What you see in the picture below is the famous Mandelbrot set. We can witness the complex shape of this set:
To understand how a neural network learns the Mandelbrot set, we first need to start with the most basic mathematical concepts: What is a function? A function is essentially a system that transforms input into output, that is, a mapping from number to number. Here, you enter an x value and the system outputs a y value. We can plot all x and y values on the chart to form a continuous line. The important thing is that as long as you know this function, you can calculate the corresponding output y for any input x.
But if we don’t know the specific function form, but only know some x and y values, can we perform reverse reasoning on this unknown function? If we can construct such a function, we can use it to estimate the y value corresponding to the x value that is not in the original data set. Even if our data contains some noise or randomness, we are still able to capture the overall pattern of the data, resulting in y-values that are close to true but not perfect. What we need is a method that can approximate the real function. More specifically, this is exactly what neural networks do.
In the process of exploring how neural networks learn, we first encountered a basic question: Can we reverse engineer a function if we don't fully know its form, only some of its input and output values? If we can construct such a function, we can use it to estimate the output values corresponding to input values that are not in the original data set. Even if the data we have has some random noise, we can still capture the overall pattern and generate output values that are close to reality, although perhaps not perfectly. What we need is a method that can approximate the real function, and this is the core function of neural networks.
A neural network is, essentially, a function approximator . It consists of a series of interconnected neurons, each of which can receive input from the previous layer and produce an output, which is passed to the next layer. The output of each neuron is not just a simple transfer of the input, it is the result of the input being weighted, summed, and processed through an activation function . These weights and biases are key to neural network learning, determining how the network responds to different inputs.
By continuously adjusting these weights and biases, the neural network can gradually learn and approximate complex functions. In this process, the network continues to try, make mistakes, and adjust, and finally finds a way to make its output as close as possible to the real function value. This is the magic of neural networks - they can learn and simulate extremely complex data patterns and functional relationships.
Before we deeply understand how neural networks approximate complex functions, we must first recognize its core - nonlinear activation functions, such as ReLU. The introduction of this activation function greatly enhances the network's ability to handle complex tasks. Neurons adjust their weighted inputs and biases to generate outputs that are passed to the next layer. This process is the basis of network learning. As learning proceeds, the network continuously adjusts its weights and biases, gradually forming decision boundaries that can distinguish different data points. This not only shows the strong adaptability of the network, but also its versatility in handling diverse data sets.
In the learning journey of neural networks, nonlinear activation functions play a crucial role. These functions provide the network with the necessary tools to handle complex problems, allowing simple tasks that can only be handled through linear combinations to handle higher-level complex patterns. By increasing the number of neurons, the network gradually builds the ability to capture complex data sets, thereby effectively approaching the required objective function. This process not only reveals the neural network's ability to handle complexity, but also demonstrates its unique way of learning and approximating the objective function.
During the learning process of the neural network, the backpropagation algorithm improves the approximation effect of the function by gradually adjusting the network parameters. Although neural networks have been proven to have universal function approximation capabilities and can approximate various functions with arbitrary accuracy, they still face some limitations in practical applications. These limitations include adequacy of data volume, feasibility of network scale, and understanding of known functional structures. Nonetheless, neural networks excel at handling some tasks that are extremely challenging for computers, especially in areas that require intuition and fuzzy logic, such as computer vision and natural language processing, where they have revolutionized the face.
A simple yet powerful way to think about the world, by combining simple calculations we can make computers construct any function we want, and neural networks can learn almost anything.