Physics-Informed Neural Networks (PINNs) incorporate physics into neural networks by embedding partial differential equations (PDEs) into their loss function. Despite their success in learning the underlying physics, PINN models remain difficult to train and interpret. In this work, a novel modeling approach is proposed, which relies on the use of Domain-aware Fourier Features (DaFFs) for the positional encoding of the input space. These features encapsulate all the domain-specific characteristics, such as the geometry and boundary conditions, and unlike Random Fourier Features (RFFs), eliminate the need for explicit boundary condition loss terms and loss balancing schemes, while simplifying the optimization process and reducing the computational cost associated with training. We further develop an LRP-based explainability framework tailored to PINNs, enabling the extraction of relevance attribution scores for the input space. It is demonstrated that PINN-DaFFs achieve orders-of-magnitude lower errors and allow faster convergence compared to vanilla PINNs and RFFs-based PINNs. Furthermore, LRP analysis reveals that the proposed leads to more physically consistent feature attributions, while PINN-RFFs and vanilla PINNs display more scattered and less physics-relevant patterns. These results demonstrate that DaFFs not only enhance PINNs' accuracy and efficiency but also improve interpretability, laying the ground for more robust and informative physics-informed learning.

翻译：物理信息神经网络（PINNs）通过将偏微分方程（PDEs）嵌入其损失函数，将物理规律融入神经网络。尽管其在学习底层物理规律方面取得了成功，PINN 模型仍然难以训练和解释。本文提出了一种新颖的建模方法，该方法依赖于使用领域感知傅里叶特征（DaFFs）对输入空间进行位置编码。这些特征封装了所有领域特定的特性，如几何形状和边界条件，并且与随机傅里叶特征（RFFs）不同，它消除了对显式边界条件损失项和损失平衡方案的需求，同时简化了优化过程并降低了训练相关的计算成本。我们进一步开发了一个专为 PINNs 设计的基于 LRP 的可解释性框架，能够提取输入空间的相关性归因分数。实验证明，与原始 PINNs 和基于 RFFs 的 PINNs 相比，PINN-DaFFs 实现了数量级更低的误差，并允许更快的收敛。此外，LRP 分析表明，所提出的方法能产生物理一致性更强的特征归因，而 PINN-RFFs 和原始 PINNs 则显示出更分散且与物理相关性较弱的模式。这些结果表明，DaFFs 不仅提高了 PINNs 的准确性和效率，还增强了其可解释性，为构建更稳健、信息更丰富的物理信息学习奠定了基础。