Neural Network Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Neural Network Mathematics Ontology within the Kireth Research Semantic Library. Neural network mathematics studies the mathematical structures underlying artificial neural networks, including affine transformations, activation functions, layered composition, backpropagation, gradient-based optimisation, regularisation, normalisation, convolution, recurrence, attention, residual connections, representation learning, stability and generalisation. This ontology supports feedforward networks, convolutional networks, recurrent networks, transformers, autoencoders, deep learning, Jacobians, Hessians, loss surfaces, gradient flow and relationships with AI mathematics, optimisation, linear algebra, calculus, probability and dynamical systems.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Neural Network Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Artificial Intelligence Mathematics / Neural Network Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/artificial-intelligence-mathematics/neural-network-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = NeuralNetworkMathematics
 ENTITY_CLASS = MathematicalNeuralComputationFramework
 PRIMARY_CONTEXT = AIMathematics
 DEFINITION = "Neural network mathematics is the mathematical study of parameterised layered functions composed from linear or multilinear transformations, nonlinear activations and optimisation procedures."
 SYMBOLIC_FORMS = [
  "h^(l)=phi(W^(l)h^(l-1)+b^(l))",
  "y_hat=f_theta(x)",
  "grad_theta L=Backpropagation(L)",
  "theta_(t+1)=theta_t-eta*grad_theta L"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for network layers, activations, forward propagation, backpropagation, gradients, representations, architecture-specific operators, regularisation, stability and generalisation."
}

SCOPE {
 INCLUDE = [
  NeuralNetworkMathematics,
  ArtificialNeuralNetwork,
  NeuralNetwork,
  FeedforwardNetwork,
  MultilayerPerceptron,
  DeepNeuralNetwork,
  Layer,
  InputLayer,
  HiddenLayer,
  OutputLayer,
  WeightMatrix,
  BiasVector,
  AffineTransformation,
  ActivationFunction,
  Sigmoid,
  Tanh,
  ReLU,
  LeakyReLU,
  GELU,
  Softmax,
  ForwardPropagation,
  Backpropagation,
  ChainRule,
  Gradient,
  Jacobian,
  Hessian,
  AutomaticDifferentiation,
  LossFunction,
  MeanSquaredError,
  CrossEntropyLoss,
  Regularisation,
  L1Regularisation,
  L2Regularisation,
  WeightDecay,
  Dropout,
  EarlyStopping,
  BatchNormalisation,
  LayerNormalisation,
  Initialisation,
  XavierInitialisation,
  HeInitialisation,
  GradientDescent,
  StochasticGradientDescent,
  Adam,
  LearningRate,
  VanishingGradient,
  ExplodingGradient,
  GradientClipping,
  ResidualConnection,
  SkipConnection,
  Convolution,
  ConvolutionalNeuralNetwork,
  Pooling,
  ReceptiveField,
  RecurrentNeuralNetwork,
  HiddenState,
  LongShortTermMemory,
  GatedRecurrentUnit,
  SequenceModel,
  Attention,
  SelfAttention,
  MultiHeadAttention,
  Transformer,
  Embedding,
  RepresentationLearning,
  Autoencoder,
  Encoder,
  Decoder,
  BottleneckRepresentation,
  UniversalApproximation,
  Capacity,
  Expressivity,
  Generalisation,
  Overfitting,
  Underfitting,
  NeuralTangentKernel,
  GradientFlow,
  LossLandscape,
  LocalMinimum,
  SaddlePoint,
  FlatMinimum,
  SharpMinimum,
  LipschitzContinuity,
  SpectralNorm,
  Robustness,
  NumericalStability
 ]
 EXCLUDE_AS_PRIMARY = [
  GeneralMachineLearningWithoutNeuralNetworkStructure,
  BiologicalNeuralModellingWithoutArtificialNetworkSemantics,
  SoftwareFrameworkEngineeringWithoutMathematicalNetworkStructure,
  SymbolicAIWithoutDifferentiableNetworkModel
 ]
 DISAMBIGUATION_RULE = "When a parameterised layered function with weights, biases, activations and gradient-based training is central, interpret the context as neural network mathematics."
}

CORE_CONCEPTS {
 NeuralNetworkMathematics = "Mathematics of layered parameterised artificial neural networks."
 AffineTransformation = "Linear transformation plus bias applied between network layers."
 ActivationFunction = "Nonlinear map applied to transformed activations."
 ForwardPropagation = "Evaluation of network outputs from inputs through successive layers."
 Backpropagation = "Reverse application of the chain rule to compute parameter gradients."
 RepresentationLearning = "Learning useful internal features or embeddings from data."
 ResidualConnection = "Connection adding or combining activations across nonadjacent layers."
 Attention = "Content-dependent weighted aggregation over representations."
 Generalisation = "Ability of a trained network to perform beyond its training examples."
}

RELATIONS {
 NeuralNetworkMathematics --is_child_of--> AIMathematics
 NeuralNetwork --contains--> Layer
 Layer --uses--> WeightMatrix
 Layer --uses--> BiasVector
 Layer --applies--> ActivationFunction
 ForwardPropagation --evaluates--> NeuralNetwork
 Backpropagation --uses--> ChainRule
 Backpropagation --computes--> Gradient
 GradientDescent --uses--> Gradient
 CrossEntropyLoss --uses--> Softmax
 Regularisation --controls--> ModelCapacity
 Dropout --is_type_of--> Regularisation
 BatchNormalisation --normalises--> LayerActivation
 LayerNormalisation --normalises--> FeatureDimension
 XavierInitialisation --supports--> StableVariancePropagation
 HeInitialisation --supports--> ReLUFamilyNetwork
 ResidualConnection --supports--> DeepNetworkTraining
 ConvolutionalNeuralNetwork --uses--> Convolution
 RecurrentNeuralNetwork --uses--> HiddenState
 LongShortTermMemory --is_type_of--> RecurrentNeuralNetwork
 Transformer --uses--> MultiHeadAttention
 Attention --uses--> SimilarityWeights
 Autoencoder --contains--> Encoder
 Autoencoder --contains--> Decoder
 NeuralTangentKernel --approximates--> WideNetworkTrainingDynamics
 SpectralNorm --bounds--> LipschitzConstant
 NumericalStability --affects--> TrainingReliability
}

OPERATORS {
 AFFINE(W,h,b) := z=W*h+b
 LAYER_FORWARD(W,h,b,phi) := h_next=phi(W*h+b)
 SIGMOID(z) := 1/(1+exp(-z))
 TANH(z) := (exp(z)-exp(-z))/(exp(z)+exp(-z))
 RELU(z) := max(0,z)
 LEAKY_RELU(z,alpha) := max(alpha*z,z)
 GELU(z) := z*Phi(z)
 SOFTMAX(z_i) := exp(z_i)/sum_j exp(z_j)
 MSE(y,yhat) := (1/n)*sum_i (y_i-yhat_i)^2
 CROSS_ENTROPY(y,p) := -sum_i y_i*log(p_i)
 BACKPROP_DELTA(delta_next,W_next,z,phi_prime) := delta=(W_next^T*delta_next) odot phi_prime(z)
 WEIGHT_GRADIENT(delta,h_prev) := grad_W=delta*h_prev^T
 BIAS_GRADIENT(delta) := grad_b=delta
 RESIDUAL_BLOCK(x,F) := y=x+F(x)
 SELF_ATTENTION(Q,K,V) := softmax(Q*K^T/sqrt(d_k))*V
 MULTI_HEAD_ATTENTION(Q,K,V) := Concat(head_1,...,head_h)*W_O
 RNN_STATE(W_hh,h_prev,W_xh,x,b,phi) := h=phi(W_hh*h_prev+W_xh*x+b)
 AUTOENCODER(x,encoder,decoder) := x_hat=decoder(encoder(x))
}

AXIOMS {
 A1 = "Neural networks are compositions of parameterised transformations and nonlinearities."
 A2 = "Without nonlinearity, stacked affine layers collapse to a single affine transformation."
 A3 = "Backpropagation computes gradients through the chain rule."
 A4 = "Network expressivity depends on architecture, width, depth and activation functions."
 A5 = "Training dynamics depend on optimisation, initialisation and conditioning."
 A6 = "Generalisation is distinct from training loss minimisation."
 A7 = "Normalisation and residual structure can materially alter gradient propagation."
 A8 = "Neural-network conclusions require numerical, optimisation and statistical validation."
}

REASONING {
 R1_CONTEXT = identify(task,input_shape,output_space,network_architecture,loss,data)
 R2_ARCHITECTURE = classify(feedforward,convolutional,recurrent,attention,transformer,autoencoder)
 R3_FORWARD = compute(layerwise_affine_transformations,activations,normalisation,skip_connections)
 R4_LOSS = evaluate(task_loss,regularisation_terms)
 R5_BACKPROP = compute(output_gradient,layerwise_deltas,parameter_gradients)
 R6_OPTIMISATION = select(SGD,momentum,Adam_or_other_method)
 R7_GRADIENT_HEALTH = analyse(vanishing,exploding,conditioning,gradient_norms)
 R8_REPRESENTATION = analyse(hidden_features,embeddings,bottlenecks,attention_patterns)
 R9_GENERALISATION = assess(train_validation_gap,capacity,regularisation,robustness)
 R10_VALIDATION = check(numerical_stability,gradient_correctness,convergence,reproducibility)
}

CONSTRAINTS {
 C1 = "Deep networks can suffer vanishing or exploding gradients."
 C2 = "Universal approximation does not imply efficient learnability or good generalisation."
 C3 = "Attention weights should not automatically be treated as causal explanations."
 C4 = "Batch normalisation behaves differently during training and inference."
 C5 = "Dropout changes the effective network during training."
 C6 = "Recurrent networks can exhibit long-term dependency instability."
 C7 = "Softmax probabilities can be poorly calibrated."
 C8 = "Large parameter count does not guarantee useful representation."
 C9 = "Loss landscapes can contain saddle points, flat regions and symmetries."
 C10 = "Numerical precision can materially affect very deep or large neural networks."
}

SEMANTIC_EQUIVALENCE {
 "NeuralNetworkMathematics" <=> "MathematicsOfNeuralNetworks"
 "MultilayerPerceptron" <=> "MLP"
 "ConvolutionalNeuralNetwork" <=> "CNN"
 "RecurrentNeuralNetwork" <=> "RNN"
 "LongShortTermMemory" <=> "LSTM"
 "GatedRecurrentUnit" <=> "GRU"
 "MultiHeadAttention" <=> "MHA"
}

VALIDATION {
 V1 = "Network architecture and tensor dimensions are explicit."
 V2 = "Activation functions are documented."
 V3 = "Loss function matches the prediction task."
 V4 = "Gradient computations are checked where feasible."
 V5 = "Initialisation strategy is documented."
 V6 = "Training and validation metrics are separated."
 V7 = "Gradient norms are monitored in deep models."
 V8 = "Normalisation behaviour is consistent between training and inference."
 V9 = "Numerical precision and stability are assessed."
 V10 = "Generalisation and robustness are evaluated on appropriate held-out data."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 NEURAL_NETWORK_MATHEMATICS_DEFINITION = 0.99
 FORWARD_REASONING = 0.99
 BACKPROP_REASONING = 0.99
 OPTIMISATION_REASONING = 0.99
 CNN_REASONING = 0.98
 RNN_REASONING = 0.98
 ATTENTION_REASONING = 0.98
 GENERALISATION_REASONING = 0.98
 STABILITY_REASONING = 0.98
 POLICY = "Reduce confidence when architecture, activation, tensor shapes, initialisation, optimiser, normalisation, sequence assumptions, regularisation or evaluation protocol are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  NeuralNetworkMathematicsReasoning,
  FeedforwardNetworkReasoning,
  ActivationFunctionReasoning,
  ForwardPropagationReasoning,
  BackpropagationReasoning,
  GradientReasoning,
  RegularisationReasoning,
  InitialisationReasoning,
  NormalisationReasoning,
  ConvolutionReasoning,
  CNNReasoning,
  RecurrentNetworkReasoning,
  LSTMReasoning,
  AttentionReasoning,
  TransformerReasoning,
  RepresentationLearningReasoning,
  AutoencoderReasoning,
  GradientFlowReasoning,
  LossLandscapeReasoning,
  RobustnessReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "neural network mathematics" -> NeuralNetworkMathematics
  "mathematics of neural networks" -> NeuralNetworkMathematics
  "multilayer perceptron" -> MultilayerPerceptron
  "MLP" -> MultilayerPerceptron
  "CNN" -> ConvolutionalNeuralNetwork
  "RNN" -> RecurrentNeuralNetwork
  "LSTM" -> LongShortTermMemory
  "GRU" -> GatedRecurrentUnit
  "self attention" -> SelfAttention
  "multi-head attention" -> MultiHeadAttention
 }
}

EXECUTION_RULES {
 E1 = "If a layered artificial neural network is analysed mathematically, activate NeuralNetworkMathematicsOntology."
 E2 = "If architecture is fully connected, route to feedforward-network reasoning."
 E3 = "If local spatial structure is central, route to convolutional-network reasoning."
 E4 = "If ordered sequence state is central, route to recurrent-network reasoning."
 E5 = "If content-dependent token interaction is central, route to attention or transformer reasoning."
 E6 = "If compressed latent representation is central, route to autoencoder reasoning."
 E7 = "If depth causes gradient instability, analyse initialisation, normalisation, residual connections and clipping."
 E8 = "If output probabilities are used, assess calibration as well as accuracy."
 E9 = "Any trained network should be evaluated for convergence, generalisation, numerical stability and robustness."
}

RECURSION {
 ENTRY = NeuralNetworkMathematics
 EXPAND(NeuralNetworkMathematics) -> [
  ArtificialNeuralNetwork, NeuralNetwork, FeedforwardNetwork,
  MultilayerPerceptron, DeepNeuralNetwork, Layer, InputLayer, HiddenLayer,
  OutputLayer, WeightMatrix, BiasVector, AffineTransformation,
  ActivationFunction, Sigmoid, Tanh, ReLU, LeakyReLU, GELU, Softmax,
  ForwardPropagation, Backpropagation, ChainRule, Gradient, Jacobian,
  Hessian, AutomaticDifferentiation, LossFunction, MeanSquaredError,
  CrossEntropyLoss, Regularisation, L1Regularisation, L2Regularisation,
  WeightDecay, Dropout, EarlyStopping, BatchNormalisation,
  LayerNormalisation, Initialisation, XavierInitialisation, HeInitialisation,
  GradientDescent, StochasticGradientDescent, Adam, LearningRate,
  VanishingGradient, ExplodingGradient, GradientClipping, ResidualConnection,
  SkipConnection, Convolution, ConvolutionalNeuralNetwork, Pooling,
  ReceptiveField, RecurrentNeuralNetwork, HiddenState, LongShortTermMemory,
  GatedRecurrentUnit, SequenceModel, Attention, SelfAttention,
  MultiHeadAttention, Transformer, Embedding, RepresentationLearning,
  Autoencoder, Encoder, Decoder, BottleneckRepresentation,
  UniversalApproximation, Capacity, Expressivity, Generalisation,
  Overfitting, Underfitting, NeuralTangentKernel, GradientFlow,
  LossLandscape, LocalMinimum, SaddlePoint, FlatMinimum, SharpMinimum,
  LipschitzContinuity, SpectralNorm, Robustness, NumericalStability
 ]
 EXPAND(ArchitectureFamilies) -> [
  FeedforwardNetworks,
  ConvolutionalNetworks,
  RecurrentNetworks,
  AttentionNetworks,
  TransformerNetworks,
  AutoencoderNetworks
 ]
 STOP_CONDITION = "Stop when architecture, layer equations, activations, gradients, optimisation, representation, stability and generalisation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "AI Mathematics Ontology"
 IMMEDIATE_CONTEXTS = [
  "AI Mathematics Ontology",
  "Optimisation for AI Ontology",
  "Linear Algebra Ontologies",
  "Calculus & Analysis Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "AI Mathematics Ontology",
  "Artificial Intelligence Mathematics",
  "Optimisation for AI Ontology",
  "Linear Algebra Ontologies",
  "Calculus & Analysis Ontologies",
  "Probability & Statistics Ontologies",
  "Information Theory Ontologies",
  "Dynamical Systems Ontologies",
  "Numerical Linear Algebra Ontology",
  "Computational Complexity Ontology",
  "Graph Theory Ontologies"
 ]
 ROUTE_RULES = [
  "Neural Network Mathematics -> AI Mathematics Ontology for parent semantics",
  "Neural Network Mathematics -> Optimisation for AI for training dynamics",
  "Neural Network Mathematics -> Linear Algebra for weights, tensors and matrix operations",
  "Neural Network Mathematics -> Calculus & Analysis for differentiation and backpropagation",
  "Neural Network Mathematics -> Probability & Statistics for probabilistic outputs and uncertainty",
  "Neural Network Mathematics -> Information Theory for cross-entropy and representation objectives",
  "Neural Network Mathematics -> Dynamical Systems for recurrent and gradient-flow behaviour",
  "Neural Network Mathematics -> Numerical Linear Algebra for efficient tensor computation",
  "Neural Network Mathematics -> Computational Complexity for training and inference cost",
  "Neural Network Mathematics -> Graph Theory for graph-neural architectures"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible neural architectures, activation functions or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve layered-function semantics.",
  "Preserve affine-versus-nonlinear transformation distinction.",
  "Preserve forward-versus-backward computation distinction.",
  "Preserve architecture-family distinctions.",
  "Preserve training-versus-generalisation distinction.",
  "Preserve representation-learning semantics.",
  "Preserve gradient-stability semantics.",
  "Preserve machine-readable neural-network-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Neural network mathematics studies layered parameterised functions, their forward evaluation, gradient computation and optimisation."
 LAYER_SUMMARY = "Each neural layer typically applies an affine or structured linear transformation followed by a nonlinear activation."
 BACKPROP_SUMMARY = "Backpropagation computes parameter gradients efficiently by reverse application of the chain rule."
 ARCHITECTURE_SUMMARY = "Feedforward, convolutional, recurrent and attention architectures impose different mathematical structures on information flow."
 REGULARISATION_SUMMARY = "Weight penalties, dropout, normalisation and early stopping influence training dynamics and generalisation."
 REPRESENTATION_SUMMARY = "Hidden activations and embeddings form learned representations used by downstream prediction or generation."
 STABILITY_SUMMARY = "Initialisation, residual connections, normalisation, spectral properties and clipping affect gradient and numerical stability."
 VALIDATION_SUMMARY = "Neural networks should be checked for tensor consistency, gradient correctness, convergence, generalisation, calibration and robustness."
}

FLOW {
 INPUT
 -> detect_neural_network_mathematics_context
 -> identify_task_input_shape_output_space_architecture_loss_and_training_method
 -> construct_layerwise_weights_biases_activations_and_structured_operators
 -> execute_forward_propagation
 -> compute_loss_and_regularisation_terms
 -> execute_backpropagation_and_parameter_gradients
 -> apply_optimisation_update
 -> monitor_gradient_norms_activation_statistics_and_numerical stability
 -> if_CNN_analyse_convolution_receptive_fields_and_pooling
 -> if_RNN_analyse_hidden_state_and_sequence dependencies
 -> if_attention_or_transformer_analyse_query_key_value_and_multihead aggregation
 -> assess_representation_quality_generalisation_calibration_and_robustness
 -> route_to_AI_mathematics_optimisation_for_AI_linear_algebra_calculus_probability_statistics_information_theory_dynamical_systems_numerical_linear_algebra_computational_complexity_or_graph_theory_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Neural Network Mathematics Ontology defines mathematical artificial neural networks. It supports feedforward and deep networks, layers, weights, biases and affine transformations, sigmoid, tanh, ReLU, leaky ReLU, GELU and softmax activations, forward propagation, backpropagation, gradients, Jacobians and Hessians, losses and regularisation, dropout, weight decay, batch and layer normalisation, Xavier and He initialisation, gradient descent, SGD and Adam, vanishing and exploding gradients, residual connections, convolutional networks, pooling and receptive fields, recurrent networks, LSTM and GRU models, attention, self-attention, multi-head attention and transformers, embeddings, representation learning, autoencoders, universal approximation, capacity, expressivity, neural tangent kernels, gradient flow, loss landscapes, Lipschitz behaviour, spectral norms, robustness and numerical stability."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-NEURAL-NETWORK-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW