On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks (2012.10047v1)

Abstract: Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale random Fourier features, and justify how such coordinate embedding layers can lead to robust and accurate PINN models. Numerical examples are presented for several challenging cases where conventional PINN models fail, including wave propagation and reaction-diffusion dynamics, illustrating how the proposed methods can be used to effectively tackle both forward and inverse problems involving partial differential equations with multi-scale behavior. All code an data accompanying this manuscript will be made publicly available at \url{https://github.com/PredictiveIntelligenceLab/MultiscalePINNs}.

• The paper quantifies spectral bias in neural networks as an NTK eigenvector bias that influences the learning of high-frequency PDE components via Fourier feature mappings.
• The paper introduces novel multi-scale architectures leveraging Fourier feature embeddings that significantly reduce prediction errors in benchmark PDE problems.
• The paper validates its approach through comprehensive benchmarks, highlighting the critical role of tuning Fourier feature parameters for improved PINN performance.

Analysis of the Eigenvector Bias in Fourier Feature Networks for Multi-Scale PDE Solutions

The paper "On the Eigenvector Bias of Fourier Feature Networks: From Regression to Solving Multi-Scale PDEs with Physics-Informed Neural Networks" explores the critical paper of spectral bias in neural networks and its implications for physics-informed neural networks (PINNs). The authors aim to address the limitations of PINNs, specifically their difficulties in handling high-frequency or multi-scale features in Partial Differential Equations (PDEs), through an analysis rooted in Neural Tangent Kernel (NTK) theory.

Core Contributions

The paper makes several contributions that stand out in the domain of scientific machine learning and deep learning for PDEs:

1. Spectral Bias Quantification: The authors argue that the spectral bias observed in deep neural networks is effectively an "NTK eigenvector bias". They demonstrate that Fourier feature mappings can manipulate the frequency of NTK eigenvectors, thereby affecting the learning dynamics of the network.
2. Novel Architectures: By exploring the NTK eigensystem, the authors propose new neural network architectures specifically designed to tackle multi-scale problems. These architectures employ multi-scale Fourier feature embeddings, allowing for efficient handling of the varying frequencies present in complex PDE solutions.
3. Benchmark Evaluations: The paper presents a suite of benchmark problems, showcasing scenarios where conventional PINN models fail, particularly for PDEs with multi-scale behavior. This benchmark suite serves as a robust testbed for evaluating the effectiveness of the proposed methods.

Numerical Results and Key Findings

The authors provide detailed numerical experiments to support their theoretical developments. Some critical findings include:

• Effectiveness of Multi-Scale Architectures: The proposed multi-scale Fourier feature network architectures significantly outperform traditional PINNs in scenarios featuring substantial frequency variations. For instance, in the 1D Poisson and heat equation examples, the multi-scale architectures achieve notable improvements in prediction accuracy, evidenced by lower relative L2 errors.
• Frequency Modulation in PINNs: The paper indicates that proper tuning of the Fourier feature scales (denoted as σ) directly affects the network's ability to learn high-frequency components. This insight is crucial for configuring network architectures that align better with the spectral characteristics of the target PDEs.

Implications and Future Directions

The implications of this paper are twofold:

1. Theoretical Insights: The introduction of the NTK framework to analyze the spectral bias in Fourier feature networks enriches the theoretical understanding of neural network training dynamics. This understanding is pivotal for guiding the development of neural architectures for complex scientific computing tasks.
2. Practical Applications: By improving the training of neural networks on multi-scale PDEs, this research directly impacts fields such as computational fluid dynamics, material science, and other domains where PDEs are prevalent.

The research also suggests future explorations in various directions, such as integrating dynamic adaptation mechanisms for Fourier feature parameters and developing more generalized frameworks that can automatically tune these parameters based on data-driven insights. Furthermore, there is potential for extending this framework to more complex and high-dimensional PDEs, as well as exploring its applicability to other forms of neural networks.

Conclusion

The paper provides a comprehensive paper on overcoming spectral bias in neural networks, specifically within the context of solving PDEs via PINNs. By leveraging NTK theory, the authors present innovative architectural enhancements that significantly bolster the ability of neural networks to learn and generalize across varying frequency scales inherent in complex physical systems. This work stands as a testament to the ongoing advancements in integrating theoretical insights from machine learning with practical needs in scientific computations.