New Neural Network Architecture Using Fourier Basis Functions Shows Promise for High-Frequency Function Approximation
Researchers have introduced the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), a novel architecture that pairs sine-type activation functions with a multi-component, multi-layer design to better approximate high-frequency functions. The model builds on existing Multi-Component Multi-Layer Neural Network (MMNN) frameworks by incorporating trainable linear combinations of fixed random sine-type basis functions, with deeper layers generating increasingly complex frequency features. The work addresses a longstanding challenge in neural network design—accurately representing oscillatory and high-frequency targets—with theoretical guarantees and empirical validation.
The paper presents FMMNN, a neural network architecture designed to overcome limitations that standard fully connected networks face when approximating high-frequency or oscillatory functions. Each component in an FMMNN is a trainable linear combination of fixed random sine-type basis functions, and multi-layer composition allows the network to build progressively more complex, adaptive high-frequency representations. The authors prove that FMMNNs retain exponential expressive power even under a low-rank architectural constraint, a result that distinguishes them from many compact network designs that sacrifice representational capacity. Analysis of the optimization landscape reveals it to be substantially more favorable than that of conventional fully connected networks, particularly for high-frequency targets, suggesting easier and more reliable training. The team also proposes a scaled random initialization scheme for first-layer weights, which is shown to accelerate convergence and improve final accuracy when sufficient training data is available. Extensive numerical experiments on oscillatory function-approximation benchmarks corroborate the theoretical findings, demonstrating strong accuracy and favorable convergence. Code and implementation details have been made publicly available by the authors.
What's missing
The study does not discuss computational cost or parameter efficiency of FMMNNs relative to baseline architectures at equivalent accuracy levels, nor does it address performance on non-oscillatory or real-world downstream tasks beyond synthetic benchmarks. It is also unclear how sensitive the scaled random initialization method is to hyperparameter choices or how it generalizes across different problem domains.
What different sources said
- arXiv stat.MLCenter
Fourier Multi-Component and Multi-Layer Neural Networks: Unlocking High-Frequency Potential
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