PILD: Physics-Informed Learning via Diffusion

Diffusion models have emerged as powerful generative tools for modeling complex data distributions, yet their purely data-driven nature limits applicability in practical engineering and scientific problems where physical laws need to be followed. This paper proposes Physics-Informed Learning via Diffusion (PILD), a framework that unifies diffusion modeling and first-principles physical constraints by introducing a virtual residual observation sampled from a Laplace distribution to supervise generation during training. To further integrate physical laws, a conditional embedding module is incorporated to inject physical information into the denoising network at multiple layers, ensuring consistent guidance throughout the diffusion process. The proposed PILD framework is concise, modular, and broadly applicable to problems governed by ordinary differential equations, partial differential equations, as well as algebraic equations or inequality constraints. Extensive experiments across engineering and scientific tasks including estimating vehicle trajectories, tire forces, Darcy flow and plasma dynamics, demonstrate that our PILD substantially improves accuracy, stability, and generalization over existing physics-informed and diffusion-based baselines.

Fourier Feature Attribution: A New Efficiency Attribution Method

The study of neural networks from the perspective of Fourier features has garnered significant attention. While existing analytical research suggests that neural networks tend to learn low-frequency features, a clear attribution method for identifying the specific learned Fourier features has remained elusive. To bridge this gap, we propose a novel Fourier feature attribution method grounded in signal decomposition theory. Additionally, we analyze the differences between game-theoretic attribution metrics for Fourier and spatial domain features, demonstrating that game-theoretic evaluation metrics are better suited for Fourier-based feature attribution. Our experiments show that Fourier feature attribution exhibits superior feature selection capabilities compared to spatial domain attribution methods. For instance, in the case of Vision Transformers (ViTs) on the ImageNet dataset, only $8\%$ of the Fourier features are required to maintain the original predictions for $80\%$ of the samples. Furthermore, we compare the specificity of features identified by our method against traditional spatial domain attribution methods. Results reveal that Fourier features exhibit greater intra-class concentration and inter-class distinctiveness, indicating their potential for more efficient classification and explainable AI algorithms.