FAN: Fourier Analysis Networks

Abstract: Despite the remarkable successes of general-purpose neural networks, such as MLPs and Transformers, we find that they exhibit notable shortcomings in modeling and reasoning about periodic phenomena, achieving only marginal performance within the training domain and failing to generalize effectively to out-of-domain (OOD) scenarios. Periodicity is ubiquitous throughout nature and science. Therefore, neural networks should be equipped with the essential ability to model and handle periodicity. In this work, we propose FAN, a novel general-purpose neural network that offers broad applicability similar to MLP while effectively addressing periodicity modeling challenges. Periodicity is naturally integrated into FAN's structure and computational processes by introducing the Fourier Principle. Unlike existing Fourier-based networks, which possess particular periodicity modeling abilities but are typically designed for specific tasks, our approach maintains the general-purpose modeling capability. Therefore, FAN can seamlessly replace MLP in various model architectures with fewer parameters and FLOPs. Through extensive experiments, we demonstrate the superiority of FAN in periodicity modeling tasks and the effectiveness and generalizability of FAN across a range of real-world tasks, e.g., symbolic formula representation, time series forecasting, language modeling, and image recognition.

• The paper introduces a novel neural architecture that embeds Fourier Analysis to inherently capture periodic patterns.
• It replaces traditional MLP layers with cosine and sine transformations to reduce model complexity while boosting performance.
• Empirical evaluations highlight FAN’s superiority in tasks like time series forecasting and language modeling over conventional models.

Overview of "FAN: Fourier Analysis Networks"

The paper "FAN: Fourier Analysis Networks" proposes an innovative neural network architecture designed to address limitations in existing models when dealing with periodic data. Unlike traditional neural networks, such as MLPs and Transformers, which often struggle to generalize periodic patterns beyond the training domain, the Fourier Analysis Network (FAN) incorporates principles from Fourier Analysis to enhance its modeling capabilities and reduce parameter usage.

Core Concept

The authors highlight that conventional neural networks, while successful across various tasks, primarily memorize rather than understand periodic data, limiting their application in domains where periodicity is fundamental. To resolve this, FAN integrates Fourier Series directly into its architecture. By doing so, it naturally embeds periodic characteristics within network computations. This integration equips FAN with an intrinsic ability to model and predict periodic phenomena more accurately, unlike its predecessors.

Architecture and Design

The FAN model is structured around the Fourier Series, with layers designed to explicitly encode periodic patterns. The architecture includes several layers, each employing both cosine and sine transformations of input data, thereby capturing the essential elements of periodic functions. The authors claim that this approach not only maintains the expressive power of traditional architectures but also outperforms them with fewer parameters and floating point operations.

A distinctive feature of FAN is its ability to replace MLP layers in various models seamlessly. By doing so, it reduces complexity while enhancing performance, especially in tasks where capturing periodicity is crucial.

Empirical Evaluation

The paper provides comprehensive experimental results demonstrating FAN's effectiveness:

• Periodicity Modeling: Compared to MLP, KAN, and Transformer models, FAN exhibits a significant improvement in modeling both simple and complex periodic functions, especially in out-of-domain scenarios. The results showcase FAN’s capacity to genuinely understand periodic data rather than merely interpolate within the training set.
• Real-World Applications: FAN excels in symbolic formula representation, time series forecasting, and language modeling tasks. It shows superiority over existing baselines in these domains. For instance, in language modeling, FAN yielded better cross-domain generalization compared to Transformer, LSTM, and Mamba, indicating enhanced robustness and adaptability.

Implications and Future Directions

The introduction of FAN presents notable theoretical and practical implications. Theoretically, it offers a new perspective on embedding explicit mathematical principles within neural architectures, paving the way for further exploration in integrating other analytical frameworks. Practically, FAN's ability to efficiently model periodic phenomena suggests potential applications in fields like signal processing, weather forecasting, and any domain requiring pattern prediction.

Future work may involve scaling FAN for larger models and exploring its application in more diverse domains. There is also potential in further refining the integration of Fourier Analysis to enhance the model’s robustness and efficiency.

In conclusion, this paper presents FAN as a promising advancement in neural network design. By addressing periodicity directly within the architecture, FAN not only demonstrates improved performance across several tasks but also offers a conceptual shift in how neural networks can be structured for specific data characteristics.