1. Overview
2. Neural Network Fundamentals
3. Model Architecture
4. Trainable Parameters
5. Linear Layers
6. Activation Functions
7. Convolutional Neural Networks
8. Recurrent Neural Networks
9. LSTM and GRU
10. Attention Mechanisms
11. Transformer Architectures
12. Loss Functions
13. Forward Propagation
14. Backward Propagation
15. Training Workflow
16. Model Evaluation
17. Initialization and Stability
18. Serialization and Checkpoints
19. Performance Considerations
20. Testing and Validation
21. Troubleshooting
22. Developer References
23. Next Steps

Neural networks are computational models composed of interconnected mathematical operations and trainable parameters.

They learn patterns from data by adjusting parameters to reduce an objective function.

EveryO includes neural network development components intended to support experimentation with deep learning architectures.

Neural network systems typically provide:

• Reusable computational layers.
• Trainable parameters.
• Activation functions.
• Model composition.
• Forward computation.
• Gradient integration.
• Loss calculation.
• Optimizer compatibility.

Input Data
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Tensor Engine
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Neural Network Layers
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Predictions
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Loss Function
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Automatic Differentiation
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Optimizer
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Updated Parameters

An artificial neuron applies a weighted transformation to input values.

A typical neuron computes:

[ z=\sum_{i=1}^{n}w_ix_i+b ]

where:

• (x_i) represents input features.
• (w_i) represents trainable weights.
• (b) represents bias.
• (z) is the linear output.

An activation function may then be applied:

[ a=\phi(z) ]

A neural network commonly consists of:

1. Input layer.
2. Hidden layers.
3. Output layer.

Input Features
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Dense Layer
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Activation
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Dense Layer
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Activation
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Output Layer
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Prediction

A model architecture defines how layers and operations are connected.

• Input dimensions.
• Output dimensions.
• Number of layers.
• Layer types.
• Activation functions.
• Parameter count.
• Computational complexity.
• Training stability.

Support for individual architectures must be verified against the installed EveryO version.

Trainable parameters determine how a neural network transforms input data.

Weights control the contribution of input features.

Biases provide additive offsets to transformations.

During training, an optimizer updates parameters using computed gradients.

A basic gradient descent update is:

[ \theta_{t+1}=\theta_t-\eta\nabla_\theta L ]

where:

• (\theta) is the parameter.
• (\eta) is the learning rate.
• (L) is the loss.

A neural network implementation should support appropriate mechanisms for:

• Parameter initialization.
• Parameter discovery.
• Gradient tracking.
• Parameter updates.
• State serialization.
• Validation of parameter shapes.

Linear layers perform affine transformations.

[ Y=XW+b ]

where:

• (X) is the input matrix.
• (W) is the weight matrix.
• (b) is the bias vector.
• (Y) is the output.

If:

[ X\in\mathbb{R}^{32\times128} ]

and:

[ W\in\mathbb{R}^{128\times64} ]

then:

[ Y\in\mathbb{R}^{32\times64} ]

The batch contains 32 examples, each transformed from 128 features to 64 features.

For a fully connected layer with bias:

[ P=n_{in}n_{out}+n_{out} ]

For 128 inputs and 64 outputs:

[ P=128(64)+64=8256 ]

Activation functions introduce nonlinearity.

[ \operatorname{ReLU}(x)=\max(0,x) ]

Commonly used in feedforward and convolutional networks.

[ \sigma(x)=\frac{1}{1+e^{-x}} ]

Maps values to the interval (0, 1).

[ \tanh(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}} ]

Maps values to (-1, 1).

For a vector (z):

\frac{e^{z_i}}{\sum_j e^{z_j}} ]

Used to convert logits into a normalized probability distribution.

Convolutional neural networks are designed to process spatially structured data.

Input Image
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Convolution
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Activation
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Pooling
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Feature Maps
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Classification Head
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Prediction

• Input channels.
• Output channels.
• Kernel size.
• Stride.
• Padding.
• Dilation.
• Feature maps.

For a convolution dimension:

[ O= \left\lfloor \frac{I+2P-D(K-1)-1}{S}+1 \right\rfloor ]

where:

• (I) = input size.
• (P) = padding.
• (D) = dilation.
• (K) = kernel size.
• (S) = stride.

Convolution implementations should be tested for:

• Output dimensions.
• Kernel behavior.
• Padding.
• Stride.
• Gradient correctness.
• Numerical precision.

Recurrent neural networks process sequential information.

A simplified recurrent update is:

[ h_t=\phi(W_xx_t+W_hh_{t-1}+b) ]

where:

• (x_t) is the current input.
• (h_t) is the current hidden state.
• (h_{t-1}) is the previous hidden state.

• Time-series modeling.
• Sequential prediction.
• Language processing.
• Sensor data analysis.

Traditional recurrent networks can experience vanishing or exploding gradients across long sequences.

LSTM and GRU architectures introduce gating mechanisms to help manage information over sequences.

Long Short-Term Memory networks use gates and a cell state to regulate information flow.

Common components include:

• Input gate.
• Forget gate.
• Output gate.
• Cell state.

Gated Recurrent Units commonly use:

• Update gate.
• Reset gate.
• Hidden state.

Attention mechanisms compute context-dependent representations by weighting relationships between inputs.

\operatorname{softmax} \left( \frac{QK^T}{\sqrt{d_k}} \right)V ]

where:

• (Q) = queries.
• (K) = keys.
• (V) = values.
• (d_k) = key dimension.

Input Representations
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Query / Key / Value
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Attention Scores
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Softmax
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Weighted Values
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Context Representation

• Tensor dimensions.
• Attention masks.
• Numerical stability.
• Memory complexity.
• Gradient propagation.

Transformers use attention-based operations to model relationships across input sequences.

• Input embeddings.
• Positional information.
• Multi-head attention.
• Feedforward networks.
• Residual connections.
• Normalization layers.

Input
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Multi-Head Attention
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Residual + Normalization
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Feedforward Network
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Residual + Normalization
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Output

Actual ordering varies across transformer architectures.

• Sequence length.
• Attention complexity.
• Model dimensions.
• Head count.
• Memory usage.
• Training stability.

Loss functions measure the difference between model predictions and target values.

\frac{1}{N} \sum_{i=1}^{N}(y_i-\hat{y}_i)^2 ]

Commonly used in regression.

\frac{1}{N} \sum_{i=1}^{N}|y_i-\hat{y}_i| ]

-\frac{1}{N} \sum_{i=1}^{N} \left[ y_i\log(p_i) + (1-y_i)\log(1-p_i) \right] ]

Probabilities should be handled using numerically stable implementations.

Cross-entropy is commonly used in classification tasks.

The appropriate implementation depends on whether inputs are logits or probabilities.

Forward propagation computes model outputs from inputs.

Input
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Layer 1
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Activation
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Layer 2
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Output

Forward operations should:

• Validate inputs.
• Apply transformations.
• Produce correctly shaped outputs.
• Preserve differentiation information when required.
• Report invalid operations.

Backward propagation calculates gradients of the loss with respect to trainable parameters.

For:

[ y=f(g(x)) ]

the derivative is:

\frac{df}{dg} \frac{dg}{dx} ]

Model Prediction
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Loss
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Backward Differentiation
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Parameter Gradients
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Optimizer Update

Related Documentation: Automatic Differentiation

1. Prepare the dataset.
2. Initialize the model.
3. Select a loss function.
4. Configure an optimizer.
5. Run forward propagation.
6. Calculate loss.
7. Compute gradients.
8. Update parameters.
9. Evaluate the model.
10. Repeat as appropriate.

# Conceptual pseudocode.
# Not a verified EveryO API example.

model = create_model()
optimizer = create_optimizer(model.parameters())

for epoch in range(num_epochs):
    for inputs, targets in training_data:
        predictions = model(inputs)
        loss = loss_function(predictions, targets)

        optimizer.zero_grad()
        loss.backward()
        optimizer.step()

Important hyperparameters include:

• Learning rate.
• Batch size.
• Number of epochs.
• Optimizer selection.
• Weight initialization.
• Regularization.
• Gradient clipping.

Model evaluation measures performance on data not used for parameter updates.

A typical workflow distinguishes:

• Training data.
• Validation data.
• Test data.

• Avoid data leakage.
• Use appropriate metrics.
• Document dataset characteristics.
• Compare against relevant baselines.
• Evaluate robustness.
• Report limitations.

Parameter initialization can affect training convergence.

• Random initialization.
• Xavier/Glorot initialization.
• He/Kaiming initialization.
• Zero initialization for appropriate bias terms.

Potential issues include:

• Exploding gradients.
• Vanishing gradients.
• Non-finite loss values.
• Poorly scaled features.
• Excessive learning rates.

Depending on architecture and implementation:

• Appropriate initialization.
• Normalization.
• Gradient clipping.
• Stable loss computation.
• Learning-rate tuning.

Model serialization allows supported model state to be stored and restored.

• Model parameters.
• Model configuration.
• Optimizer state, if supported.
• Training progress.
• Relevant metadata.

• Version compatibility.
• State integrity.
• Parameter shapes.
• Error handling.
• File validation.
• Security of untrusted artifacts.

Neural network performance depends on architecture, numerical operations, memory use, and hardware.

Benchmark results should document:

• Hardware.
• Python version.
• Framework version.
• Model architecture.
• Input shape.
• Data type.
• Batch size.
• Measurement methodology.

Neural network modules should be tested for both forward behavior and gradients.

• Layer output shapes.
• Parameter initialization.
• Forward computation.
• Backward computation.
• Gradient correctness.
• Loss calculations.
• Serialization behavior.
• Invalid inputs.
• Backend compatibility.

python -m pytest

Gradient checks can compare analytical gradients against finite-difference approximations.

For a scalar function:

[ f'(x)\approx \frac{f(x+h)-f(x-h)}{2h} ]

Appropriate numerical tolerances are necessary.

Continue with the following EveryO Wiki pages.

Some Wiki links will become active after the corresponding pages are created.