{"slug": "deep-learning-and-transformers", "title": "Deep Learning and Transformers", "summary": "A developer explains modern AI through a mathematical lens, framing neural networks as parameterized mathematical transformations and training as high-dimensional optimization. The writeup describes the artificial neuron as a weighted sum with a bias passed through an activation function, and shows how stacking purely linear layers collapses into a single linear transformation until nonlinearities like ReLU are introduced. It further characterizes the attention mechanism inside a Transformer as a learned, input-dependent interaction matrix.", "body_md": "Artificial intelligence is often introduced with phrases like:\n\n“Neural networks imitate the human brain.”\n\nThat analogy can be useful, but if you come from physics, mathematics, engineering, or scientific computing, there is another way to think about modern AI that may feel much more natural.\n\nA neural network is fundamentally a **parameterized mathematical transformation**.\n\nTraining that network is an **optimization problem in a very high-dimensional space**.\n\nAnd the attention mechanism inside a Transformer can be interpreted as a **learned, input-dependent interaction matrix**.\n\nOnce we look at AI from this perspective, much of the mystery starts to disappear.\n\nLet’s build the idea from the ground up.\n\nThe basic computational element of a neural network is an **artificial neuron**.\n\nSuppose we have several inputs:\n\n```\nx₁, x₂, …, xₙ\n```\n\nEach input is associated with a weight:\n\n```\nw₁, w₂, …, wₙ\n```\n\nThe neuron calculates a weighted sum:\n\n```\nz = Σᵢ wᵢxᵢ + b\n```\n\nwhere `b` is called the **bias**.\n\nThe result then passes through an activation function:\n\n```\na = f(z)\n```\n\nIn vector notation, we can write the same basic idea as:\n\n```\nz = w · x + b\n```\n\nSo despite the biological name, an artificial neuron is not literally a microscopic brain cell.\n\nIt is a mathematical operation.\n\nThe interesting behavior begins when many of these operations are connected together.\n\nSuppose we create several neural-network layers but use only linear transformations.\n\nThe first layer might be:\n\n```\nh₁ = W₁x\n```\n\nThe second layer could be:\n\n```\nh₂ = W₂h₁\n```\n\nSubstituting the first expression into the second gives:\n\n```\nh₂ = W₂W₁x\n```\n\nBut `W₂W₁` is simply another matrix.\n\nSo no matter how many purely linear layers we stack, the entire system can still collapse into one larger linear transformation.\n\nWe have added depth, but not much expressive power.\n\nThat changes when we introduce **nonlinearity**.\n\nNow suppose the first layer becomes:\n\n```\nh₁ = f(W₁x + b₁)\n```\n\nand the next layer becomes:\n\n```\nh₂ = f(W₂h₁ + b₂)\n```\n\nThe activation function `f` prevents the whole network from reducing to a single linear transformation.\n\nOne of the most common activation functions is ReLU:\n\n```\nReLU(x) = max(0, x)\n```\n\nIf the input is positive, ReLU keeps it.\n\nIf the input is negative, ReLU returns zero.\n\nThis simple nonlinearity allows networks to represent much more complicated relationships.\n\nThat is one of the foundations of deep learning.\n\nA deep neural network contains many transformation stages.\n\nConceptually, information moves through something like this:\n\n```\nx\n↓\nf(W₁x + b₁)\n↓\nf(W₂h₁ + b₂)\n↓\n...\n↓\ny\n```\n\nEach layer transforms the representation created by the previous layer.\n\nFor an image-processing system, early layers may respond to relatively simple structures such as edges and local color changes.\n\nLater layers can combine those structures into larger patterns.\n\nThose patterns can then be combined again into increasingly useful internal representations.\n\nThe entire network can be thought of as one large parameterized function:\n\n```\ny = F(x; θ)\n```\n\nHere, `θ` represents all the parameters inside the model.\n\nThose parameters include weights and biases.\n\nA modern neural network can contain millions or billions of them.\n\nThis gives us a useful mental model:\n\n**A deep neural network is a very high-dimensional parameterized function.**\n\nThe architecture determines the structure of the function.\n\nTraining determines the numerical values of its parameters.\n\nNow suppose the network makes a prediction:\n\n```\nŷ\n```\n\nwhile the desired answer is:\n\n```\ny\n```\n\nWe define a loss function:\n\n```\nL(ŷ, y)\n```\n\nThe loss measures how far the prediction is from the desired result.\n\nTraining then asks a very mathematical question:\n\nWhich parameter values make the loss smaller?\n\nConceptually, we are trying to find:\n\n```\nθ* = arg minθ L(θ)\n```\n\nIn other words, we want a parameter configuration that minimizes the loss.\n\nNow imagine the loss as a function of every parameter:\n\n```\nL(θ₁, θ₂, …, θₙ)\n```\n\nIf the network contains a billion parameters, then this loss is defined over a billion-dimensional parameter space.\n\nWe cannot visualize that space directly.\n\nBut conceptually, we can still imagine a landscape containing regions of higher and lower loss.\n\nFor someone coming from physics, this is a very useful perspective:\n\n**Neural-network training ≈ optimization in a huge-dimensional landscape**\n\nInstead of explicitly programming every rule the system should follow, we search for parameter values that allow the model to reproduce useful patterns in data.\n\nHow do we know which direction to move in this enormous parameter space?\n\nWe calculate the gradient.\n\nThe gradient of the loss can be written as:\n\n```\n∇θ L\n```\n\nIt tells us how the loss changes when we make small changes to the parameters.\n\nA basic gradient-descent update looks like this:\n\n```\nθₜ₊₁ = θₜ − η∇θL\n```\n\nHere:\n\n`θₜ` represents the current parameters`∇θL` represents the gradient of the loss`η` is the learning rate`θₜ₊₁` represents the updated parameters\nThe learning rate controls the size of each step.\n\nIf the step is too large, the optimizer may jump past useful regions.\n\nIf the step is too small, training may become extremely slow.\n\nConceptually, gradient descent repeatedly asks:\n\nWhich small change in the parameters should reduce the loss?\n\nThen it makes that change and repeats the process.\n\nModern training algorithms are more sophisticated than basic gradient descent, but this core idea remains central.\n\nA deep neural network is a composition of many functions.\n\nConceptually:\n\n```\nF = fₙ ∘ fₙ₋₁ ∘ ... ∘ f₁\n```\n\nTo train the network, we need to know how the final loss depends on parameters buried deep inside those functions.\n\nFor example:\n\n```\n∂L / ∂Wᵢ\n```\n\nBackpropagation gives us an efficient way to calculate these derivatives.\n\nAt its core, backpropagation is an application of the chain rule.\n\nSuppose:\n\n```\ny = f(g(x))\n```\n\nThen:\n\n```\ndy/dx = (df/dg)(dg/dx)\n```\n\nA deep neural network may contain thousands of connected mathematical operations.\n\nBackpropagation applies this principle repeatedly through the computational graph.\n\nDuring the **forward pass**, information moves through the model and produces a prediction.\n\nThe loss is calculated.\n\nThen the derivatives are propagated backward through the computation so the system can determine how changes in earlier parameters would affect that loss.\n\nThose gradients are then used by the optimizer to update the parameters.\n\nThis gives us another useful way to think about neural networks:\n\n**The model is a computational graph, and backpropagation computes derivatives through that graph.**\n\nDeep learning existed long before Transformers.\n\nBut sequence problems such as language create a special challenge.\n\nWords do not exist independently.\n\nThe meaning of one word often depends on other words that appeared earlier — sometimes much earlier — in the sequence.\n\nTransformers introduced an especially powerful mechanism for handling these relationships:\n\n**attention**.\n\nInstead of forcing information to move only step-by-step through the sequence, attention allows different elements of the sequence to interact directly.\n\nThe central operation is:\n\n```\nAttention(Q, K, V) = softmax(QKᵀ / √dₖ)V\n```\n\nThis equation may look intimidating at first.\n\nBut each part has a clear role.\n\nLet’s unpack it.\n\nEach token representation is transformed into three vectors:\n\n```\nQ = queries\nK = keys\nV = values\n```\n\nA useful intuition is:\n\n**Query**\n\nWhat information am I looking for?\n\n**Key**\n\nWhat kind of information do I contain?\n\n**Value**\n\nWhat information should I contribute if I am relevant?\n\nThe model compares queries with keys using:\n\n```\nQKᵀ\n```\n\nThe result is a matrix of interaction scores.\n\nThose scores describe how strongly different elements of the sequence relate to one another.\n\nIf there are `N` tokens, we can describe an individual score as:\n\n```\nAᵢⱼ\n```\n\nThis represents how strongly token `i` should attend to token `j`.\n\nThe important point is that these relationships are calculated from the current input.\n\nThey are not simply fixed in advance.\n\nThis is where the physics intuition becomes especially interesting.\n\nThe operation:\n\n```\nQKᵀ\n```\n\ncreates a matrix describing relationships between elements of the sequence.\n\nConceptually, imagine something like:\n\n```\n       Token1  Token2  Token3  ...  TokenN\n      ┌                                  ┐\nToken1│ a₁₁     a₁₂     a₁₃     ...  a₁ₙ│\nToken2│ a₂₁     a₂₂     a₂₃     ...  a₂ₙ│\nToken3│ a₃₁     a₃₂     a₃₃     ...  a₃ₙ│\n  ⋮   │  ⋮       ⋮       ⋮       ⋱    ⋮  │\nTokenN│ aₙ₁     aₙ₂     aₙ₃     ...  aₙₙ│\n      └                                  ┘\n```\n\nThe raw scores are then normalized with softmax.\n\n```\nPᵢⱼ = exp(Aᵢⱼ) / Σⱼ exp(Aᵢⱼ)\n```\n\nThese normalized weights determine how strongly information from one token contributes to another token's updated representation.\n\nThe value vectors are mixed according to those weights.\n\nThis leads to one of my favorite physics-inspired interpretations of Transformers:\n\n**Attention ≈ a learned, input-dependent interaction matrix**\n\nThere is an important difference from a fixed physical interaction matrix, however.\n\nThe attention matrix depends on the current input.\n\nA new sequence creates new interactions.\n\nThe system is effectively asking:\n\nWhich elements should interact strongly in this particular configuration?\n\nConsider this sentence:\n\nThe animal didn't cross the street because **it** was tired.\n\nWhat does **it** refer to?\n\nMost likely, the animal.\n\nThe model must connect information from different positions in the sequence.\n\nAttention allows the representation associated with `it` to interact strongly with the representation associated with `animal`.\n\nNow consider:\n\nThe truck couldn't cross the bridge because **it** was broken.\n\nThis time, `it` most likely refers to the bridge.\n\nThe token `it` is unchanged.\n\nBut the context is different.\n\nTherefore the attention pattern can also be different.\n\nThat is one of the fundamental strengths of Transformers.\n\nThe relationships among elements are not completely hard-coded.\n\nThey are calculated dynamically from the current input.\n\nTransformers usually do not calculate just one attention pattern.\n\nThey calculate several attention patterns in parallel.\n\nThis is called **multi-head attention**.\n\nAn individual attention head can be written conceptually as:\n\n```\nheadᵢ = Attention(Qᵢ, Kᵢ, Vᵢ)\n```\n\nSeveral heads are then combined:\n\n```\nMultiHead(Q, K, V)\n    = Concat(head₁, head₂, …, headₕ) Wᴼ\n```\n\nDifferent attention heads can capture different relationships.\n\nOne may become useful for relatively local structure.\n\nAnother may capture longer-range dependencies.\n\nAnother may respond to different semantic or structural patterns.\n\nBut we should be careful not to assume that every attention head always has one neat, human-readable job.\n\nNeural-network representations are often distributed across many components.\n\nStill, multi-head attention gives the Transformer multiple interaction channels through which information can flow.\n\nAttention is central to the Transformer architecture.\n\nBut attention alone is not the entire Transformer.\n\nA simplified Transformer block looks roughly like this:\n\n```\nInput Representations\n        ↓\nSelf-Attention\n        ↓\nFeed-Forward Network\n        ↓\nNext Transformer Layer\n```\n\nModern Transformer blocks also use important components such as residual connections and normalization.\n\nA slightly more realistic conceptual picture looks like:\n\n```\nInput\n  ↓\nSelf-Attention\n  ↓\nResidual Connection + Normalization\n  ↓\nFeed-Forward Network\n  ↓\nResidual Connection + Normalization\n  ↓\nOutput\n```\n\nThis process is repeated across many layers.\n\nWe can imagine the internal representations evolving like this:\n\n```\nX⁽⁰⁾ → X⁽¹⁾ → X⁽²⁾ → ... → X⁽ᴸ⁾\n```\n\nAt each stage, the representation of each token can change.\n\nInformation from other tokens can influence it through attention.\n\nThe feed-forward network then performs additional nonlinear transformations.\n\nLayer after layer, the model builds increasingly rich representations of the input.\n\nA language model begins with tokens:\n\n```\nt₁, t₂, …, tₙ\n```\n\nEach token is mapped into a numerical representation.\n\nThose representations pass through many Transformer layers.\n\nEventually, the model produces numerical scores for possible next tokens.\n\nThose scores are converted into a probability distribution.\n\n```\nP(tₙ₊₁ | t₁, t₂, …, tₙ)\n```\n\nFor example, the model might produce something like:\n\n```\nP(\"physics\")    = 0.35\nP(\"science\")    = 0.21\nP(\"experiment\") = 0.08\n```\n\nA decoding strategy then chooses the next token.\n\nThat token becomes part of the context.\n\nThen the process happens again.\n\n```\nt₁, t₂, …, tₙ\n        ↓\n      tₙ₊₁\n        ↓\n      tₙ₊₂\n        ↓\n       ...\n```\n\nAt its core, a language model repeatedly predicts what token is likely to come next given the context.\n\nThat may sound surprisingly simple.\n\nBut when this objective is scaled across enormous datasets, large models, and powerful computing infrastructure, remarkably sophisticated behavior can emerge.\n\nA common misconception is that a large language model is simply an enormous database containing billions of stored sentences.\n\nThat is not the best way to think about it.\n\nThe model learns statistical structure through its parameters.\n\n```\nP(next token | context; θ)\n```\n\nThe parameter set `θ` contains the numerical structure learned during training.\n\nKnowledge is distributed through these parameters rather than being stored as a clean collection of sentences waiting to be retrieved.\n\nWhen you provide a prompt, the model performs **inference**.\n\nThe text is represented as tokens.\n\nThose tokens become numerical vectors.\n\nThe Transformer repeatedly transforms those vectors.\n\nAttention allows information to flow between relevant parts of the context.\n\nLayer after layer modifies the internal representations.\n\nFinally, the model produces a probability distribution over possible next tokens.\n\nSo an LLM is better understood as a huge nonlinear transformation than as a conventional lookup database.\n\nNow the pieces fit together.\n\nAn artificial neuron performs a simple transformation:\n\n```\na = f(w · x + b)\n```\n\nMany artificial neurons form a neural network.\n\nMany layers give us deep learning.\n\nTraining searches a high-dimensional parameter space:\n\n```\nθ* = arg minθ L(θ)\n```\n\nBackpropagation calculates the derivatives needed for optimization.\n\nTransformers introduce attention:\n\n```\nAttention(Q, K, V) = softmax(QKᵀ / √dₖ)V\n```\n\nAttention creates dynamic interactions between elements of the input.\n\nSo we can summarize the architecture like this:\n\n```\nNEURAL NETWORKS\nParameterized nonlinear transformations\n\n        ↓\n\nDEEP LEARNING\nMany transformations composed together\n\n        ↓\n\nTRAINING\nOptimization in high-dimensional parameter space\n\n        ↓\n\nATTENTION\nLearned, input-dependent interactions\n\n        ↓\n\nTRANSFORMERS\nDeep architectures built around attention\n```\n\nFrom this perspective, modern AI stops looking like one mysterious invention.\n\nIt becomes a collection of mathematical ideas working together.\n\nAI often feels mysterious because we encounter the finished system first.\n\nWe type a sentence into a chatbot and receive a remarkably coherent response.\n\nBut underneath that interface are familiar ideas:\n\n**linear algebra, nonlinear functions, probability, optimization, derivatives, matrix multiplication, high-dimensional representations, and enormous amounts of computation.**\n\nFor someone coming from physics, mathematics, engineering, or scientific computing, perhaps the most useful shift in perspective is this:\n\n**Don't begin by asking whether the machine \"thinks.\"**\n\nBegin by asking:\n\nWhat mathematical transformation is being performed?\n\nWhat quantity is being optimized?\n\nWhat information is interacting?\n\nHow does the representation evolve through the system?\n\nThose questions bring the subject back onto familiar ground.\n\nModern AI may be enormous.\n\nIt may contain billions of parameters.\n\nIts behavior may sometimes surprise us.\n\nBut underneath it all, the system is still built from mathematical transformations, interactions, optimization, and probability.\n\nOnce we start looking at it that way, artificial intelligence becomes much less mysterious —\n\nand much more interesting.\n\n*This article is part of my work exploring how complex artificial-intelligence concepts can be explained from first principles — starting with simple building blocks and gradually connecting them to modern AI systems.*", "url": "https://wpnews.pro/news/deep-learning-and-transformers", "canonical_source": "https://dev.to/p_ym_n/deep-learning-and-transformers-bii", "published_at": "2026-09-10 03:05:21+00:00", "updated_at": "2026-09-10 03:18:27.493785+00:00", "lang": "en", "topics": ["neural-networks", "machine-learning", "artificial-intelligence", "large-language-models"], "entities": ["Transformer", "ReLU"], "alternates": {"html": 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