Mathematical Modelling Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Mathematical Modelling Ontology within the Kireth Research Semantic Library. It represents mathematical modelling as a structured ontology of systems, purposes, assumptions, variables, parameters, governing relations, constraints, conditions, calibration, verification, validation, sensitivity, uncertainty and model reduction. It sits beneath the Mathematical Modelling parent layer and provides a machine-readable reasoning framework for constructing, classifying, analysing and evaluating mathematical models.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Mathematical Modelling Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Modelling / Core Modelling Ontology"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-modelling/mathematical-modelling-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = MathematicalModellingOntology
 ENTITY_CLASS = FormalModelRepresentationOntology
 PRIMARY_CONTEXT = MathematicalModelling
 DEFINITION = "The Mathematical Modelling Ontology is a formal semantic framework for representing the components, assumptions, structures, methods and evaluation processes of mathematical models."
 SEMANTIC_ROLE = "Provides a machine-readable ontology for identifying model purpose, system boundaries, variables, parameters, equations, conditions, constraints, uncertainty, calibration, verification, validation and interpretation."
}

SCOPE {
 INCLUDE = [
  MathematicalModellingOntology,
  MathematicalModel,
  ModelPurpose,
  ModelScope,
  System,
  Subsystem,
  Environment,
  Assumption,
  SimplifyingAssumption,
  StructuralAssumption,
  ModelVariable,
  StateVariable,
  InputVariable,
  OutputVariable,
  LatentVariable,
  Parameter,
  Constant,
  GoverningEquation,
  AlgebraicEquation,
  DifferentialEquation,
  IntegralEquation,
  DifferenceEquation,
  Constraint,
  InitialCondition,
  BoundaryCondition,
  ConstitutiveRelation,
  ConservationLaw,
  ModelClass,
  DeterministicModel,
  StochasticModel,
  StaticModel,
  DynamicModel,
  ContinuousModel,
  DiscreteModel,
  LinearModel,
  NonlinearModel,
  Calibration,
  Verification,
  Validation,
  SensitivityAnalysis,
  UncertaintyQuantification,
  Identifiability,
  ModelReduction,
  Simulation
 ]

 EXCLUDE_AS_PRIMARY = [
  PureMathematicalObjectWithoutModelPurpose,
  EmpiricalDatasetWithoutModelStructure,
  NumericalSolverWithoutModelSemantics,
  ScientificNarrativeWithoutFormalVariablesOrRelations
 ]

 DISAMBIGUATION_RULE =
  "When a mathematical representation is being described in terms of purpose, assumptions, variables, parameters, governing equations, conditions, calibration or validation, interpret the context through the Mathematical Modelling Ontology."
}

CORE_CONCEPTS {

 MathematicalModellingOntology {
  definition = "The ontology organising the semantic components and reasoning relations of mathematical models."
 }

 MathematicalModel {
  definition = "A purpose-driven mathematical representation of selected aspects of a system."
 }

 ModelPurpose {
  definition = "The intended descriptive, explanatory, predictive, control, estimation or optimisation use of the model."
 }

 ModelScope {
  definition = "The declared system boundaries, processes, scales and regimes represented by the model."
 }

 System {
  definition = "The phenomenon, process or abstract structure represented by the model."
 }

 Assumption {
  definition = "A declared premise constraining model interpretation."
 }

 ModelVariable {
  definition = "A quantity whose value may vary within the model."
 }

 StateVariable {
  definition = "A variable forming part of the model state."
 }

 Parameter {
  definition = "A quantity controlling model behaviour but treated as fixed during an individual model run."
 }

 GoverningEquation {
  definition = "A mathematical relation encoding the principal behaviour of the model."
 }

 Constraint {
  definition = "A condition restricting allowable states, variables, parameters or solutions."
 }

 Calibration {
  definition = "The process of estimating model parameters from data."
 }

 Verification {
  definition = "Assessment of whether the intended mathematics is implemented and solved correctly."
 }

 Validation {
  definition = "Assessment of whether the model is adequate for its stated purpose."
 }

 SensitivityAnalysis {
  definition = "Analysis of the dependence of model outputs on inputs, parameters or assumptions."
 }

 UncertaintyQuantification {
  definition = "Representation and propagation of uncertainty through the model."
 }

 Identifiability {
  definition = "The extent to which model parameters or structures can be uniquely inferred from available information."
 }
}

RELATIONS {
 MathematicalModellingOntology --is_child_of--> MathematicalModelling
 MathematicalModel --represents--> System
 MathematicalModel --has_purpose--> ModelPurpose
 MathematicalModel --has_scope--> ModelScope
 MathematicalModel --depends_on--> Assumption
 MathematicalModel --contains--> ModelVariable
 MathematicalModel --contains--> Parameter
 MathematicalModel --is_governed_by--> GoverningEquation
 MathematicalModel --is_restricted_by--> Constraint
 DynamicModel --may_require--> InitialCondition
 SpatialModel --may_require--> BoundaryCondition
 GoverningEquation --may_encode--> ConservationLaw
 GoverningEquation --may_use--> ConstitutiveRelation
 Calibration --estimates--> Parameter
 Verification --checks--> MathematicalImplementation
 Validation --evaluates--> FitnessForPurpose
 SensitivityAnalysis --measures--> OutputDependence
 UncertaintyQuantification --propagates--> InputAndParameterUncertainty
 Identifiability --constrains--> ParameterEstimation
 ModelReduction --approximates--> HigherComplexityModel
 Simulation --executes--> MathematicalModel
}

OPERATORS {
 DEFINE_MODEL_PURPOSE(system,goal) :=
  formal_statement_of_intended_model_use

 DEFINE_SCOPE(system,boundaries,processes,scales) :=
  model_scope_description

 DEFINE_VARIABLE(name,domain,unit,role) :=
  typed_model_variable

 DEFINE_PARAMETER(name,value_or_distribution,unit) :=
  typed_model_parameter

 FORM_GOVERNING_RELATION(variables,parameters,laws) :=
  algebraic_differential_integral_or_discrete equation

 CLASSIFY_MODEL(model) :=
  [
   deterministic_or_stochastic,
   static_or_dynamic,
   continuous_or_discrete,
   linear_or_nonlinear
  ]

 NONDIMENSIONALISE(model,scales) :=
  dimensionless_equivalent_model

 CALIBRATE(model,data,criterion) :=
  estimated_parameter_set

 VERIFY_IMPLEMENTATION(model,solver,test_cases) :=
  implementation_correctness_result

 VALIDATE_MODEL(model,reference_data,criteria) :=
  fitness_for_purpose_result

 SENSITIVITY(output,input) :=
  local_or_global_input_output dependence

 PROPAGATE_UNCERTAINTY(model,input_uncertainty) :=
  output_uncertainty_description

 ASSESS_IDENTIFIABILITY(model,observations) :=
  structural_or_practical_identifiability status
}

DEFINITIONS {
 D1 = "A mathematical model is always interpreted relative to purpose and scope."
 D2 = "Model variables and parameters have distinct semantic roles."
 D3 = "Assumptions define the validity regime of the model."
 D4 = "Governing equations encode the formal model structure."
 D5 = "Constraints restrict the admissible solution space."
 D6 = "Initial and boundary conditions complete many dynamic or spatial models."
 D7 = "Calibration estimates parameters from data."
 D8 = "Verification checks implementation correctness."
 D9 = "Validation checks adequacy for intended use."
 D10 = "Identifiability determines whether parameters can be uniquely inferred."
}

AXIOMS {
 A1_PURPOSE_DEPENDENCE:
  Model adequacy is defined relative to an intended purpose

 A2_SCOPE_DEPENDENCE:
  Model interpretation is valid only within its declared scope and regime

 A3_ASSUMPTION_DEPENDENCE:
  Model conclusions are conditional on assumptions

 A4_ROLE_DISTINCTION:
  Variables parameters constants and observations have distinct semantic roles

 A5_DIMENSIONAL_CONSISTENCY:
  Dimensioned governing equations must be dimensionally consistent

 A6_VERIFICATION_VALIDATION_DISTINCTION:
  Correct implementation does not imply empirical validity

 A7_CALIBRATION_VALIDATION_DISTINCTION:
  Parameter fitting does not itself constitute independent validation

 A8_UNCERTAINTY_PRESERVATION:
  Material uncertainty must be retained through model interpretation
}

REASONING {

 R1_MODEL_ENTITY_EXTRACTION {
  INPUT = model_description
  IDENTIFY = purpose_scope_system_assumptions_variables_parameters_equations_conditions_constraints
  RETURN = structured_model_entities
 }

 R2_MODEL_CLASSIFICATION {
  INPUT = structured_model
  TEST = deterministic_vs_stochastic
  TEST = static_vs_dynamic
  TEST = continuous_vs_discrete
  TEST = linear_vs_nonlinear
  RETURN = model_classification
 }

 R3_COMPLETENESS_CHECK {
  INPUT = model_classification
  IF dynamic
   CHECK = initial_conditions
  IF spatial_boundary_value
   CHECK = boundary_conditions
  CHECK = units_constraints_and_parameter_domains
  RETURN = completeness_status
 }

 R4_DIMENSIONAL_VALIDATION {
  INPUT = governing_equations
  COMPUTE = dimensions_of_terms
  VERIFY = consistency_across_equalities_and_sums
  RETURN = valid_or_invalid
 }

 R5_CALIBRATION_REASONING {
  INPUT = model_data_parameters
  DEFINE = fitting_criterion
  ESTIMATE = parameters
  CHECK = identifiability
  RETURN = calibrated_parameter_set_with_limitations
 }

 R6_VERIFICATION_REASONING {
  INPUT = model_implementation
  COMPARE = against analytical_limits_manufactured_solutions_or_reference_cases
  RETURN = verification_status
 }

 R7_VALIDATION_REASONING {
  INPUT = model_predictions_and_reference_data
  DEFINE = validation_metrics
  ASSESS = agreement_relative_to_intended_use
  RETURN = validation_status
 }

 R8_SENSITIVITY_REASONING {
  INPUT = model_input_output_relation
  COMPUTE = local_derivative_or_global_influence_measure
  RETURN = sensitivity_profile
 }

 R9_UNCERTAINTY_REASONING {
  INPUT = uncertain_inputs_parameters_and_structure
  PROPAGATE = uncertainty_through_model
  RETURN = output_uncertainty_and_confidence context
 }

 R10_MODEL_REDUCTION {
  INPUT = full_model_and_preservation_goal
  IDENTIFY = removable_states_or_modes
  CONSTRUCT = reduced_model
  VERIFY = preservation_of_target_behaviour
  RETURN = reduced_model
 }
}

CONSTRAINTS {
 C1 = "Purpose must be specified before adequacy is judged."
 C2 = "Scope and regime limits must be explicit."
 C3 = "Assumptions must not be hidden."
 C4 = "Variables and parameters must not be conflated."
 C5 = "Calibration fit quality does not establish causal correctness."
 C6 = "Parameter identifiability must be assessed when fitting."
 C7 = "Validation should use evidence relevant to intended use."
 C8 = "Numerical error, parameter uncertainty and model-form uncertainty are distinct."
 C9 = "Model reduction may invalidate behaviours outside its preservation target."
 C10 = "Extrapolation beyond validated regimes requires reduced confidence."
}

SEMANTIC_EQUIVALENCE {
 "MathematicalModellingOntology" <=> "FormalModelSemanticsOntology"
 "MathematicalModel" <=> "PurposeDrivenMathematicalRepresentation"
 "Calibration" <=> "ParameterEstimationFromData"
 "Verification" <=> "ImplementationCorrectnessAssessment"
 "Validation" <=> "FitnessForPurposeAssessment"
 "Identifiability" <=> "ParameterUniquenessRecoverability"
}

VALIDATION {
 V1 = "Model purpose is identified."
 V2 = "Model scope is identified."
 V3 = "Assumptions are explicit."
 V4 = "Variable and parameter roles are defined."
 V5 = "Governing equations are dimensionally consistent."
 V6 = "Initial and boundary conditions are present when required."
 V7 = "Calibration and validation are distinguished."
 V8 = "Identifiability is considered before strong parameter claims."
 V9 = "Uncertainty sources are represented."
 V10 = "Predictions remain within supported model regime or are flagged as extrapolations."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MODELLING_ONTOLOGY_DEFINITION = 0.99
 ENTITY_EXTRACTION_REASONING = 0.99
 MODEL_CLASSIFICATION_REASONING = 0.99
 COMPLETENESS_REASONING = 0.98
 CALIBRATION_REASONING = 0.97
 VERIFICATION_REASONING = 0.98
 VALIDATION_REASONING = 0.97
 IDENTIFIABILITY_REASONING = 0.96
 UNCERTAINTY_REASONING = 0.97

 POLICY =
  "Reduce confidence when model purpose, scope, assumptions, parameter identifiability, data quality, conditions, uncertainty structure or validation domain are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MathematicalModellingOntologyReasoning,
  ModelEntityExtraction,
  ModelClassification,
  ModelCompletenessChecking,
  AssumptionReasoning,
  VariableParameterReasoning,
  GoverningEquationReasoning,
  DimensionalValidation,
  CalibrationReasoning,
  VerificationReasoning,
  ValidationReasoning,
  IdentifiabilityReasoning,
  SensitivityAnalysis,
  UncertaintyQuantification,
  ModelReduction,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "mathematical modelling ontology" -> MathematicalModellingOntology
  "mathematical modeling ontology" -> MathematicalModellingOntology
  "mathematical model" -> MathematicalModel
  "model purpose" -> ModelPurpose
  "model scope" -> ModelScope
  "model variable" -> ModelVariable
  "state variable" -> StateVariable
  "parameter" -> Parameter WHEN modelling_context
  "governing equation" -> GoverningEquation
  "calibration" -> Calibration
  "verification" -> Verification
  "validation" -> Validation
  "identifiability" -> Identifiability
 }

 INTERPRETATION_ORDER = [
  detect_mathematical_modelling_ontology_context,
  extract purpose_scope_system_assumptions_variables_parameters_equations_conditions_constraints_and_data,
  classify model_structure,
  test model_completeness_and_dimensional_consistency,
  identify calibration_verification_validation_sensitivity_uncertainty_or_reduction goal,
  apply model_semantic reasoning,
  verify scope_identifiability_validation_and_uncertainty constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("mathematical modelling ontology")
  OR formal_model_semantics_context
  THEN activate(MathematicalModellingOntology)

 E2:
  IF model_description_given
  THEN extract typed entities_and_relations

 E3:
  IF dynamic_model
  THEN require or query initial_conditions when solution depends_on_them

 E4:
  IF spatial_boundary_value_model
  THEN require boundary_conditions

 E5:
  IF calibration_requested
  THEN assess parameter_identifiability

 E6:
  IF implementation_correctness_requested
  THEN route to verification

 E7:
  IF real_system_adequacy_requested
  THEN route to validation

 E8:
  IF uncertain_inputs_or_parameters
  THEN preserve and propagate uncertainty

 E9:
  IF prediction_outside_supported_scope
  THEN mark as extrapolation_and_reduce confidence
}

RECURSION {
 ENTRY = MathematicalModellingOntology

 EXPAND(MathematicalModellingOntology) -> [
  MathematicalModel,
  ModelPurpose,
  ModelScope,
  System,
  Subsystem,
  Environment,
  Assumption,
  ModelVariable,
  StateVariable,
  InputVariable,
  OutputVariable,
  LatentVariable,
  Parameter,
  Constant,
  GoverningEquation,
  InitialCondition,
  BoundaryCondition,
  Constraint,
  ConservationLaw,
  ConstitutiveRelation,
  ModelClass,
  Calibration,
  Verification,
  Validation,
  SensitivityAnalysis,
  UncertaintyQuantification,
  Identifiability,
  ModelReduction,
  Simulation
 ]

 EXPAND(ModelClass) -> [
  DeterministicModel,
  StochasticModel,
  StaticModel,
  DynamicModel,
  ContinuousModel,
  DiscreteModel,
  LinearModel,
  NonlinearModel
 ]

 EXPAND(ModelAssessment) -> [
  Calibration,
  Verification,
  Validation,
  SensitivityAnalysis,
  UncertaintyQuantification,
  Identifiability
 ]

 EXPAND(ModelStructure) -> [
  Variable,
  Parameter,
  Equation,
  Constraint,
  InitialCondition,
  BoundaryCondition,
  ConstitutiveRelation,
  ConservationLaw
 ]

 STOP_CONDITION =
  "Stop when the model purpose, scope, entities, equations, assumptions, classification, calibration, validation, identifiability, uncertainty and interpretation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Modelling"

 IMMEDIATE_CONTEXTS = [
  "Mathematical Modelling",
  "Applied Mathematics Ontologies",
  "Differential Equations Ontologies",
  "Numerical Analysis Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Mathematical Modelling",
  "Applied Mathematics Ontologies",
  "Differential Equations Ontologies",
  "Dynamical Systems Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies",
  "Optimisation Ontologies",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Scientific Measurement Ontologies",
  "Linear Algebra Ontologies",
  "Calculus & Analysis Ontologies",
  "Engineering Ontologies",
  "Physics Ontologies"
 ]

 ROUTE_RULES = [
  "Mathematical Modelling Ontology -> Mathematical Modelling for parent process semantics",
  "Mathematical Modelling Ontology -> Applied Mathematics for wider application context",
  "Mathematical Modelling Ontology -> Differential Equations for dynamic governing models",
  "Mathematical Modelling Ontology -> Numerical Analysis for solver and approximation semantics",
  "Mathematical Modelling Ontology -> Optimisation for calibration and parameter estimation",
  "Mathematical Modelling Ontology -> Probability & Statistics for uncertainty and stochastic inference",
  "Mathematical Modelling Ontology -> Measurement Ontologies for units and observed data",
  "Mathematical Modelling Ontology -> Engineering and Physics for domain-specific system laws"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible model entities, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve purpose-dependent model semantics.",
  "Preserve model-scope semantics.",
  "Preserve assumption transparency.",
  "Preserve variable-parameter distinction.",
  "Preserve calibration-verification-validation distinctions.",
  "Preserve identifiability semantics.",
  "Preserve uncertainty-aware interpretation.",
  "Preserve machine-readable mathematical-modelling ontology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The Mathematical Modelling Ontology formally represents the entities, relations and reasoning processes that define a mathematical model."

 PURPOSE_SUMMARY =
  "Every model is interpreted relative to a declared purpose."

 SCOPE_SUMMARY =
  "Model scope defines the represented system boundaries, processes and regimes."

 STRUCTURE_SUMMARY =
  "A model is composed of variables, parameters, governing equations, assumptions, conditions and constraints."

 CLASSIFICATION_SUMMARY =
  "Models may be deterministic or stochastic, static or dynamic, continuous or discrete, linear or nonlinear."

 CALIBRATION_SUMMARY =
  "Calibration estimates parameter values from data."

 VERIFICATION_SUMMARY =
  "Verification checks that the intended mathematical model is implemented and solved correctly."

 VALIDATION_SUMMARY =
  "Validation assesses whether the model is adequate for its intended use."

 IDENTIFIABILITY_SUMMARY =
  "Identifiability determines whether parameters or structures can be uniquely inferred."

 UNCERTAINTY_SUMMARY =
  "Uncertainty quantification propagates imperfect knowledge through model predictions."
}

FLOW {
 INPUT
 -> detect_mathematical_modelling_ontology_context
 -> identify model_purpose_scope_system_assumptions_variables_parameters_equations_conditions_constraints_data_and_uncertainty
 -> classify model_structure
 -> verify completeness_dimensional_consistency_and_scope
 -> identify calibration_verification_validation_identifiability_sensitivity_uncertainty_simulation_or_reduction goal
 -> apply formal_modelling ontology reasoning
 -> verify assumptions_parameter_roles_conditions_validation_domain_and_uncertainty constraints
 -> route to mathematical_modelling applied_mathematics differential_equations numerical_analysis optimisation probability_statistics measurement engineering_or_physics ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Mathematical Modelling Ontology provides the formal semantic layer beneath mathematical modelling. It represents model purpose, scope, systems, assumptions, variables, parameters, governing equations, constraints, initial and boundary conditions, model classes, calibration, verification, validation, identifiability, sensitivity, uncertainty, model reduction and simulation in a machine-readable reasoning structure."
}

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

}

END_FLOW