AI Computational Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the AI Computational Mathematics parent ontology within the Kireth Research Semantic Library. AI computational mathematics concerns the use of artificial intelligence, machine learning, symbolic reasoning, optimisation, numerical methods and learned representations to analyse, approximate, discover, accelerate or control mathematical computation. This ontology supports neural numerical methods, learned solvers, surrogate models, operator learning, differentiable programming, automatic differentiation, symbolic-regression relationships, theorem-reasoning relationships, scientific machine learning, physics-informed models, uncertainty, verification, robustness, computational complexity and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = AIComputationalMathematics
 ENTITY_CLASS = ArtificialIntelligenceMathematicalComputationFramework
 PRIMARY_CONTEXT = ComputationalMathematicsOntologies
 DEFINITION = "AI computational mathematics is the application of artificial intelligence and machine learning methods to mathematical modelling, numerical computation, symbolic reasoning, optimisation, scientific simulation and algorithm discovery."
 SYMBOLIC_FORMS = [
  "y_hat = model_theta(x)",
  "theta* = argmin_theta L(model_theta,data,constraints)",
  "u_hat = neural_operator(f)",
  "algorithm_hat = learned_computation(problem_state)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for AI-assisted mathematical computation, learned solvers, scientific machine learning, differentiable computation, surrogate modelling, operator learning, symbolic reasoning, verification and reproducibility."
}

SCOPE {
 INCLUDE = [
  AIComputationalMathematics,
  ArtificialIntelligence,
  MachineLearning,
  DeepLearning,
  ScientificMachineLearning,
  SciML,
  MathematicalMachineLearning,
  LearnedComputation,
  LearnedAlgorithm,
  AlgorithmLearning,
  NeuralComputation,
  NeuralNumericalMethod,
  LearnedNumericalMethod,
  LearnedSolver,
  NeuralSolver,
  SurrogateModel,
  ReducedOrderModelRelationship,
  Emulator,
  MetaModel,
  ApproximationModel,
  FunctionApproximator,
  UniversalApproximationRelationship,
  NeuralNetwork,
  FeedforwardNetwork,
  MultilayerPerceptron,
  MLP,
  RecurrentNeuralNetworkRelationship,
  TransformerRelationship,
  GraphNeuralNetwork,
  GNN,
  ConvolutionalNeuralNetworkRelationship,
  NeuralOperator,
  OperatorLearning,
  DeepONetRelationship,
  FourierNeuralOperator,
  FNO,
  GraphNeuralOperatorRelationship,
  PhysicsInformedNeuralNetwork,
  PINN,
  PhysicsInformedMachineLearning,
  PDELearning,
  ODELearning,
  DifferentialEquationLearning,
  DataDrivenDynamics,
  SystemIdentificationRelationship,
  KoopmanOperatorLearningRelationship,
  NeuralODE,
  NeuralPDErelationship,
  DifferentialProgramming,
  DifferentiableProgramming,
  AutomaticDifferentiation,
  ForwardModeAD,
  ReverseModeAD,
  ComputationalGraph,
  Gradient,
  Jacobian,
  Hessian,
  Backpropagation,
  AdjointMethodRelationship,
  SensitivityAnalysis,
  GradientBasedOptimisation,
  StochasticGradientDescent,
  AdamRelationship,
  QuasiNewtonRelationship,
  LearnedOptimiser,
  MetaOptimisation,
  HyperparameterOptimisation,
  BayesianOptimisationRelationship,
  ReinforcementLearningRelationship,
  ControlPolicyRelationship,
  AlgorithmSelection,
  SolverSelection,
  MethodRecommendation,
  AutoMLRelationship,
  SymbolicRegression,
  EquationDiscovery,
  SparseIdentificationRelationship,
  SINDyRelationship,
  MathematicalExpressionGenerationRelationship,
  ProgramSynthesisRelationship,
  AlgorithmSynthesis,
  AutomatedTheoremReasoningRelationship,
  AutomatedTheoremProvingRelationship,
  FormalReasoningRelationship,
  ProofSearchRelationship,
  NeuroSymbolicReasoning,
  SymbolicComputation,
  ComputerAlgebraSystem,
  CAS,
  ExpressionSimplification,
  SymbolicDifferentiation,
  SymbolicIntegration,
  SymbolicEquationSolving,
  NumericalMethods,
  NumericalLinearAlgebra,
  MatrixComputation,
  VectorComputation,
  TensorComputation,
  SparseDataStructure,
  DenseDataStructure,
  ComputationalGraphOntologyRelationship,
  GraphAlgorithms,
  NumericalDifferentiation,
  NumericalIntegration,
  Quadrature,
  DifferentialEquationSolver,
  ODESolver,
  PDESolver,
  EulerMethod,
  RungeKutta,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  BoundaryElementMethod,
  SpectralMethod,
  ComputationalOptimisation,
  LinearProgrammingSolver,
  NonlinearOptimisationSolver,
  GradientDescent,
  StochasticGradientDescent,
  NewtonOptimisation,
  SimulatedAnnealing,
  GeneticAlgorithmMathematics,
  EvolutionaryOptimisation,
  ComputationalGeometry,
  ScientificSimulation,
  PhysicalSimulation,
  ParticleSimulation,
  MolecularSimulation,
  FluidSimulation,
  HeatTransferSimulation,
  StructuralSimulation,
  AgentBasedSimulation,
  ComputationalStatistics,
  MonteCarloMethod,
  MarkovChainMonteCarlo,
  BootstrapMethod,
  RandomNumberGeneration,
  StatisticalSampling,
  BayesianComputation,
  TimeSeriesComputation,
  HighPerformanceComputing,
  ParallelComputing,
  DistributedComputing,
  GPUComputing,
  VectorProcessing,
  DataDrivenApproximation,
  InterpolationLearningRelationship,
  Regression,
  GaussianProcessRelationship,
  KernelMethodRelationship,
  BasisLearning,
  DictionaryLearningRelationship,
  ManifoldLearningRelationship,
  DimensionalityReduction,
  PrincipalComponentAnalysisRelationship,
  AutoencoderRelationship,
  LatentRepresentation,
  FeatureLearning,
  RepresentationLearning,
  Embedding,
  LearnedBasis,
  SparseRepresentationLearning,
  CompressionLearningRelationship,
  MeshLearningRelationship,
  AdaptiveMeshRelationship,
  ErrorEstimatorLearning,
  APosterioriErrorEstimationRelationship,
  ResidualLearning,
  CorrectionModel,
  PreconditionerLearning,
  LearnedPreconditioner,
  MultigridLearningRelationship,
  IterativeSolverAcceleration,
  KrylovAccelerationRelationship,
  MatrixInverseApproximationRelationship,
  LowRankApproximation,
  TensorDecompositionRelationship,
  UncertaintyQuantification,
  AleatoricUncertainty,
  EpistemicUncertainty,
  BayesianNeuralNetworkRelationship,
  EnsembleModel,
  Calibration,
  ConfidenceEstimate,
  OutOfDistributionDetectionRelationship,
  Robustness,
  AdversarialRobustnessRelationship,
  Stability,
  NumericalStability,
  Generalisation,
  Extrapolation,
  Interpolation,
  DistributionShift,
  DataQuality,
  TrainingData,
  ValidationData,
  TestData,
  SyntheticData,
  SimulationData,
  ExperimentalDataRelationship,
  SupervisedLearning,
  UnsupervisedLearning,
  SelfSupervisedLearningRelationship,
  ReinforcementLearning,
  OnlineLearning,
  ActiveLearningRelationship,
  TransferLearning,
  MultiTaskLearningRelationship,
  MetaLearning,
  FewShotLearningRelationship,
  ZeroShotRelationship,
  LossFunction,
  ObjectiveFunction,
  ConstraintLoss,
  PhysicsLoss,
  DataLoss,
  Regularisation,
  WeightDecayRelationship,
  SparsityRegularisation,
  L1Regularisation,
  L2Regularisation,
  ConstraintEnforcement,
  HardConstraint,
  SoftConstraint,
  BoundaryCondition,
  InitialCondition,
  ConservationLaw,
  SymmetryConstraint,
  Invariant,
  EquivarianceRelationship,
  PermutationInvarianceRelationship,
  TranslationEquivarianceRelationship,
  RotationalEquivarianceRelationship,
  DataAssimilationRelationship,
  InverseProblem,
  ParameterEstimation,
  SystemCalibration,
  ModelCalibration,
  ModelDiscovery,
  ModelSelection,
  SurrogateAssistedOptimisation,
  ExperimentalDesignRelationship,
  ActiveSubspaceRelationship,
  Verification,
  Validation,
  Benchmark,
  ReferenceSolution,
  KnownAnswerTest,
  ResidualCheck,
  ConstraintCheck,
  ConservationCheck,
  StabilityCheck,
  ErrorAnalysis,
  GeneralisationError,
  ApproximationError,
  DiscretisationError,
  OptimisationError,
  TrainingError,
  ValidationError,
  ComputationalComplexity,
  SampleComplexity,
  ModelComplexity,
  TimeComplexity,
  SpaceComplexity,
  TrainingCost,
  InferenceCost,
  MemoryCost,
  EnergyCostRelationship,
  Scalability,
  Reproducibility,
  Repeatability,
  Provenance,
  DatasetVersion,
  ModelVersion,
  TrainingConfiguration,
  OptimiserVersionRelationship,
  LibraryVersion,
  HardwareVersionRelationship,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  PureMachineLearningWithoutMathematicalComputationContext,
  TraditionalNumericalComputationWithoutAIOrLearnedComponent,
  GeneralArtificialIntelligenceWithoutMathematicalProblemSolving,
  SymbolicMathematicsWithoutAIOrLearningComponent
 ]
 DISAMBIGUATION_RULE = "When artificial intelligence or machine learning is used to perform, accelerate, approximate, discover, select or reason about mathematical computation, interpret the context as AI computational mathematics."
}

CORE_CONCEPTS {
 AIComputationalMathematics = "Use of AI and machine learning for mathematical computation."
 LearnedSolver = "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 operator representations."
 PhysicsInformedNeuralNetwork = "Neural model trained using physical or mathematical governing equations as constraints."
 DifferentiableProgramming = "Programming paradigm in which computations are differentiable and gradients are propagated automatically."
 SymbolicRegression = "Discovery of explicit mathematical expressions from data."
 Verification = "Assessment that AI-assisted mathematical outputs satisfy mathematical and computational requirements."
}

RELATIONS {
 AIComputationalMathematics --is_child_of--> ComputationalMathematicsOntologies
 AIComputationalMathematics --uses--> ArtificialIntelligence
 AIComputationalMathematics --uses--> MachineLearning
 LearnedSolver --approximates--> NumericalSolver
 SurrogateModel --approximates--> ExpensiveModel
 NeuralOperator --learns--> OperatorMapping
 PhysicsInformedNeuralNetwork --uses--> GoverningEquationConstraints
 DifferentiableProgramming --uses--> AutomaticDifferentiation
 AutomaticDifferentiation --produces--> Gradients
 SymbolicRegression --discovers--> MathematicalExpressions
 LearnedPreconditioner --accelerates--> IterativeSolver
 AIComputationalMathematics --requires--> Verification
 Reproducibility --requires--> DatasetModelAndTrainingMetadata
}

OPERATORS {
 TRAIN(model,data,loss) := optimise_parameters(model,data,loss)
 PREDICT(model,x) := model(x)
 LEARN_OPERATOR(theta,f) := neural_operator_theta(f)
 PHYSICS_LOSS(u) := residual_of_governing_equations(u)
 TOTAL_LOSS := data_loss + physics_loss + regularisation
 AD_GRADIENT(f,x) := automatic_derivative(f,x)
 SURROGATE_ERROR(y,y_hat) := measure(y,y_hat)
 VERIFY_CONSTRAINTS(output) := check_mathematical_constraints(output)
}

AXIOMS {
 A1 = "AI computational mathematics combines learned or AI-based mechanisms with mathematical computation."
 A2 = "A learned approximation must be distinguished from an exact mathematical algorithm."
 A3 = "Training error is distinct from mathematical solution error."
 A4 = "Physics or structural constraints can be imposed as hard or soft conditions."
 A5 = "Generalisation outside the training distribution cannot be assumed."
 A6 = "AI acceleration does not remove the need for numerical or symbolic verification."
 A7 = "Model, data and optimisation choices influence computational behaviour."
 A8 = "AI-computational-mathematics results require explicit model, data, loss, constraints, error, verification and reproducibility semantics."
}

REASONING {
 R1_CONTEXT = identify(mathematical_problem,traditional_method,AI_role,data_availability,accuracy_goal,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,invariants,symmetries_and_data_constraints)
 R6_TRAIN = choose(loss,optimiser,sampling_strategy,precision_and_hardware)
 R7_VALIDATE = evaluate(reference_solution,residuals,constraints,generalisation_and_uncertainty)
 R8_COMPARE = compare_with_traditional_numerical_or_symbolic_baseline)
 R9_SCALE = if_large use_GPU_parallel_distributed_or_HPC_execution)
 R10_REPRODUCE = document_data_model_training_configuration_versions_and_provenance)
}

CONSTRAINTS {
 C1 = "Low training loss does not guarantee mathematically correct solutions."
 C2 = "Learned solvers can fail outside their training distribution."
 C3 = "Physics-informed losses can be poorly balanced against data losses."
 C4 = "Neural operator accuracy depends on discretisation, training distribution and resolution handling."
 C5 = "Symbolic regression can return spurious expressions without validation."
 C6 = "Automatic differentiation computes derivatives of implemented computations, not necessarily of intended mathematical models."
 C7 = "Surrogate models can hide conservation or stability violations."
 C8 = "Extrapolation can be substantially less reliable than interpolation."
 C9 = "AI models can be computationally expensive to train even if inference is fast."
 C10 = "Benchmark superiority on one problem family does not establish 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 = "AI model architecture and representation are documented."
 V3 = "Training, validation and test data are documented."
 V4 = "Loss terms and mathematical constraints are documented."
 V5 = "Results are compared with trusted numerical, symbolic or analytical references."
 V6 = "Residuals and governing constraints are checked."
 V7 = "Generalisation and distribution-shift behaviour are assessed."
 V8 = "Uncertainty or confidence measures are documented where relevant."
 V9 = "Training and inference precision, hardware and software are documented."
 V10 = "Results are reproducible under documented data, model, seed, optimiser and implementation versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.98
 AI_COMPUTATIONAL_MATHEMATICS_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, reference baseline, constraints, uncertainty, extrapolation regime or validation methodology are underspecified."
}

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

EXECUTION_RULES {
 E1 = "If AI or machine learning is used to perform or assist mathematical computation, activate AIComputationalMathematics."
 E2 = "Define the mathematical problem and reference computational method before selecting an AI approach."
 E3 = "Distinguish learned approximation from exact numerical or symbolic computation."
 E4 = "Encode governing equations, constraints, symmetries or invariants explicitly when they are available."
 E5 = "Validate against analytical, numerical or symbolic reference solutions."
 E6 = "Assess residuals, constraint violations, uncertainty and out-of-distribution behaviour."
 E7 = "Use automatic differentiation only with clearly defined computational semantics."
 E8 = "Use GPU, parallel, distributed or HPC execution where training or inference scale requires it."
 E9 = "Document datasets, architectures, losses, seeds, optimiser settings and versions for reproducibility."
}

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

ROUTING {
 PARENT_ONTOLOGY = "Computational Mathematics Ontologies"
 RELATED_ONTOLOGIES = [
  "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 -> Symbolic Computation for neuro-symbolic and symbolic-regression relationships",
  "AI Computational Mathematics -> Computational Data Structures for tensor, matrix, graph and sparse representations",
  "AI Computational Mathematics -> Numerical Methods for learned numerical approximation",
  "AI Computational Mathematics -> Numerical Linear Algebra for learned solvers and preconditioners",
  "AI Computational Mathematics -> Differential Equation Solvers for PINN and neural differential-equation methods",
  "AI Computational Mathematics -> Optimisation Computation for model training and learned optimisation",
  "AI Computational Mathematics -> Computational Statistics for Bayesian and uncertainty-aware AI methods",
  "AI Computational Mathematics -> Scientific Simulation for surrogate and emulator relationships",
  "AI Computational Mathematics -> HPC, Parallel, Distributed and GPU Computing for scalable training and inference",
  "AI Computational Mathematics -> 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 machine-readable AI-computational-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "AI computational mathematics applies artificial intelligence and machine learning to mathematical computation, modelling, reasoning and algorithm design."
 SOLVER_SUMMARY = "Learned solvers, preconditioners and surrogate models can approximate or accelerate conventional numerical computation."
 SCIENTIFIC_ML_SUMMARY = "Scientific machine learning combines data with governing equations, physical constraints and mathematical structure."
 OPERATOR_SUMMARY = "Neural operators learn mappings between functions or discretised fields for repeated solution of related problems."
 SYMBOLIC_SUMMARY = "Symbolic regression and neuro-symbolic methods connect learned representations with explicit mathematical expressions."
 VALIDATION_SUMMARY = "AI-assisted mathematical computation should be checked against trusted references, residuals, constraints, uncertainty, distribution shift and reproducible model settings."
}

FLOW {
 INPUT
 -> detect_AI_computational_mathematics_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_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_and_distribution_shift
 -> if_large_use_GPU_parallel_distributed_or_HPC_execution
 -> document_data_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-COMPUTATIONAL-MATHEMATICS-AI-COMPUTATIONAL-MATHEMATICS-V1"
}

}

END_FLOW