-SA-PINNs to achieve higher accuracy. In numerical experiments, we conduct comparative and ablation experiments to verify the performance of the model on Helmholtz, Maxwell, Burgers and high-dimensional Poisson equations. The comparative experiment results show that our model can achieve higher accuracy and fewer iterations in test cases, and the ablation experiments demonstrate the positive impact of every improvement.