Mathematical Modelling

AI INTERPRETATION NOTE
This Flow Algebra defines the Mathematical Modelling layer within the Kireth Research Semantic Library. Mathematical modelling converts real, observed, engineered or abstract systems into mathematical representations that can be analysed, simulated, calibrated, validated and used for prediction, explanation or optimisation. This ontology supports problem formulation, assumptions, variables, parameters, governing equations, state spaces, initial and boundary conditions, deterministic and stochastic models, continuous and discrete models, dimensional analysis, nondimensionalisation, calibration, validation, sensitivity analysis, uncertainty quantification, model reduction and relationships with applied mathematics, numerical analysis, differential equations, probability and scientific computing.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = MathematicalModelling
 ENTITY_CLASS = AppliedMathematicalRepresentationFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Mathematical modelling is the process of constructing, analysing, calibrating and validating mathematical representations of systems or phenomena."
 SEMANTIC_ROLE = "Provides the semantic framework connecting real or abstract systems to variables, parameters, assumptions, equations, computational methods, data, uncertainty and model validation."
}

SCOPE {
 INCLUDE = [
  MathematicalModelling,
  MathematicalModel,
  ModelPurpose,
  System,
  Phenomenon,
  Assumption,
  SimplifyingAssumption,
  ModelVariable,
  StateVariable,
  InputVariable,
  OutputVariable,
  Parameter,
  Constant,
  GoverningEquation,
  AlgebraicModel,
  DifferentialEquationModel,
  IntegralModel,
  DiscreteModel,
  ContinuousModel,
  DeterministicModel,
  StochasticModel,
  StaticModel,
  DynamicModel,
  LinearModel,
  NonlinearModel,
  InitialCondition,
  BoundaryCondition,
  Constraint,
  ConstitutiveRelation,
  ConservationLaw,
  DimensionalAnalysis,
  Nondimensionalisation,
  ScalingLaw,
  Calibration,
  ParameterEstimation,
  Validation,
  Verification,
  SensitivityAnalysis,
  UncertaintyQuantification,
  ModelReduction,
  SurrogateModel,
  Simulation
 ]

 EXCLUDE_AS_PRIMARY = [
  PureMathematicalTheoryWithoutModelContext,
  EmpiricalDescriptionWithoutMathematicalRepresentation,
  NumericalComputationWithoutModelSemantics,
  StatisticalSummaryWithoutSystemModel
 ]

 DISAMBIGUATION_RULE =
  "When a system or phenomenon is represented using mathematical variables, parameters, equations, assumptions and constraints for analysis, prediction or simulation, interpret the context as mathematical modelling."
}

CORE_CONCEPTS {

 MathematicalModelling {
  definition = "The process of translating a system or phenomenon into a mathematical representation."
 }

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

 ModelPurpose {
  definition = "The intended explanatory, predictive, descriptive, control or optimisation role of a model."
 }

 System {
  definition = "The real, engineered, biological, economic, physical or abstract object being modelled."
 }

 Assumption {
  definition = "A declared condition simplifying or constraining the relationship between model and system."
 }

 ModelVariable {
  definition = "A mathematical quantity representing a changing system property."
 }

 StateVariable {
  definition = "A variable sufficient, together with inputs and parameters, to characterise model state."
 }

 Parameter {
  definition = "A model quantity treated as fixed during an instance but potentially estimated or varied between instances."
 }

 GoverningEquation {
  definition = "An equation or system encoding the principal mathematical relationships of the model."
 }

 InitialCondition {
  definition = "Data specifying model state at an initial time or parameter value."
 }

 BoundaryCondition {
  definition = "Data specifying model behaviour on a domain boundary."
 }

 Calibration {
  definition = "Adjustment or estimation of model parameters using reference data."
 }

 Validation {
  definition = "Assessment of whether model outputs are adequate for their intended use."
 }

 Verification {
  definition = "Assessment of whether the model equations and numerical methods are implemented and solved correctly."
 }
}

RELATIONS {
 MathematicalModelling --is_branch_of--> AppliedMathematics
 MathematicalModel --represents--> System
 MathematicalModel --has--> ModelPurpose
 MathematicalModel --depends_on--> Assumption
 MathematicalModel --contains--> ModelVariable
 MathematicalModel --contains--> Parameter
 MathematicalModel --is_governed_by--> GoverningEquation
 DynamicModel --may_require--> InitialCondition
 SpatialModel --may_require--> BoundaryCondition
 GoverningEquation --may_encode--> ConservationLaw
 GoverningEquation --may_include--> ConstitutiveRelation
 Calibration --estimates--> Parameter
 Validation --evaluates--> MathematicalModel
 Verification --evaluates--> ModelImplementation
 SensitivityAnalysis --measures--> ParameterInfluence
 UncertaintyQuantification --propagates--> InputUncertainty
 Simulation --executes--> MathematicalModel
 ModelReduction --approximates--> HighDimensionalModel
}

OPERATORS {
 FORMULATE_MODEL(system,purpose,assumptions) :=
  define_variables_parameters_equations_constraints_and_outputs

 CLASSIFY_MODEL(model) :=
  deterministic_or_stochastic,
  static_or_dynamic,
  discrete_or_continuous,
  linear_or_nonlinear

 NONDIMENSIONALISE(model,characteristic_scales) :=
  transform_to_dimensionless_variables_and_parameters

 CALIBRATE(model,data,objective) :=
  argmin_or_argmax_parameters_against_data_fit criterion

 VERIFY(model,implementation,test_cases) :=
  implementation_correctness_assessment

 VALIDATE(model,reference_data,criteria) :=
  fitness_for_intended_use_assessment

 SENSITIVITY(model,parameter,output) :=
  derivative_or_global_dependence_of_output_on_parameter

 PROPAGATE_UNCERTAINTY(model,input_distribution) :=
  output_distribution_or_uncertainty_description

 REDUCE_MODEL(model,criterion) :=
  lower_complexity_approximation_preserving_selected_behaviour

 SIMULATE(model,initial_conditions,inputs,parameters) :=
  numerical_or_analytical_execution_of_model
}

DEFINITIONS {
 D1 = "A model is purpose-dependent and need not reproduce every feature of the represented system."
 D2 = "Assumptions define the domain in which model interpretation is meaningful."
 D3 = "Variables describe changing quantities; parameters control model behaviour."
 D4 = "Governing equations encode mathematical structure."
 D5 = "Initial conditions specify starting state."
 D6 = "Boundary conditions constrain model behaviour at spatial or abstract boundaries."
 D7 = "Calibration fits parameters; validation assesses predictive or descriptive adequacy."
 D8 = "Verification asks whether the equations are solved correctly; validation asks whether they are suitable for the intended system."
 D9 = "Sensitivity analysis studies output dependence on inputs and parameters."
 D10 = "Uncertainty quantification preserves imperfect knowledge through model calculations."
}

AXIOMS {
 A1_PURPOSE:
  Every meaningful model has an intended purpose

 A2_ASSUMPTIONS:
  Model interpretation is conditional on declared assumptions

 A3_REPRESENTATION:
  A mathematical model represents selected system behaviour rather than the system in full

 A4_DIMENSIONAL_CONSISTENCY:
  Physical model equations must be dimensionally consistent

 A5_WELL_POSEDNESS:
  Model solution claims should consider existence uniqueness and stability when relevant

 A6_VERIFICATION_VALIDATION_DISTINCTION:
  Verification and validation are distinct activities

 A7_CALIBRATION_VALIDATION_DISTINCTION:
  Data used for parameter fitting should be distinguished from independent validation evidence when possible

 A8_UNCERTAINTY:
  Materially uncertain inputs must not be represented as exact without justification
}

REASONING {

 R1_PROBLEM_DEFINITION {
  INPUT = application_question
  IDENTIFY = system
  IDENTIFY = purpose
  IDENTIFY = outputs_of_interest
  IDENTIFY = domain_and_constraints
  RETURN = modelling_problem_statement
 }

 R2_ASSUMPTION_SELECTION {
  INPUT = system_and_purpose
  IDENTIFY = dominant_processes
  IDENTIFY = negligible_processes
  STATE = simplifying_assumptions
  RETURN = model_scope
 }

 R3_VARIABLE_PARAMETER_SELECTION {
  INPUT = scoped_system
  DEFINE = states_inputs_outputs_parameters
  ASSIGN = units_and_domains
  RETURN = model_quantities
 }

 R4_GOVERNING_STRUCTURE {
  INPUT = system_processes
  IF conservation_dominant
   DERIVE = conservation_laws
  IF empirical_relation_required
   INCLUDE = constitutive_relation
  IF stochastic_variation_material
   INCLUDE = stochastic_terms
  RETURN = governing_model
 }

 R5_MODEL_CLASSIFICATION {
  INPUT = governing_model
  CLASSIFY = static_or_dynamic
  CLASSIFY = deterministic_or_stochastic
  CLASSIFY = continuous_or_discrete
  CLASSIFY = linear_or_nonlinear
  RETURN = model_class
 }

 R6_NONDIMENSIONALISATION {
  INPUT = dimensional_model
  CHOOSE = characteristic_scales
  SUBSTITUTE = dimensionless_variables
  SIMPLIFY = equations
  IDENTIFY = dimensionless_groups
  RETURN = nondimensional_model
 }

 R7_CALIBRATION {
  INPUT = model_and_training_data
  DEFINE = objective_or_likelihood
  ESTIMATE = parameters
  ASSESS = identifiability
  RETURN = calibrated_model
 }

 R8_VERIFICATION {
  INPUT = implementation
  TEST = conservation_limits_manufactured_solutions_or_reference_cases
  RETURN = verified_or_issue_detected
 }

 R9_VALIDATION {
  INPUT = calibrated_model_and_validation_data
  COMPARE = predictions_to_observations
  COMPUTE = validation_metrics
  ASSESS = intended_use_thresholds
  RETURN = validation_status
 }

 R10_UNCERTAINTY {
  INPUT = uncertain_inputs_parameters_or_structure
  QUANTIFY = uncertainty_model
  PROPAGATE = through_model
  RETURN = output_uncertainty
 }
}

CONSTRAINTS {
 C1 = "Model purpose must be identified before judging adequacy."
 C2 = "Assumptions should not be hidden."
 C3 = "Variables and parameters must not be conflated."
 C4 = "Calibration data and validation data should be distinguished where possible."
 C5 = "A good data fit does not prove model correctness."
 C6 = "Parameter non-identifiability can make fitted values nonunique."
 C7 = "Nondimensionalisation must preserve mathematical equivalence."
 C8 = "Numerical error, parameter uncertainty and structural model error are distinct."
 C9 = "Model reduction may sacrifice behaviours not included in its preservation criterion."
 C10 = "Extrapolation beyond calibration or validation domains requires reduced confidence."
}

SEMANTIC_EQUIVALENCE {
 "MathematicalModelling" <=> "FormalQuantitativeSystemRepresentation"
 "MathematicalModel" <=> "PurposeDependentMathematicalRepresentation"
 "Calibration" <=> "ParameterFitting"
 "Verification" <=> "CorrectSolutionAndImplementationAssessment"
 "Validation" <=> "FitnessForPurposeAssessment"
 "Nondimensionalisation" <=> "DimensionlessModelTransformation"
}

VALIDATION {
 V1 = "System and model purpose are identified."
 V2 = "Assumptions are explicit."
 V3 = "Variables and parameters are defined."
 V4 = "Units and dimensions are consistent."
 V5 = "Initial and boundary conditions are sufficient when required."
 V6 = "Calibration objective is stated."
 V7 = "Verification and validation evidence are distinguished."
 V8 = "Parameter identifiability is considered."
 V9 = "Uncertainty sources are identified."
 V10 = "Prediction domain is consistent with validation evidence."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MATHEMATICAL_MODELLING_DEFINITION = 0.99
 PROBLEM_FORMULATION_REASONING = 0.99
 MODEL_CLASSIFICATION_REASONING = 0.99
 GOVERNING_EQUATION_REASONING = 0.98
 CALIBRATION_REASONING = 0.97
 VALIDATION_REASONING = 0.97
 UNCERTAINTY_REASONING = 0.97
 MODEL_REDUCTION_REASONING = 0.96

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

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MathematicalModellingReasoning,
  ModelFormulation,
  ModelClassification,
  AssumptionReasoning,
  VariableParameterReasoning,
  GoverningEquationReasoning,
  DimensionalAnalysis,
  Nondimensionalisation,
  CalibrationReasoning,
  VerificationReasoning,
  ValidationReasoning,
  SensitivityAnalysis,
  UncertaintyQuantification,
  ModelReduction,
  SimulationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "mathematical modelling" -> MathematicalModelling
  "mathematical modeling" -> MathematicalModelling
  "mathematical model" -> MathematicalModel
  "model assumption" -> Assumption
  "state variable" -> StateVariable
  "model parameter" -> Parameter
  "governing equation" -> GoverningEquation
  "calibration" -> Calibration
  "verification" -> Verification
  "validation" -> Validation
  "sensitivity analysis" -> SensitivityAnalysis
  "uncertainty quantification" -> UncertaintyQuantification
 }

 INTERPRETATION_ORDER = [
  detect_mathematical_modelling_context,
  identify system_purpose_scope_assumptions_data_and_outputs,
  identify variables_parameters_units_conditions_and_constraints,
  classify model_structure,
  formulate governing_relations,
  select analysis_calibration_verification_validation_or_simulation pathway,
  verify dimensional_well_posedness_uncertainty_and_scope constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("mathematical modelling")
  OR applied_mathematics_context_with_system_representation
  THEN activate(MathematicalModelling)

 E2:
  IF model_formulation_requested
  THEN identify purpose_before equations

 E3:
  IF dynamic_model
  THEN identify initial_conditions

 E4:
  IF spatial_or_boundary_value model
  THEN identify boundary_conditions

 E5:
  IF model_fitted_to_data
  THEN label process as calibration_or_parameter estimation

 E6:
  IF implementation_accuracy_question
  THEN route to verification

 E7:
  IF real_system_adequacy_question
  THEN route to validation

 E8:
  IF dimensioned_model
  THEN perform dimensional_consistency check

 E9:
  IF uncertain_inputs_or_parameters
  THEN preserve uncertainty_and_propagate when relevant
}

RECURSION {
 ENTRY = MathematicalModelling

 EXPAND(MathematicalModelling) -> [
  MathematicalModel,
  ModelPurpose,
  System,
  Assumption,
  ModelVariable,
  StateVariable,
  InputVariable,
  OutputVariable,
  Parameter,
  GoverningEquation,
  InitialCondition,
  BoundaryCondition,
  Constraint,
  ConservationLaw,
  ConstitutiveRelation,
  DimensionalAnalysis,
  Nondimensionalisation,
  Calibration,
  Verification,
  Validation,
  SensitivityAnalysis,
  UncertaintyQuantification,
  ModelReduction,
  SurrogateModel,
  Simulation
 ]

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

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

 EXPAND(ModelSimplification) -> [
  Nondimensionalisation,
  ScalingAnalysis,
  ModelReduction,
  SurrogateModel,
  AsymptoticApproximation
 ]

 STOP_CONDITION =
  "Stop when the model purpose, assumptions, variables, parameters, governing structure, solution method, calibration, validation, uncertainty and interpretation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Differential Equations Ontologies",
  "Numerical Analysis Ontologies",
  "Probability & Statistics Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "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 -> Applied Mathematics Ontologies for parent semantics",
  "Mathematical Modelling -> Differential Equations for continuous dynamic models",
  "Mathematical Modelling -> Dynamical Systems for state-space evolution",
  "Mathematical Modelling -> Numerical Analysis for approximate solution",
  "Mathematical Modelling -> Scientific Computing for simulation",
  "Mathematical Modelling -> Optimisation for parameter estimation and design",
  "Mathematical Modelling -> Probability & Statistics for uncertainty and data fitting",
  "Mathematical Modelling -> Measurement Ontologies for units and observations",
  "Mathematical Modelling -> Engineering and Physics for domain-specific governing laws"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve model-purpose semantics.",
  "Preserve assumption transparency.",
  "Preserve variable-parameter distinction.",
  "Preserve dimensional consistency.",
  "Preserve verification-versus-validation distinction.",
  "Preserve calibration-versus-validation distinction.",
  "Preserve uncertainty-aware interpretation.",
  "Preserve machine-readable mathematical-modelling semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Mathematical modelling is the process of constructing and analysing mathematical representations of systems for a defined purpose."

 PURPOSE_SUMMARY =
  "A model should be judged relative to its intended descriptive, explanatory, predictive, control or optimisation role."

 STRUCTURE_SUMMARY =
  "A model combines assumptions, variables, parameters, equations, conditions and constraints."

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

 CALIBRATION_SUMMARY =
  "Calibration estimates model parameters using data."

 VERIFICATION_SUMMARY =
  "Verification assesses whether equations and numerical methods are implemented and solved correctly."

 VALIDATION_SUMMARY =
  "Validation assesses whether the model is adequate for its intended representation or prediction task."

 UNCERTAINTY_SUMMARY =
  "Sensitivity and uncertainty analysis preserve information about imperfectly known inputs and parameters."

 SCALING_SUMMARY =
  "Nondimensionalisation exposes dimensionless structure and dominant parameter groups."
}

FLOW {
 INPUT
 -> detect_mathematical_modelling_context
 -> identify system_purpose_scope_assumptions_data_outputs_and_constraints
 -> define variables_parameters_units_initial_conditions_and_boundary_conditions
 -> classify deterministic_stochastic_static_dynamic_continuous_discrete_linear_or_nonlinear structure
 -> formulate governing_equations_and_constitutive relations
 -> perform dimensional_analysis_and_nondimensionalisation when applicable
 -> select analytical_numerical_calibration_simulation_or_optimisation method
 -> verify implementation_and_solution
 -> validate against intended_use
 -> analyse sensitivity_uncertainty_and_extrapolation limits
 -> route to applied_mathematics differential_equations dynamical_systems numerical_analysis scientific_computing 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 layer defines the process of translating systems into mathematical representations. It supports model purpose, assumptions, variables, parameters, governing equations, initial and boundary conditions, deterministic and stochastic structures, dimensional analysis, nondimensionalisation, calibration, verification, validation, sensitivity analysis, uncertainty quantification, model reduction and simulation."
}

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

}

END_FLOW