The Mathematical Magic Behind LLMs: How They Actually Work

Large language models (LLMs) like ChatGPT have transformed artificial intelligence, enabling machines to generate coherent text, answer questions, and write code. Behind their seemingly magical abilities lies a foundation of elegant mathematics and sophisticated architectures. In this comprehensive guide, we'll explore both the high-level concepts and the mathematical underpinnings that power modern LLMs.

Understanding Artificial Intelligence and Machine Learning

Artificial intelligence refers to the field aimed at developing machines capable of tasks that typically require human intelligence—such as speech recognition, image identification, and decision-making. Within AI, machine learning focuses on teaching computers to learn patterns from data rather than relying solely on programmed rules.

Traditional programming often follows an "if-then" logic, while machine learning empowers systems to derive rules from examples. At the heart of many machine learning systems are neural networks—computational structures inspired by the human brain, where small units (neurons) process numerical inputs and pass signals through weighted connections.

The Mathematical Heartbeat of AI

When we interact with AI systems that seem almost magically intelligent, we're actually engaging with elegant mathematical structures. Behind every language model, image generator, and recommendation system lies a foundation of mathematical functions that transform raw data into understanding.

Euler's Number: The Magical Constant That Smooths Growth

At the core of many neural networks sits a seemingly random number: 2.71828... This is Euler's number e, a mathematical constant fundamental to AI and natural phenomena like population growth and compound interest.

What makes e so special? Its defining property is that the derivative of e^x is itself. This means that at any point on the curve, the rate of change exactly equals the value of the function at that point. No other non-zero function has this extraordinary property.

This self-replicating quality creates the smoothest possible growth pattern, enabling neural networks to make granular adjustments during training. Differentiability plays a critical role in the effectiveness of AI learning systems, ensuring that the output of a neural network changes smoothly in response to input changes.

Tokenization and Embeddings: From Words to Numbers

Before any neural network can process language, the raw text must be converted into a numerical format. This conversion happens in two critical steps:

Tokenization: The text is broken into smaller units called tokens. Tokens may represent whole words, subwords, or characters, depending on the tokenizer. For example, the sentence "The fat cat sat on the mat" might be split into tokens like "The", "fat", "cat", "sat", "on", "the", and "mat."

Embeddings: Each token is then mapped to a list of numbers, known as an embedding. These embeddings capture semantic relationships between words. For instance, words with similar meanings such as "cat" and "kitten" have embeddings that are close in the numerical space.

In this high-dimensional vector space, relationships between concepts become geometric relationships. Words with similar meanings cluster together. The distance between vectors represents semantic difference, while the direction can encode specific types of relationships.

This mathematical representation enables one of the most famous examples in word embeddings: "king" minus "man" plus "woman" approximately equals "queen." The vector arithmetic captures the gender relationship independent of the royal status.

The Building Blocks of Neural Networks

The S-Curve of Decision - Sigmoid Functions

When an AI needs to make a binary decision – Is this email spam? Is this image a cat? – it often relies on the sigmoid function:

σ(x) = 1/(1 + e^(-x))

This S-shaped curve elegantly transforms any input into a value between 0 and 1. It's the mathematical equivalent of taking the chaotic complexity of the world and distilling it down to a probability.

The sigmoid's graceful curve leverages Euler's number to create a smooth transition between states. The sigmoid's S-shape also mimics many natural phenomena: population growth curves, learning rates, and even how opinions spread through social networks.

ReLU: The Power of Simplicity

Sometimes the most powerful solutions are the simplest. The Rectified Linear Unit (ReLU) function is disarmingly straightforward:

If the input is positive, let it through. If the input is negative, output zero.

This piecewise linear function sparked a revolution in deep learning because it solved the vanishing gradient problem while being computationally efficient.

The beauty of ReLU lies in its binary gradient behavior - it's either 0 (for negative inputs) or 1 (for positive inputs). This property makes gradient flow much more stable in deep networks, allowing for faster and more reliable training.

The Softmax Function: The Heart of AI Decision-Making

Softmax is a mathematical function that transforms a vector of raw numbers into a probability distribution. The resulting probabilities all sum to exactly 1, making softmax essential for multi-class classification problems.

For an input vector z with n elements, the softmax function is defined as:

softmax(z)_i = e^(z_i) / Σ e^(z_j)

Softmax appears at the final layer of classification networks and in text generation when selecting the next token. It converts raw neural outputs into statements like: "This image is 87% likely to be a cat, 12% likely to be a dog, and 1% likely to be something else."

The Transformer Architecture and Attention Mechanism

ChatGPT and other modern LLMs are based on the transformer architecture, which marked a significant departure from earlier models that processed text sequentially. Transformers can process entire sequences of text in parallel, thanks to the attention mechanism.

Attention Mechanism

This mechanism enables the model to weigh the importance of different words in a sentence based on their context. For example, in the sentence "He sat on the bank and watched the river flow," the model uses attention to understand that "bank" refers to the side of a river, not a financial institution.

At its mathematical core lies the self-attention mechanism, powered by the softmax function:

softmax(QK^T/√d_k)V

This elegant formula relies fundamentally on Euler's number. When the model calculates attention scores between words, each score undergoes an exponential transformation, creating precisely the right mathematical dynamics for language understanding.

Training and Fine-Tuning: The Learning Process

Next-Token Prediction

The core training process of LLMs involves next-token prediction. The model is presented with a vast dataset composed of internet text—including books, articles, and code—and learns to predict the next token in a sequence. With every prediction, the model calculates an error based on the difference between its guess and the actual token. By iteratively adjusting its internal parameters (often numbering in the billions), the model gradually learns the statistical relationships and patterns within the language.

Post-Training Enhancements

After the initial training phase, a secondary process known as post-training fine-tunes the model:

Instruction Tuning: The model is provided with examples of user instructions alongside correct responses, teaching it to better follow human commands.

Reinforcement Learning from Human Feedback (RLHF): In this phase, human evaluators rate the quality of the model's responses. These ratings are then used to further adjust the model, ensuring that its outputs are more aligned with human expectations and values.

While these processes improve the model's performance, it is important to note that LLMs operate on probabilities and patterns. They do not "understand" facts in the traditional sense and may occasionally produce plausible but incorrect responses—a phenomenon commonly referred to as hallucination.

Putting It All Together: BERT as an Example

BERT (Bidirectional Encoder Representations from Transformers) demonstrates how the mathematical building blocks we've explored work in concert. What makes BERT revolutionary is its bidirectional approach to context, analyzing words from both directions simultaneously.

BERT employs a clever training approach called masked language modeling. The model randomly masks tokens in its input and learns to predict them based on bidirectional context:

"The [MASK] fox jumps over the lazy dog" → BERT predicts "quick"

This prediction task creates a challenging mathematical optimization problem, forcing the model to develop a deep contextual understanding that works in both directions simultaneously.

Inference and Real-World Applications

Once training and fine-tuning are complete, the model enters the inference phase. This is when users interact with the model and receive responses generated in real time. The inference process involves:

1. Converting the user's input into tokens
2. Mapping these tokens into embeddings
3. Processing the embeddings through multiple layers of the transformer network to predict the most likely next tokens
4. Returning the generated text to the user, one token at a time

These capabilities enable modern LLMs to perform a wide range of tasks—from generating creative writing to answering technical queries. However, because the model bases its responses on learned patterns rather than a factual database, users should always verify critical information.

Conclusion

The journey of transforming raw text into coherent, context-aware language generation involves a series of sophisticated processes—tokenization, embedding, transformer-based processing with attention mechanisms, and extensive training with both next-token prediction and human feedback.

What makes these mathematical functions remarkable is how they work in concert to create systems that can perform tasks once thought to be the exclusive domain of human cognition. The special properties of Euler's number ripple throughout these systems, creating the mathematical symphony that powers modern artificial intelligence.

As AI continues to evolve, understanding the underlying mechanisms helps demystify how these models operate and underscores the importance of responsible use and continuous improvement. Whether you are new to AI or an experienced developer, the concepts discussed here provide a foundation for exploring the fascinating world of large language models.

I believe that the AI community has hidden the maths from you by talking human like to describe these artefacts, this has been going on since the 1950’s -- all to make the topic interesting

So the first thing to do is to load up all the data that can be loaded, human text, books, fiction, journals, religious texts, website, reddit. You name they bring it in. is a massive database.

What they call Machine Learning is actually loading this data fitting patterns to the sets , they use techniques like least-squares and error detection


They gather a lot of data together they accumulate integers against floating point numbers, and map all varieties of the integers and their relationships into an array, they then combine all the arrays together and each row in the combined work is known as a vector

The machine then uses mathematica algorithms against each column of floats looking for patterns, sometimes with many runs against the data, each time the pattern changes a parameter is created to say change occurred here.


When the machine has learnt from the data the it moves to the training phase

Training means feeding the machine with data it has never seen before and a matching correct answer


The algorithms then predict the value from the parameters, and double check the answer. If the answer is wrong it adjusts the parameters until it is close enough, sometimes adding new parameters This is training. Unsupervised training.

The machine then focus in a human reinforced learning phase where the answers to regular questions are fed to humans (probably working in a 3rd world country) where they speak enough english cheaply to serif the answers are good enough, thes humans approve adjust and reject the output feeding back to the machine which then changes is parameters to head toward the answer desired.

Then the ‘Training ends’ all of the questions and answers are kept fo the next upgrade of the AI and fed in at the unsupervised learning stage and fresh questions generated for HRL

Next comes alignment, ‘alignment with human values; this is where the AI is told not to tell people how to make atomic bombs, how not to write horror stories or create sadistic material or generate pornographic material -- this where the machine is censored.

There are a number of standard tests that AI is expected to be put to. So the companies that create the machines (know as models) ensure that the AI will pass these tests by running the questions and answers through the machine after alignment

Finally smarty pants academics will create questions that try to confound the machines, with answers. These are then ‘learnt’ by adapting parameters

Then they test it.