AI Computational Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the AI Computational Mathematics Ontology within the Kireth Research Semantic Library. It formalises the use of artificial intelligence and machine learning as computational mechanisms for solving, approximating, accelerating, discovering or reasoning about mathematical problems. This ontology supports learned numerical methods, scientific machine learning, neural operators, physics-informed models, surrogate models, differentiable programming, automatic differentiation, symbolic-regression relationships, solver selection, learned optimisation, verification, uncertainty, robustness, scalability and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = AIComputationalMathematicsOntology
 ENTITY_CLASS = AIMathematicalComputationOntology
 PRIMARY_CONTEXT = AIComputationalMathematics
 DEFINITION = "The AI Computational Mathematics Ontology defines entities, relationships, methods, constraints and validation semantics for applying artificial intelligence and machine learning to mathematical computation."
 SYMBOLIC_FORMS = [
  "problem -> AI_model -> mathematical_output",
  "theta* = argmin_theta L_math(theta)",
  "u_hat = learned_solver(problem)",
  "verify(u_hat,reference,constraints)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for AI-assisted mathematical solving, approximation, discovery, optimisation, differentiation, scientific modelling, validation and reproducibility."
}

SCOPE {
 INCLUDE = [
  AIComputationalMathematicsOntology,
  AIComputationalMathematics,
  ArtificialIntelligence,
  MachineLearning,
  DeepLearning,
  ScientificMachineLearning,
  SciML,
  MathematicalMachineLearning,
  LearnedComputation,
  LearnedAlgorithm,
  LearnedNumericalMethod,
  NeuralNumericalMethod,
  LearnedSolver,
  NeuralSolver,
  SurrogateModel,
  Emulator,
  MetaModel,
  ReducedOrderModelRelationship,
  ApproximationModel,
  FunctionApproximator,
  NeuralNetwork,
  MultilayerPerceptron,MLP,
  GraphNeuralNetwork,GNN,
  TransformerRelationship,
  RecurrentNeuralNetworkRelationship,
  NeuralOperator,
  OperatorLearning,
  FourierNeuralOperator,FNO,
  DeepONetRelationship,
  GraphNeuralOperatorRelationship,
  PhysicsInformedNeuralNetwork,PINN,
  PhysicsInformedMachineLearning,
  NeuralODE,
  NeuralPDERelationship,
  PDELearning,
  ODELearning,
  DifferentialEquationLearning,
  DataDrivenDynamics,
  SystemIdentificationRelationship,
  KoopmanLearningRelationship,
  DifferentiableProgramming,
  DifferentialProgramming,
  AutomaticDifferentiation,
  ForwardModeAD,
  ReverseModeAD,
  ComputationalGraph,
  Gradient,
  Jacobian,
  Hessian,
  Backpropagation,
  AdjointMethodRelationship,
  SensitivityAnalysis,
  Optimisation,
  GradientBasedOptimisation,
  GradientDescent,
  StochasticGradientDescent,
  AdamRelationship,
  NewtonOptimisationRelationship,
  QuasiNewtonRelationship,
  LearnedOptimiser,
  MetaOptimisation,
  HyperparameterOptimisation,
  BayesianOptimisationRelationship,
  SolverSelection,
  AlgorithmSelection,
  MethodRecommendation,
  AutoMLRelationship,
  SymbolicRegression,
  EquationDiscovery,
  SparseIdentificationRelationship,
  SINDyRelationship,
  ProgramSynthesisRelationship,
  AlgorithmSynthesis,
  AutomatedTheoremReasoningRelationship,
  AutomatedTheoremProvingRelationship,
  NeuroSymbolicReasoning,
  SymbolicComputation,
  SymbolicMathematics,
  SymbolicAlgebra,
  SymbolicDifferentiation,
  SymbolicIntegration,
  SymbolicEquationSolver,
  ComputerAlgebraSystem,
  NumericalMethods,
  NumericalLinearAlgebra,
  MatrixComputation,
  VectorComputation,
  TensorComputation,
  SparseDataStructure,
  DenseDataStructure,
  ComputationalGraph,
  GraphAlgorithms,
  DifferentialEquationSolver,
  OrdinaryDifferentialEquationSolver,
  PartialDifferentialEquationSolver,
  EulerMethod,
  RungeKutta,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  BoundaryElementMethod,
  SpectralMethod,
  ComputationalOptimisation,
  LinearProgrammingSolver,
  NonlinearOptimisationSolver,
  SimulatedAnnealing,
  GeneticAlgorithmMathematics,
  EvolutionaryOptimisation,
  ComputationalStatistics,
  MonteCarloMethod,
  MarkovChainMonteCarlo,
  BootstrapMethod,
  RandomNumberGeneration,
  StatisticalSampling,
  BayesianComputation,
  TimeSeriesComputation,
  ScientificSimulation,
  PhysicalSimulation,
  ParticleSimulation,
  MolecularSimulation,
  FluidSimulation,
  HeatTransferSimulation,
  StructuralSimulation,
  AgentBasedSimulation,
  HighPerformanceComputing,
  ParallelComputing,
  DistributedComputing,
  GPUComputing,
  VectorProcessing,
  DataDrivenApproximation,
  Regression,
  GaussianProcessRelationship,
  KernelMethodRelationship,
  RepresentationLearning,
  FeatureLearning,
  LatentRepresentation,
  BasisLearning,
  DictionaryLearningRelationship,
  DimensionalityReduction,
  AutoencoderRelationship,
  LowRankApproximation,
  TensorDecompositionRelationship,
  LearnedBasis,
  LearnedPreconditioner,
  PreconditionerLearning,
  MultigridLearningRelationship,
  IterativeSolverAcceleration,
  ErrorEstimatorLearning,
  ResidualLearning,
  CorrectionModel,
  UncertaintyQuantification,
  AleatoricUncertainty,
  EpistemicUncertainty,
  BayesianNeuralNetworkRelationship,
  EnsembleModel,
  Calibration,
  ConfidenceEstimate,
  OutOfDistributionDetectionRelationship,
  Robustness,
  Stability,
  NumericalStability,
  Generalisation,
  Extrapolation,
  Interpolation,
  DistributionShift,
  TrainingData,
  ValidationData,
  TestData,
  SyntheticData,
  SimulationData,
  SupervisedLearning,
  UnsupervisedLearning,
  SelfSupervisedLearningRelationship,
  ReinforcementLearning,
  OnlineLearning,
  ActiveLearningRelationship,
  TransferLearning,
  MultiTaskLearningRelationship,
  MetaLearning,
  LossFunction,
  ObjectiveFunction,
  DataLoss,
  PhysicsLoss,
  ConstraintLoss,
  Regularisation,
  L1Regularisation,
  L2Regularisation,
  BoundaryCondition,
  InitialCondition,
  ConservationLaw,
  SymmetryConstraint,
  Invariant,
  EquivarianceRelationship,
  InverseProblem,
  ParameterEstimation,
  ModelCalibration,
  ModelDiscovery,
  ModelSelection,
  Verification,
  Validation,
  Benchmark,
  ReferenceSolution,
  KnownAnswerTest,
  ResidualCheck,
  ConstraintCheck,
  ConservationCheck,
  StabilityCheck,
  ErrorAnalysis,
  ApproximationError,
  GeneralisationError,
  TrainingError,
  ValidationError,
  ComputationalComplexity,
  SampleComplexity,
  ModelComplexity,
  TrainingCost,
  InferenceCost,
  TimeComplexity,
  SpaceComplexity,
  Reproducibility,
  Repeatability,
  Provenance,
  DatasetVersion,
  ModelVersion,
  TrainingConfiguration,
  LibraryVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  GenericAIWithoutMathematicalComputationContext,
  PureNumericalMethodWithoutLearnedOrAIComponent,
  PureSymbolicComputationWithoutAIOrLearningComponent,
  GeneralMachineLearningApplicationWithoutMathematicalProblemSemantics
 ]
 DISAMBIGUATION_RULE = "When AI or machine learning directly participates in solving, approximating, accelerating, discovering or validating mathematical computation, interpret the context as AI computational mathematics."
}

CORE_CONCEPTS {
 AIComputationalMathematicsOntology = "Ontology describing AI-assisted mathematical computation."
 LearnedSolver = "AI model trained to approximate or accelerate a mathematical solution process."
 SurrogateModel = "Learned approximation replacing or supplementing an expensive mathematical model."
 NeuralOperator = "Learned mapping between function spaces or discretised operators."
 PhysicsInformedNeuralNetwork = "Neural model constrained by governing mathematical or physical equations."
 DifferentiableProgramming = "Programming in which gradients are propagated through computations."
 SymbolicRegression = "Discovery of explicit symbolic mathematical expressions from data."
 Verification = "Mathematical and computational checking of AI-produced outputs."
}

RELATIONS {
 AIComputationalMathematicsOntology --is_child_of--> AIComputationalMathematics
 AIComputationalMathematicsOntology --models--> AIMathematicalComputation
 LearnedSolver --approximates--> NumericalSolver
 SurrogateModel --approximates--> ExpensiveMathematicalModel
 NeuralOperator --learns--> OperatorMapping
 PhysicsInformedNeuralNetwork --uses--> GoverningEquationConstraints
 DifferentiableProgramming --uses--> AutomaticDifferentiation
 SymbolicRegression --discovers--> MathematicalExpression
 LearnedPreconditioner --accelerates--> IterativeSolver
 SolverSelection --chooses--> ComputationalMethod
 Verification --checks--> AIProducedMathematicalOutput
 Reproducibility --requires--> DataModelTrainingAndVersionMetadata
}

OPERATORS {
 TRAIN(model,data,loss) := optimise_model_parameters(model,data,loss)
 PREDICT(model,input) := model(input)
 LEARN_SOLVER(problem_class,data) := train_solver_model(problem_class,data)
 LEARN_OPERATOR(theta,f) := neural_operator_theta(f)
 PHYSICS_RESIDUAL(u) := governing_equation_residual(u)
 TOTAL_LOSS := data_loss + physics_loss + constraint_loss + regularisation
 AD_GRADIENT(f,x) := automatic_derivative(f,x)
 VERIFY(output,reference,constraints) := mathematical_validation(output,reference,constraints)
}

AXIOMS {
 A1 = "AI computational mathematics combines mathematical computation with AI or machine learning."
 A2 = "Learned approximations are distinct from exact analytical or deterministic numerical algorithms."
 A3 = "Training loss is distinct from mathematical solution error."
 A4 = "Mathematical constraints can be encoded as hard constraints, soft losses or architectural structure."
 A5 = "Generalisation beyond training support cannot be assumed."
 A6 = "AI acceleration does not remove the requirement for mathematical verification."
 A7 = "Model architecture, data and optimisation procedure affect mathematical behaviour."
 A8 = "AI computational mathematics requires explicit model, data, objective, constraints, error and reproducibility semantics."
}

REASONING {
 R1_CONTEXT = identify(mathematical_problem,reference_method,AI_role,data,accuracy_goal,computational_budget)
 R2_TASK = classify(approximation,solver_acceleration,operator_learning,surrogate_modelling,symbolic_discovery,algorithm_selection_or_reasoning)
 R3_REPRESENTATION = choose(vector,matrix,tensor,graph,function,mesh_or_symbolic_representation)
 R4_MODEL = choose(neural_network,operator_model,GNN,surrogate,symbolic_regression_or_hybrid_model)
 R5_CONSTRAINTS = define(governing_equations,boundary_conditions,initial_conditions,invariants,symmetries_and_data_constraints)
 R6_TRAINING = configure(loss,optimiser,sampling_strategy,precision,seed_and_hardware)
 R7_EXECUTION = train_or_execute_AI_model)
 R8_VALIDATION = compare_with_reference_solution_residuals_constraints_generalisation_and_uncertainty)
 R9_SCALE = if_large use_GPU_parallel_distributed_or_HPC_execution)
 R10_REPRODUCE = document_dataset_model_training_configuration_versions_and_provenance)
}

CONSTRAINTS {
 C1 = "Low training error does not guarantee mathematically correct solutions."
 C2 = "Learned solvers can fail under distribution shift."
 C3 = "Physics-informed objectives can suffer from imbalanced loss terms."
 C4 = "Neural operator accuracy can depend strongly on discretisation and resolution."
 C5 = "Symbolic regression can produce spurious or overfitted expressions."
 C6 = "Automatic differentiation differentiates implemented computation rather than intended mathematics."
 C7 = "Surrogate models can violate conservation or invariants unless constrained."
 C8 = "Extrapolation is generally less reliable than interpolation."
 C9 = "Training cost can exceed the cost of the baseline method for limited-use problems."
 C10 = "Performance on a benchmark does not imply universal solver superiority."
}

SEMANTIC_EQUIVALENCE {
 "ScientificMachineLearning" <=> "SciML"
 "PhysicsInformedNeuralNetwork" <=> "PINN"
 "FourierNeuralOperator" <=> "FNO"
 "GraphNeuralNetwork" <=> "GNN"
 "MultilayerPerceptron" <=> "MLP"
 "ComputerAlgebraSystem" <=> "CAS"
}

VALIDATION {
 V1 = "Mathematical problem and target quantity are documented."
 V2 = "Reference numerical, symbolic or analytical method is documented."
 V3 = "Model architecture and representation are documented."
 V4 = "Training, validation and test data are documented."
 V5 = "Loss components and mathematical constraints are documented."
 V6 = "Outputs are checked against references, residuals and constraints."
 V7 = "Generalisation and distribution-shift behaviour are assessed."
 V8 = "Uncertainty or confidence is assessed where relevant."
 V9 = "Training and inference hardware, precision and software versions are documented."
 V10 = "Results are reproducible under documented data, seed, model, optimiser and implementation settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.98
 AI_COMPUTATIONAL_MATHEMATICS_ONTOLOGY_DEFINITION = 0.99
 LEARNED_SOLVER_REASONING = 0.98
 SCIENTIFIC_ML_REASONING = 0.98
 PINN_REASONING = 0.98
 OPERATOR_LEARNING_REASONING = 0.98
 SYMBOLIC_REGRESSION_REASONING = 0.97
 GENERALISATION_REASONING = 0.96
 POLICY = "Reduce confidence when training distribution, mathematical constraints, baseline comparison, uncertainty, extrapolation regime or validation method are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,AIComputationalMathematicsOntologyReasoning,
  LearnedSolverReasoning,ScientificMachineLearningReasoning,
  SurrogateModelReasoning,NeuralOperatorReasoning,
  PINNReasoning,DifferentiableProgrammingReasoning,
  AutomaticDifferentiationReasoning,
  SymbolicRegressionReasoning,EquationDiscoveryReasoning,
  SolverSelectionReasoning,LearnedOptimisationReasoning,
  LearnedPreconditionerReasoning,UncertaintyReasoning,
  GeneralisationReasoning,VerificationReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "AI computational mathematics ontology" -> AIComputationalMathematicsOntology
  "AI computational mathematics" -> AIComputationalMathematics
  "SciML" -> ScientificMachineLearning
  "PINN" -> PhysicsInformedNeuralNetwork
  "FNO" -> FourierNeuralOperator
  "GNN" -> GraphNeuralNetwork
  "MLP" -> MultilayerPerceptron
 }
}

EXECUTION_RULES {
 E1 = "If AI or machine learning directly participates in mathematical computation, activate AIComputationalMathematicsOntology."
 E2 = "Define the mathematical problem and trusted reference method before selecting an AI method."
 E3 = "Distinguish learned approximation from exact numerical or symbolic computation."
 E4 = "Encode known governing equations, constraints, symmetries and invariants explicitly where possible."
 E5 = "Validate AI outputs against analytical, numerical or symbolic reference solutions."
 E6 = "Assess residuals, constraint violations, uncertainty, robustness and distribution shift."
 E7 = "Use automatic differentiation only with explicit computational-graph semantics."
 E8 = "Use GPU, parallel, distributed or HPC execution where scale requires it."
 E9 = "Document datasets, architectures, losses, seeds, optimiser settings and versions for reproducibility."
}

RECURSION {
 ENTRY = AIComputationalMathematicsOntology
 EXPAND(AIComputationalMathematicsOntology) -> [
  AIComputationalMathematics,ScientificMachineLearning,SciML,
  LearnedComputation,LearnedSolver,NeuralSolver,SurrogateModel,
  NeuralNetwork,MLP,GNN,NeuralOperator,OperatorLearning,
  FourierNeuralOperator,FNO,PhysicsInformedNeuralNetwork,PINN,
  NeuralODE,DifferentiableProgramming,AutomaticDifferentiation,
  ForwardModeAD,ReverseModeAD,Gradient,Jacobian,Hessian,
  SolverSelection,AlgorithmSelection,LearnedOptimiser,
  SymbolicRegression,EquationDiscovery,NeuroSymbolicReasoning,
  SymbolicComputation,NumericalMethods,NumericalLinearAlgebra,
  DifferentialEquationSolver,ComputationalOptimisation,
  ComputationalStatistics,ScientificSimulation,
  LearnedPreconditioner,IterativeSolverAcceleration,
  UncertaintyQuantification,Robustness,Generalisation,DistributionShift,
  PhysicsLoss,ConstraintLoss,BoundaryCondition,InitialCondition,
  ConservationLaw,SymmetryConstraint,InverseProblem,
  ParameterEstimation,ModelDiscovery,Verification,Validation,
  Benchmark,ResidualCheck,ComputationalComplexity,
  TrainingCost,InferenceCost,HighPerformanceComputing,
  ParallelComputing,DistributedComputing,GPUComputing,
  Reproducibility,Provenance
 ]
 EXPAND(AIComputationalMathematicsFamilies) -> [
  LearnedNumericalMethods,
  ScientificMachineLearning,
  PhysicsInformedModels,
  NeuralOperatorMethods,
  DifferentiableProgrammingAndAutomaticDifferentiation,
  SymbolicRegressionAndEquationDiscovery,
  AISolverSelectionAndOptimisation,
  VerificationUncertaintyAndScalableAIComputation
 ]
 STOP_CONDITION = "Stop when mathematical problem, AI role, representation, model, data, constraints, validation, uncertainty and reproducibility are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "AI Computational Mathematics"
 RELATED_ONTOLOGIES = [
  "AI Computational Mathematics",
  "Symbolic Computation",
  "Computational Data Structures",
  "Numerical Methods Ontology 2",
  "Numerical Linear Algebra Ontology 2",
  "Differential Equation Solvers",
  "Optimisation Computation",
  "Computational Statistics",
  "Scientific Simulation",
  "High Performance Computing Ontology 2",
  "Parallel Computing Ontology",
  "Distributed Computing Ontology",
  "GPU Computing Ontology",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "AI Computational Mathematics Ontology -> AI Computational Mathematics for parent semantics",
  "AI Computational Mathematics Ontology -> Symbolic Computation for neuro-symbolic and symbolic-regression relationships",
  "AI Computational Mathematics Ontology -> Computational Data Structures for tensor, matrix, graph and sparse representations",
  "AI Computational Mathematics Ontology -> Numerical Methods for learned numerical approximation",
  "AI Computational Mathematics Ontology -> Numerical Linear Algebra for learned solvers and preconditioners",
  "AI Computational Mathematics Ontology -> Differential Equation Solvers for PINN and neural differential-equation methods",
  "AI Computational Mathematics Ontology -> Optimisation Computation for training and learned optimisation",
  "AI Computational Mathematics Ontology -> Computational Statistics for uncertainty-aware AI methods",
  "AI Computational Mathematics Ontology -> Scientific Simulation for surrogate and emulator relationships",
  "AI Computational Mathematics Ontology -> HPC, Parallel, Distributed and GPU Computing for scalable training and inference",
  "AI Computational Mathematics Ontology -> Computational Complexity for training, inference and sample-cost semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible AI mathematical methods, model families or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve learned-approximation-versus-exact-algorithm distinction.",
  "Preserve training-error-versus-mathematical-solution-error distinction.",
  "Preserve interpolation-versus-extrapolation distinction.",
  "Preserve data-driven-versus-physics-constrained distinction.",
  "Preserve model-confidence-versus-mathematical-verification distinction.",
  "Preserve training-cost-versus-inference-cost distinction.",
  "Preserve parent-topic-versus-child-ontology distinction.",
  "Preserve machine-readable AI-computational-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "The AI Computational Mathematics Ontology formalises AI-assisted mathematical solving, approximation, discovery, optimisation and validation."
 SOLVER_SUMMARY = "Learned solvers, preconditioners and surrogate models can approximate or accelerate traditional mathematical computation."
 SCIENTIFIC_ML_SUMMARY = "Scientific machine learning combines training data with governing equations, boundary conditions, conservation laws and mathematical structure."
 OPERATOR_SUMMARY = "Neural operators learn mappings between functions or discretised fields for repeated families of mathematical problems."
 SYMBOLIC_SUMMARY = "Symbolic regression and neuro-symbolic methods connect learned representations with explicit mathematical expressions and reasoning."
 VALIDATION_SUMMARY = "AI mathematical outputs should be checked against trusted references, residuals, constraints, uncertainty, robustness, distribution shift and reproducible training settings."
}

FLOW {
 INPUT
 -> detect_AI_computational_mathematics_ontology_context
 -> identify_mathematical_problem_reference_method_AI_role_data_accuracy_goal_and_budget
 -> classify_approximation_solver_acceleration_operator_learning_surrogate_symbolic_discovery_algorithm_selection_or_reasoning_task
 -> choose_vector_matrix_tensor_graph_function_mesh_or_symbolic_representation
 -> choose_neural_network_operator_model_GNN_surrogate_symbolic_regression_or_hybrid_method
 -> define_data_loss_physics_loss_constraints_boundary_conditions_initial_conditions_invariants_and_regularisation
 -> configure_training_sampling_optimiser_precision_seed_and_hardware
 -> train_or_execute_AI_mathematical_model
 -> compare_with_analytical_numerical_symbolic_or_reference_baseline
 -> evaluate_residuals_constraints_uncertainty_generalisation_robustness_and_distribution_shift
 -> if_large_use_GPU_parallel_distributed_or_HPC_execution
 -> document_dataset_model_training_configuration_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-ONTOLOGY-V1"
}

}

END_FLOW