Enhancing Physics-Informed Neural Networks with Domain-aware Fourier Features: Towards Improved Performance and Interpretable Results

2026-03-03

AI Summary

This work addresses the well-known challenges of traditional physics-informed neural networks (PINNs)—including difficult training dynamics, slow convergence, and limited interpretability—by introducing domain-aware Fourier features (DaFFs). DaFFs uniquely embed geometric and boundary information directly into the input encoding, thereby eliminating the need for explicit boundary loss terms and complex loss balancing strategies. Furthermore, the authors integrate a Layer-wise Relevance Propagation (LRP) framework tailored for PINNs to enable physically consistent feature attribution. Experimental results demonstrate that the proposed approach reduces prediction errors by several orders of magnitude compared to conventional PINNs and methods based on random Fourier features, while also achieving significantly faster convergence and producing explanations that align more closely with underlying physical principles.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Computer Vision: Low Level & Physics-based Vision
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Physics-Informed Neural Networks
• interpretability
• training difficulty
• boundary conditions
• partial differential equations

Innovation

• Domain-aware Fourier Features
• Physics-Informed Neural Networks
• Interpretability
• Boundary Condition Encoding
• Layer-wise Relevance Propagation