VOL. I NEURAL VISUALIZATION LAB
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The Topology
of Learning.

An interactive field guide to neural networks, gradient descent, and the high-dimensional geographies of loss where mathematics becomes landscape, and optimization becomes a journey.

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CHAPTER 01 FORWARD PROPAGATION

How a network thinks.

Each neuron receives weighted signals from the layer before, sums them, and decides through a non-linear squashing function whether its signal is worth passing forward. Toggle the input features below and watch activations cascade through the network, layer by layer, until a prediction crystallizes at the output.

Network · 3 → 5 → 4 → 2 IDLE
INPUTS
Loss Curve · Live BCE
PREDICTION ŷ
[—, —]
LOSS ℒ
FORWARD PASS
$$a^{(l)} = \sigma\!\left(W^{(l)} a^{(l-1)} + b^{(l)}\right)$$
CHAIN RULE · BACKPROP
$$\frac{\partial \mathcal{L}}{\partial w_{ij}^{(l)}} = \frac{\partial \mathcal{L}}{\partial a_j^{(l)}} \cdot \frac{\partial a_j^{(l)}}{\partial z_j^{(l)}} \cdot \frac{\partial z_j^{(l)}}{\partial w_{ij}^{(l)}}$$
CHAPTER 02 GRADIENT DESCENT

Following the gradient downhill.

The loss landscape stretches in every direction. To descend, we compute the partial derivative of loss with respect to each parameter, then step in the opposite direction. Learning rate controls step size; momentum accumulates velocity to power through shallow valleys. Drag the sliders and watch the optimizer's path snap to convergence.

Loss Contour · 2D Projection READY
STEP 0
Hyperparameters TUNE
Learning Rate · η 0.050
Momentum · β 0.00
UPDATE RULE
$$\theta_{t+1} = \theta_t - \eta \,\nabla_\theta \mathcal{L}(\theta_t)$$
WITH MOMENTUM
$$v_{t+1} = \beta v_t + \eta\, \nabla_\theta \mathcal{L}(\theta_t), \quad \theta_{t+1} = \theta_t - v_{t+1}$$
CHAPTER 03 BIAS–VARIANCE

The shape of overfitting.

A model too simple cannot capture the data; a model too complex memorizes noise. Between these failures lies the bias–variance frontier. Drag the complexity slider and watch the decision boundary flex from a straight line to a writhing membrane, while training and validation errors diverge.

Decision Boundary · Polynomial Classifier DEGREE 3
CLASS A CLASS B
Bias–Variance Tradeoff TUNE
Model Complexity · Polynomial Degree 3
TRAIN ERROR
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VALIDATION ERROR
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← UNDERFITTING OVERFITTING →
ERROR DECOMPOSITION
$$\mathbb{E}\!\left[(y - \hat{f}(x))^2\right] = \underbrace{\text{Bias}^2[\hat{f}]}_{\text{underfit}} + \underbrace{\text{Var}[\hat{f}]}_{\text{overfit}} + \sigma^2$$
CHAPTER 04 PLAYGROUND

Train your own classifier.

Paint data points onto the canvas, choose a preset dataset, then train a small neural network in real time. Watch the loss curve descend and the decision boundary adapt to your data. Everything runs in your browser — no servers, no APIs, just gradients.

Dataset · Click to Add Points
CLASS A 0 PTS
Training · 2-H-H-1 MLP READY
Learning Rate 0.030
Hidden Units 8
EPOCH
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ACCURACY