Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space

arXiv — stat.MLMonday, December 22, 2025 at 5:00:00 AM
  • A new approach to operator learning in Sobolev space has been introduced through Regularized Random Fourier Features (RRFF) and Finite Element Reconstruction (RRFF-FEM), which aims to improve the accuracy of approximating mappings between infinite-dimensional function spaces, particularly in the context of partial differential equations (PDEs). This method leverages random features from multivariate Student's t distributions and incorporates frequency-weighted Tikhonov regularization to mitigate noise effects.
  • The development of RRFF-FEM is significant as it addresses the computational challenges associated with traditional kernel-based operator learning methods, especially when dealing with large datasets and noisy data. By enhancing the robustness and efficiency of operator learning, this approach could lead to more reliable solutions in various applications, including structural mechanics and fluid dynamics.
  • This advancement reflects a broader trend in artificial intelligence and machine learning, where researchers are increasingly focused on developing methods that can effectively handle noise and computational complexity in data-driven modeling. The integration of techniques such as Tikhonov regularization and the exploration of probabilistic frameworks for PDEs highlight the ongoing efforts to refine operator learning methodologies and improve their applicability across diverse scientific and engineering domains.
— via World Pulse Now AI Editorial System

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