Most neural networks have a major limitation: they have no memory of the past. If you show a standard network an image and then another one, it processes each completely independently. It doesn't remember the first image when it sees the second.

But what if the order of information matters? Think about reading a sentence. The meaning of a word often depends on the words that came before it. The same is true for understanding speech, predicting stock prices, or analyzing DNA. This is sequential data, where the sequence itself contains crucial information.

Recurrent Neural Networks, or RNNs, are a type of neural network designed specifically for this kind of data. They have a form of memory that allows them to use information from prior inputs to influence the current input and output. An RNN is called recurrent because it performs the same task for every element in a sequence, with the output at each step depending on the computations from the previous steps.

The core idea behind an RNN is a loop. The network has a "hidden state," which acts as its memory. At each step in a sequence, the network takes in a new input and combines it with the hidden state from the previous step. This process updates the hidden state, which is then used to generate an output. The updated hidden state is also passed along to the very next step in the sequence.

This structure allows the network to maintain a summary of the information it has seen so far. You can visualize this by "unrolling" the network through time. For a sentence with five words, you can imagine five copies of the network, each one passing its memory to the next.

In this diagram, at any given time step $t$, the hidden state $h_t$ is a function of the input at that step, $x_t$, and the hidden state from the previous step, $h_{t-1}$. The output $y_t$ is then calculated based on this new hidden state $h_t$. This chain-like nature is what allows RNNs to connect previous information to the present task.

Because they excel at processing sequences, RNNs are used in a variety of powerful applications.

Natural Language Processing (NLP): This is the most common use case. RNNs can predict the next word in a sentence, which is essential for autocomplete features and language generation. They are also used for machine translation, sentiment analysis (determining if a review is positive or negative), and text summarization.

Speech Recognition: Audio is a sequence of soundwaves over time. RNNs can process these sequences to transcribe spoken language into text, which is the technology behind virtual assistants like Siri and Alexa.

Time Series Prediction: RNNs can analyze historical data points in a sequence to forecast future values. This is useful for predicting stock market prices, forecasting weather patterns, or estimating energy demand.

While powerful, basic RNNs have a significant weakness: they struggle with long-term dependencies. This means they have a hard time connecting information over long sequences.

Imagine trying to predict the last word of this sentence: "He grew up in France and has many friends there; he speaks fluent ____." The word "France" gives us a strong clue that the last word should be "French." But if the sentence were much longer, with many clauses in between, a simple RNN might forget the crucial context provided by "France."

This happens because of a technical issue during training known as the vanishing gradient problem. As information flows through the recurrent connections, its influence can diminish exponentially. It's like a game of telephone; the further back in the sequence a piece of information is, the more its signal fades by the time it reaches the end. This makes it very difficult for the network to learn the connection between distant events in a sequence.

Because of this challenge, researchers developed more advanced types of RNNs to better handle long-range dependencies.