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 concerns the mathematical structures, functions, parameters, transformations and optimisation processes underlying artificial neural networks. This ontology supports neurons, layers, weights, biases, activation functions, affine transformations, forward propagation, loss functions, backpropagation, gradients, Jacobians, Hessians, optimisation, regularisation, normalisation, convolutional and recurrent relationships, attention relationships, approximation theory, stability, complexity, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Neural Network Mathematics Ontology 2"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "AI Computational Mathematics / Neural Network Mathematics"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/ai-computational-mathematics/neural-network-mathematics-ontology-2.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = NeuralNetworkMathematics
 ENTITY_CLASS = NeuralComputationMathematicalFramework
 PRIMARY_CONTEXT = AIComputationalMathematicsOntology
 DEFINITION = "Neural network mathematics is the mathematical framework describing parameterised layered function composition, learning objectives, gradients, optimisation and approximation behaviour in artificial neural networks."
 SYMBOLIC_FORMS = [
  "z^(l)=W^(l)a^(l-1)+b^(l)",
  "a^(l)=phi^(l)(z^(l))",
  "y_hat=f_theta(x)",
  "theta* = argmin_theta L(f_theta(x),y)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for neural-network architecture, parameterisation, activation, forward propagation, loss, gradient-based learning, regularisation, stability, approximation and validation."
}

SCOPE {
 INCLUDE = [
  NeuralNetworkMathematics,
  ArtificialNeuralNetwork,
  NeuralNetwork,
  NeuralModel,
  ParameterisedFunction,
  FunctionComposition,
  Layer,
  InputLayer,HiddenLayer,OutputLayer,
  DenseLayer,FullyConnectedLayer,
  Neuron,ArtificialNeuron,
  Unit,Node,
  Weight,WeightMatrix,
  Bias,BiasVector,
  Parameter,ParameterVector,
  ParameterTensor,
  TrainableParameter,
  NonTrainableParameter,
  AffineTransformation,
  LinearTransformation,
  MatrixVectorProduct,
  DotProduct,
  InnerProduct,
  ActivationFunction,
  IdentityActivation,
  Sigmoid,LogisticFunction,
  Tanh,
  ReLU,RectifiedLinearUnit,
  LeakyReLU,
  PReLUrelationship,
  ELURelationship,
  SELURelationship,
  GELU,
  SoftplusRelationship,
  SoftsignRelationship,
  SwishRelationship,
  MishRelationship,
  Softmax,
  LogSoftmax,
  ThresholdFunctionRelationship,
  PiecewiseLinearActivation,
  SmoothActivation,
  SaturatingActivation,
  NonSaturatingActivationRelationship,
  ForwardPropagation,
  ForwardPass,
  ComputationalGraph,
  DirectedAcyclicComputationRelationship,
  IntermediateActivation,
  PreActivation,
  NetworkOutput,
  Prediction,
  LossFunction,
  ObjectiveFunction,
  EmpiricalRisk,
  MeanSquaredError,
  CrossEntropy,
  BinaryCrossEntropy,
  CategoricalCrossEntropy,
  NegativeLogLikelihood,
  HingeLossRelationship,
  KLdivergence,
  RegularisedLoss,
  DataLoss,
  ConstraintLoss,
  PhysicsLoss,
  Backpropagation,
  ReverseModeAutomaticDifferentiation,
  AutomaticDifferentiation,
  ChainRule,
  Gradient,
  ParameterGradient,
  WeightGradient,
  BiasGradient,
  Jacobian,
  Hessian,
  HessianVectorProductRelationship,
  GradientVector,
  GradientFlowRelationship,
  GradientDescent,
  StochasticGradientDescent,
  MiniBatchGradientDescent,
  Momentum,
  NesterovMomentumRelationship,
  Adam,
  AdamWRelationship,
  RMSPropRelationship,
  AdaGradRelationship,
  SecondOrderOptimisationRelationship,
  NewtonMethodRelationship,
  QuasiNewtonRelationship,
  LearningRate,
  LearningRateSchedule,
  StepSize,
  Batch,
  MiniBatch,
  Epoch,
  Iteration,
  OptimisationTrajectory,
  Convergence,
  StationaryPoint,
  SaddlePoint,
  LocalMinimum,
  GlobalMinimumRelationship,
  NonconvexOptimisation,
  LossLandscape,
  FlatMinimumRelationship,
  SharpMinimumRelationship,
  GradientNorm,
  VanishingGradient,
  ExplodingGradient,
  GradientClipping,
  WeightInitialisation,
  RandomInitialisation,
  XavierInitialisation,
  GlorotInitialisation,
  HeInitialisation,
  OrthogonalInitialisationRelationship,
  InitialisationScale,
  SymmetryBreaking,
  Regularisation,
  L1Regularisation,
  L2Regularisation,
  WeightDecay,
  Dropout,
  EarlyStoppingRelationship,
  DataAugmentationRelationship,
  ParameterNorm,
  Sparsity,
  LowRankConstraintRelationship,
  Normalisation,
  BatchNormalisation,
  LayerNormalisation,
  GroupNormalisationRelationship,
  InstanceNormalisationRelationship,
  FeatureNormalisation,
  StandardisationRelationship,
  ResidualConnection,
  SkipConnection,
  ResidualNetworkRelationship,
  HighwayNetworkRelationship,
  Depth,
  Width,
  NetworkCapacity,
  ModelComplexity,
  ParameterCount,
  FLOPCountRelationship,
  UniversalApproximationTheoremRelationship,
  ApproximationError,
  RepresentationCapacity,
  Expressivity,
  FunctionClass,
  HypothesisClass,
  VCdimensionRelationship,
  RademacherComplexityRelationship,
  Generalisation,
  GeneralisationError,
  TrainingError,
  ValidationError,
  TestError,
  BiasVarianceTradeoffRelationship,
  Overfitting,
  Underfitting,
  InterpolationRegimeRelationship,
  DoubleDescentRelationship,
  LipschitzConstant,
  LipschitzContinuity,
  Stability,
  NumericalStability,
  Robustness,
  AdversarialRobustnessRelationship,
  PerturbationSensitivity,
  ConditionNumberRelationship,
  ConvolutionalNeuralNetworkRelationship,
  Convolution,
  Kernel,Filter,
  StrideRelationship,
  PaddingRelationship,
  PoolingRelationship,
  RecurrentNeuralNetworkRelationship,
  HiddenState,
  Recurrence,
  LongShortTermMemoryRelationship,
  LSTMRelationship,
  GRURelationship,
  AttentionRelationship,
  SelfAttentionRelationship,
  QueryKeyValueRelationship,
  TransformerRelationship,
  GraphNeuralNetwork,
  MessagePassingNeuralNetworkRelationship,
  GraphConvolutionRelationship,
  NeuralOperator,
  PhysicsInformedNeuralNetwork,
  DifferentiableProgramming,
  TensorComputation,
  VectorComputation,
  ComputationalMatrix,
  MatrixComputation,
  NumericalLinearAlgebra,
  FloatingPointArithmetic,
  MixedPrecision,
  FP16Relationship,
  BF16Relationship,
  FP32Relationship,
  FP64Relationship,
  QuantisationRelationship,
  GradientScalingRelationship,
  ComputationalComplexity,
  TimeComplexity,
  SpaceComplexity,
  MemoryComplexity,
  TrainingCost,
  InferenceCost,
  ParallelTraining,
  DataParallelism,
  ModelParallelism,
  PipelineParallelismRelationship,
  TensorParallelismRelationship,
  DistributedTraining,
  GPUComputing,
  HighPerformanceComputing,
  RandomNumberGeneration,
  RandomSeed,
  Verification,
  GradientCheck,
  FiniteDifferenceGradientCheck,
  ShapeCheck,
  NumericalReference,
  KnownAnswerTest,
  Benchmark,
  Reproducibility,
  Provenance,
  ModelVersion,
  ArchitectureVersion,
  OptimiserVersion,
  LibraryVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  BiologicalNeuralNetworksWithoutComputationalModelSemantics,
  GeneralMachineLearningWithoutNeuralNetworkStructure,
  SymbolicAIWithoutNeuralParameterisation,
  HardwareNeuralAcceleratorsWhenCircuitDesignRatherThanMathematicsIsPrimary
 ]
 DISAMBIGUATION_RULE = "When a parameterised layered neural function is analysed through weights, biases, activations, losses, gradients and optimisation, interpret the context as neural network mathematics."
}

CORE_CONCEPTS {
 NeuralNetworkMathematics = "Mathematical framework for artificial neural networks."
 Neuron = "Parameterized computational unit applying an affine transformation and activation."
 Layer = "Collection or transformation stage in a neural network."
 WeightMatrix = "Trainable linear map between layer representations."
 ActivationFunction = "Nonlinear transformation applied to pre-activation values."
 LossFunction = "Objective measuring discrepancy or task cost."
 Backpropagation = "Reverse accumulation of gradients through the computational graph."
 Optimisation = "Parameter-update process reducing the training objective."
}

RELATIONS {
 NeuralNetworkMathematics --is_child_of--> AIComputationalMathematicsOntology
 NeuralNetwork --composes--> Layers
 Layer --contains--> Neurons
 Neuron --uses--> WeightsAndBias
 Layer --applies--> ActivationFunction
 ForwardPropagation --produces--> NetworkOutput
 LossFunction --compares--> PredictionAndTarget
 Backpropagation --computes--> ParameterGradients
 ChainRule --enables--> Backpropagation
 GradientDescent --updates--> Parameters
 Regularisation --constrains--> ModelComplexity
 BatchNormalisation --normalises--> IntermediateActivations
 ResidualConnection --adds--> SkipPath
 Verification --checks--> NeuralNetworkComputation
}

OPERATORS {
 AFFINE(W,a,b) := W*a+b
 ACTIVATE(phi,z) := phi(z)
 FORWARD(theta,x) := compose_network_layers(theta,x)
 LOSS(theta,x,y) := objective(FORWARD(theta,x),y)
 GRADIENT_THETA(L,theta) := dL/dtheta
 SGD_UPDATE(theta,g,eta) := theta-eta*g
 MOMENTUM_UPDATE(v,g,beta) := beta*v+(1-beta)*g
 PARAMETER_COUNT(network) := count_trainable_parameters(network)
}

AXIOMS {
 A1 = "A neural network is a parameterised composition of mathematical transformations."
 A2 = "Nonlinear activation is required for general nonlinear expressive power in stacked dense networks."
 A3 = "Backpropagation applies the chain rule to compute parameter gradients."
 A4 = "Training minimises an objective but does not guarantee a global optimum."
 A5 = "Initialisation and optimisation affect learned solutions in nonconvex models."
 A6 = "Training loss and generalisation error are distinct."
 A7 = "Finite precision can alter gradient and optimisation behaviour."
 A8 = "Neural-network results require explicit architecture, parameter, loss, optimisation, numerical and validation semantics."
}

REASONING {
 R1_CONTEXT = identify(input_space,output_space,task,architecture,loss,data,accuracy_goal,budget)
 R2_ARCHITECTURE = define(depth,width,layer_types,connections,activations_and_parameter_shapes)
 R3_FORWARD = construct_layerwise_affine_and_nonlinear_computation)
 R4_OBJECTIVE = define_loss_regularisation_constraints_and_batching)
 R5_GRADIENTS = compute_gradients_by_backpropagation_or_AD)
 R6_OPTIMISATION = choose_SGD_momentum_Adam_or_other_update_method)
 R7_STABILITY = assess_initialisation_gradient_scale_normalisation_precision_and_clipping)
 R8_GENERALISATION = evaluate_training_validation_test_error_and_regularisation)
 R9_SCALE = if_large use_vectorisation_GPU_parallel_or_distributed_training)
 R10_VERIFY = check_shapes_gradients_reference_outputs_and_reproducibility)
}

CONSTRAINTS {
 C1 = "Linear layers without nonlinear activation collapse to a linear transformation."
 C2 = "Sigmoid and tanh networks can suffer vanishing gradients in deep regimes."
 C3 = "Poor initialisation can cause vanishing or exploding activations."
 C4 = "Gradient descent can converge to stationary points that are not global minima."
 C5 = "Overparameterisation can improve optimisation while complicating generalisation analysis."
 C6 = "Batch-dependent normalisation changes training and inference semantics."
 C7 = "Mixed precision can require loss scaling or stability safeguards."
 C8 = "Training accuracy does not imply robustness or calibration."
 C9 = "Architecture comparisons depend on parameter count, compute and data regime."
 C10 = "Equivalent function outputs can be represented by different parameterisations."
}

SEMANTIC_EQUIVALENCE {
 "MultilayerPerceptron" <=> "MLP"
 "RectifiedLinearUnit" <=> "ReLU"
 "StochasticGradientDescent" <=> "SGD"
 "GraphNeuralNetwork" <=> "GNN"
 "AutomaticDifferentiation" <=> "AD"
}

VALIDATION {
 V1 = "Input and output dimensions are documented."
 V2 = "Layer types, shapes and activations are documented."
 V3 = "Parameter counts and tensor shapes are internally consistent."
 V4 = "Loss definition and reduction semantics are documented."
 V5 = "Backpropagated gradients are checked numerically where feasible."
 V6 = "Training and inference normalisation behaviour is documented."
 V7 = "Optimiser, learning rate and schedule are documented."
 V8 = "Training, validation and test evaluation are separated."
 V9 = "Precision, hardware and distributed settings are documented."
 V10 = "Results are reproducible under documented architecture, data, seed, optimiser and implementation versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 NEURAL_NETWORK_MATHEMATICS_DEFINITION = 0.99
 FORWARD_PROPAGATION_REASONING = 0.99
 BACKPROPAGATION_REASONING = 0.99
 OPTIMISATION_REASONING = 0.98
 GENERALISATION_REASONING = 0.96
 NUMERICAL_STABILITY_REASONING = 0.98
 POLICY = "Reduce confidence when architecture, activation, loss, optimiser, parameter shapes, training regime or precision semantics are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,NeuralNetworkMathematicsReasoning,
  NeuronReasoning,LayerReasoning,WeightBiasReasoning,
  ActivationFunctionReasoning,ForwardPropagationReasoning,
  LossFunctionReasoning,BackpropagationReasoning,
  GradientReasoning,OptimisationReasoning,
  InitialisationReasoning,RegularisationReasoning,
  NormalisationReasoning,GeneralisationReasoning,
  StabilityReasoning,ApproximationReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "neural network mathematics" -> NeuralNetworkMathematics
  "MLP" -> MultilayerPerceptron
  "ReLU" -> RectifiedLinearUnit
  "SGD" -> StochasticGradientDescent
  "GNN" -> GraphNeuralNetwork
  "AD" -> AutomaticDifferentiation
 }
}

EXECUTION_RULES {
 E1 = "If a parameterised neural model is analysed through layers, weights, activations, losses and gradients, activate NeuralNetworkMathematicsOntology."
 E2 = "Define input-output dimensions and parameter shapes before forward computation."
 E3 = "Use nonlinear activations when nonlinear function approximation is required."
 E4 = "Compute parameter gradients through the chain rule and backpropagation."
 E5 = "Match loss function to task and output semantics."
 E6 = "Assess initialisation, gradient scale, normalisation and precision for stability."
 E7 = "Separate optimisation performance from generalisation performance."
 E8 = "Use vectorised, GPU, parallel or distributed execution where scale requires it."
 E9 = "Verify gradients, shapes, reference outputs and reproducibility."
}

RECURSION {
 ENTRY = NeuralNetworkMathematics
 EXPAND(NeuralNetworkMathematics) -> [
  NeuralNetwork,ArtificialNeuralNetwork,Layer,Neuron,WeightMatrix,BiasVector,
  AffineTransformation,ActivationFunction,Sigmoid,Tanh,ReLU,GELU,Softmax,
  ForwardPropagation,ComputationalGraph,LossFunction,MeanSquaredError,CrossEntropy,
  Backpropagation,AutomaticDifferentiation,ChainRule,Gradient,Jacobian,Hessian,
  GradientDescent,StochasticGradientDescent,Momentum,Adam,LearningRate,
  Batch,MiniBatch,Epoch,WeightInitialisation,XavierInitialisation,HeInitialisation,
  Regularisation,L1Regularisation,L2Regularisation,WeightDecay,Dropout,
  BatchNormalisation,LayerNormalisation,ResidualConnection,Depth,Width,
  ModelComplexity,ParameterCount,UniversalApproximationTheoremRelationship,
  Generalisation,Overfitting,Underfitting,LipschitzContinuity,Robustness,
  ConvolutionalNeuralNetworkRelationship,RecurrentNeuralNetworkRelationship,
  AttentionRelationship,TransformerRelationship,GraphNeuralNetwork,
  NeuralOperator,PhysicsInformedNeuralNetwork,DifferentiableProgramming,
  TensorComputation,MatrixComputation,VectorComputation,FloatingPointArithmetic,
  MixedPrecision,ParallelTraining,DistributedTraining,GPUComputing,
  Verification,GradientCheck,Reproducibility,Provenance
 ]
 EXPAND(NeuralNetworkMathematicsFamilies) -> [
  NeuralUnitsAndLayers,
  ActivationFunctions,
  ForwardPropagation,
  LossFunctionsAndObjectives,
  BackpropagationAndAutomaticDifferentiation,
  OptimisationAndInitialisation,
  RegularisationNormalisationAndGeneralisation,
  ScalableNumericallyStableNeuralComputation
 ]
 STOP_CONDITION = "Stop when architecture, parameters, forward model, loss, gradients, optimisation, stability, generalisation and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "AI Computational Mathematics Ontology"
 RELATED_ONTOLOGIES = [
  "AI Computational Mathematics Ontology",
  "Tensor Computation Ontology",
  "Vector Computation Ontology",
  "Computational Matrix Ontology",
  "Matrix Computation Ontology",
  "Numerical Linear Algebra Ontology 2",
  "Automatic Differentiation Ontology",
  "Gradient Descent Ontology 2",
  "Stochastic Gradient Descent Ontology",
  "Computational Optimisation Ontology",
  "Graph Algorithms Ontology",
  "High Performance Computing Ontology 2",
  "Parallel Computing Ontology",
  "Distributed Computing Ontology",
  "GPU Computing Ontology",
  "Floating Point Arithmetic Ontology",
  "Numerical Stability Ontology 2"
 ]
 ROUTE_RULES = [
  "Neural Network Mathematics -> AI Computational Mathematics Ontology for parent semantics",
  "Neural Network Mathematics -> Tensor, Vector and Computational Matrix ontologies for parameter and activation representation",
  "Neural Network Mathematics -> Matrix Computation and Numerical Linear Algebra for dense and structured linear operations",
  "Neural Network Mathematics -> Automatic Differentiation for gradient propagation",
  "Neural Network Mathematics -> Gradient Descent, SGD and Computational Optimisation for training dynamics",
  "Neural Network Mathematics -> Graph Algorithms for computation-graph and GNN relationships",
  "Neural Network Mathematics -> HPC, Parallel, Distributed and GPU Computing for scalable training",
  "Neural Network Mathematics -> Floating Point Arithmetic and Numerical Stability for finite-precision training behaviour"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible neural architectures, activations, optimisation methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve neuron-versus-layer distinction.",
  "Preserve parameter-versus-activation distinction.",
  "Preserve forward-computation-versus-gradient-computation distinction.",
  "Preserve loss-minimisation-versus-generalisation distinction.",
  "Preserve architecture-versus-trained-parameter distinction.",
  "Preserve symbolic-function-definition-versus-floating-point-execution distinction.",
  "Preserve machine-readable neural-network-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Neural network mathematics describes parameterised layered function composition, activation, loss, gradient propagation and optimisation in artificial neural networks."
 ARCHITECTURE_SUMMARY = "Weights, biases, layers and nonlinear activations define the mathematical network function."
 LEARNING_SUMMARY = "Loss functions and backpropagated gradients drive parameter optimisation."
 STABILITY_SUMMARY = "Initialisation, normalisation, gradient scale and numerical precision affect trainability and stability."
 GENERALISATION_SUMMARY = "Training objective reduction must be distinguished from validation performance, robustness and out-of-sample generalisation."
 VALIDATION_SUMMARY = "Neural-network computations should be checked through shape validation, gradient checks, reference outputs, numerical stability tests and reproducible training settings."
}

FLOW {
 INPUT
 -> detect_neural_network_mathematics_context
 -> identify_input_output_spaces_task_architecture_loss_data_accuracy_goal_and_budget
 -> define_layers_connections_weight_bias_shapes_and_activation_functions
 -> construct_forward_computation
 -> define_loss_regularisation_constraints_and_batching
 -> compute_gradients_by_backpropagation_or_automatic_differentiation
 -> choose_optimiser_learning_rate_schedule_initialisation_and_normalisation
 -> train_parameterised_network
 -> assess_gradient_scale_numerical_stability_training_validation_and_test_error
 -> if_large_use_vectorised_GPU_parallel_or_distributed_training
 -> verify_tensor_shapes_gradients_reference_outputs_and_loss_semantics
 -> document_architecture_data_seed_optimiser_precision_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-AI-COMPUTATIONAL-MATHEMATICS-NEURAL-NETWORK-MATHEMATICS-ONTOLOGY-2-V1"
}

}

END_FLOW