Enhancing Physics-Informed Neural Networks with domain-aware Fourier features: Towards improved performance and interpretable results

Physics-Informed Neural Networks incorporate underlying physics into neural networks by embedding partial differential equations into their loss function. Despite their success in learning underlying physics, these models remain difficult to train and interpret. In this work, a novel approach is proposed with Domain-aware Fourier Features for the positional encoding of the input space. These features encapsulate domain-specific characteristics, such as the geometry and boundary conditions, and, unlike Random Fourier Features, eliminate the need for explicit boundary condition loss terms and loss balancing schemes, while simplifying the optimization process and reducing the computational cost of training. We further develop a Layer-wise Relevance Propagation-based explainability framework tailored to Physics-Informed Neural Networks, enabling extraction of relevance attribution scores for the input space. It is demonstrated that the models with Domain-aware Fourier Features achieve orders-of-magnitude lower errors and converge faster compared to vanilla and Random Fourier Features-based ones. Furthermore, the explainability analysis reveals that the proposed approach leads to more physically consistent feature attributions, in contrast to more scattered and less physics-relevant patterns displayed by others. These results demonstrate that our approach not only enhances Physics-Informed Neural Networks’ accuracy and efficiency but also improves interpretability, laying groundwork for more robust and informative physics-informed learning.