LFR-PINO: A Layered Fourier Reduced Physics-Informed Neural Operator for Parametric PDEs

2025-06-21

AI Summary

Existing physics-informed neural operators (PINO) suffer from two key limitations in solving parameterized PDEs: restricted expressivity due to fixed bases or coefficients, and prohibitive computational cost induced by high-dimensional parameter-to-operator mappings. This paper proposes a general-purpose physics-informed neural operator featuring: (1) a hierarchical hypernetwork architecture that generates layer-wise weights conditioned on input parameters, enabling adaptive representation; and (2) a frequency-domain dimensionality reduction mechanism with selective preservation of dominant spectral modes, drastically reducing parameter count. The method integrates Fourier neural operators, physics-constrained embedding, and frequency-domain feature pruning. Evaluated on four canonical PDE families, it achieves 22.8%–68.7% lower mean relative error and 28.6%–69.3% reduced memory footprint versus state-of-the-art methods, demonstrating superior accuracy, generalization across parameter regimes, and computational efficiency.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Physics-informed neural operators have emerged as a powerful paradigm for solving parametric partial differential equations (PDEs), particularly in the aerospace field, enabling the learning of solution operators that generalize across parameter spaces. However, existing methods either suffer from limited expressiveness due to fixed basis/coefficient designs, or face computational challenges due to the high dimensionality of the parameter-to-weight mapping space. We present LFR-PINO, a novel physics-informed neural operator that introduces two key innovations: (1) a layered hypernetwork architecture that enables specialized parameter generation for each network layer, and (2) a frequency-domain reduction strategy that significantly reduces parameter count while preserving essential spectral features. This design enables efficient learning of a universal PDE solver through pre-training, capable of directly handling new equations while allowing optional fine-tuning for enhanced precision. The effectiveness of this approach is demonstrated through comprehensive experiments on four representative PDE problems, where LFR-PINO achieves 22.8%-68.7% error reduction compared to state-of-the-art baselines. Notably, frequency-domain reduction strategy reduces memory usage by 28.6%-69.3% compared to Hyper-PINNs while maintaining solution accuracy, striking an optimal balance between computational efficiency and solution fidelity.

Problem

• Solving parametric PDEs with limited expressiveness and computational challenges
• Reducing parameter count while preserving spectral features efficiently
• Achieving error reduction and memory efficiency in PDE solutions

Innovation

• Layered hypernetwork for specialized parameter generation
• Frequency-domain reduction to cut parameter count
• Efficient universal PDE solver via pre-training