The text delves into the idea of using Taylor Series and Fourier Series as alternatives to neural networks. It emphasizes their application in approximating functions and their similarities to neural network structures. The author discusses the limitations of Taylor and Fourier Series and why neural networks are still essential. The piece also promotes the author’s newsletter and provides further reading references.
### **Alternatives to Neural Networks: Taylor Series & Fourier Series**
Neural networks are powerful because of their universal approximation theorem, but there are alternatives like Taylor Series and Fourier Series that offer practical solutions for complex function approximation.
#### **Taylor Series**
The Taylor Series represents a function as an infinite sum of terms calculated from its derivatives at a single point. It’s used to approximate complex functions and can be simplified into the Maclaurin series for even greater efficiency.
#### **Practical Application**
In machine learning, Taylor features can be passed to a neural network to learn coefficients using backpropagation, making it a valuable tool for function approximation.
#### **Fourier Series**
The Fourier Series breaks down periodic functions into a sum of sine and cosine waves, allowing the construction of complex patterns from simple functions.
#### **Value**
It’s useful for modeling complex seasonal patterns in time series, making it a valuable tool for harmonic regression and periodic function approximation.
### **Why Do We Even Have Neural Networks?**
While Taylor and Fourier series are universal function approximators, they have limitations in generalization and complexity. Neural networks can accurately model high dimensional functions without exponentially increasing input dimensions.
#### **Practical AI Solutions**
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For further reading on forecasting and understanding AI concepts, check out [Forecasting: Principles and Practice](https://otexts.com/fpp2/) and join the discussion on [Neural Networks vs Taylor Series vs Fourier Series](https://www.reddit.com/r/MachineLearning/comments/nnj18c/d_why_do_we_even_have_neural_networks_a_deep_dive/).
