Conventional physics-informed extreme learning machine (PIELM) often faces challenges in solving partial differential equations (PDEs) involving high-frequency and variable-frequency behaviors. To address these challenges, we propose a general Fourier feature physics-informed extreme learning machine (GFF-PIELM). We demonstrate that directly concatenating multiple Fourier feature mappings (FFMs) and an extreme learning machine (ELM) network makes it difficult to determine frequency-related hyperparameters. Fortunately, we find an alternative to establish the GFF-PIELM in three main steps. First, we integrate a variation of FFM into ELM as the Fourier-based activation function, so there is still one hidden layer in the GFF-PIELM framework. Second, we assign a set of frequency coefficients to the hidden neurons, which enables ELM network to capture diverse frequency components of target solutions. Finally, we develop an innovative, straightforward initialization method for these hyperparameters by monitoring the distribution of ELM output weights. GFF-PIELM not only retains the high accuracy, efficiency, and simplicity of the PIELM framework but also inherits the ability of FFMs to effectively handle high-frequency problems. We carry out five case studies with a total of ten numerical examples to highlight the feasibility and validity of the proposed GFF-PIELM, involving high frequency, variable frequency, multi-scale behaviour, irregular boundary and inverse problems. Compared to conventional PIELM, the GFF-PIELM approach significantly improves predictive accuracy without additional cost in training time and architecture complexity. Our results confirm that that PIELM can be extended to solve high-frequency and variable-frequency PDEs with high accuracy, and our initialization strategy may further inspire advances in other physics-informed machine learning (PIML) frameworks.

翻译：传统的物理信息极限学习机（PIELM）在求解涉及高频和变频率行为的偏微分方程（PDEs）时常常面临挑战。为应对这些挑战，本文提出了一种广义傅里叶特征物理信息极限学习机（GFF-PIELM）。我们发现，直接将多个傅里叶特征映射（FFMs）与一个极限学习机（ELM）网络简单拼接，难以确定与频率相关的超参数。幸运的是，我们找到了一种替代方案，通过三个主要步骤构建GFF-PIELM。首先，我们将一种变体的FFM作为基于傅里叶的激活函数集成到ELM中，因此GFF-PIELM框架中仍只包含一个隐藏层。其次，我们为隐藏层神经元分配一组频率系数，使ELM网络能够捕捉目标解的不同频率分量。最后，通过监测ELM输出权重的分布，我们为这些超参数开发了一种创新且简单的初始化方法。GFF-PIELM不仅保持了PIELM框架的高精度、高效性和简洁性，还继承了FFMs有效处理高频问题的能力。我们通过五个案例研究（共包含十个数值算例）来验证所提GFF-PIELM的可行性与有效性，涉及高频、变频率、多尺度行为、不规则边界及反问题。与传统PIELM相比，GFF-PIELM方法在训练时间和架构复杂度未增加的情况下，显著提高了预测精度。我们的结果表明，PIELM能够扩展用于高精度求解高频和变频率PDEs，且我们的初始化策略可能进一步启发其他物理信息机器学习（PIML）框架的改进。