Physical Model Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Physical Model Ontology within the Kireth Research Semantic Library. A physical model is a mathematical representation of a physical system, process or phenomenon grounded in physical quantities, conservation laws, constitutive relations, geometry, material properties, initial and boundary conditions and measurable observables. This ontology supports mechanics, thermodynamics, electromagnetism, fluid dynamics, heat transfer, waves, transport, multiphysics coupling, dimensional analysis, calibration, verification, validation and uncertainty while preserving the distinction between physical law, constitutive assumption, numerical approximation and empirical parameterisation.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = PhysicalModel
 ENTITY_CLASS = PhysicsGroundedMathematicalModel
 PRIMARY_CONTEXT = MathematicalModelling
 DEFINITION = "A physical model is a mathematical representation of a physical system whose variables, equations, constraints and parameters are grounded in physical quantities, laws and constitutive relations."
 SEMANTIC_ROLE = "Provides the machine-readable framework for representing physically grounded models, including conservation laws, constitutive behaviour, geometry, material properties, fields, units, conditions, couplings, observables and validation."
}

SCOPE {
 INCLUDE = [
  PhysicalModel,
  PhysicalSystem,
  PhysicalQuantity,
  StateVariable,
  FieldVariable,
  InputVariable,
  OutputVariable,
  Parameter,
  MaterialProperty,
  Geometry,
  Domain,
  ConservationLaw,
  ConstitutiveRelation,
  BalanceLaw,
  GoverningEquation,
  InitialCondition,
  BoundaryCondition,
  InterfaceCondition,
  SourceTerm,
  Flux,
  Force,
  Energy,
  Momentum,
  Mass,
  Charge,
  Temperature,
  Pressure,
  Velocity,
  Displacement,
  Stress,
  Strain,
  ElectricField,
  MagneticField,
  Density,
  Diffusivity,
  Conductivity,
  Viscosity,
  DimensionalAnalysis,
  Nondimensionalisation,
  CoupledPhysics,
  MultiphysicsModel,
  Calibration,
  Verification,
  Validation,
  SensitivityAnalysis,
  UncertaintyQuantification,
  NumericalApproximation
 ]

 EXCLUDE_AS_PRIMARY = [
  AbstractMathematicalModelWithoutPhysicalInterpretation,
  PureStatisticalModelWithoutPhysicalStructure,
  NumericalSolverWithoutPhysicalModel,
  ConceptualPhysicalDescriptionWithoutMathematicalFormalisation
 ]

 DISAMBIGUATION_RULE =
  "When a mathematical model represents a physical system using physical quantities, units, conservation laws, constitutive relations, geometry, material properties or field equations, interpret it as a physical model."
}

CORE_CONCEPTS {

 PhysicalModel {
  definition = "A mathematical model whose semantics are grounded in physical laws, quantities and measurable properties."
 }

 PhysicalSystem {
  definition = "The material, field-based or energetic system represented by the model."
 }

 PhysicalQuantity {
  definition = "A measurable quantity with a physical dimension and unit."
 }

 FieldVariable {
  definition = "A physical quantity defined over space, time or spacetime."
 }

 MaterialProperty {
  definition = "A parameter characterising material response, such as density, conductivity, stiffness or viscosity."
 }

 ConservationLaw {
  definition = "A physical principle expressing conservation of a quantity such as mass, momentum, energy or charge."
 }

 BalanceLaw {
  definition = "A local or global relation balancing storage, transport, source and sink terms."
 }

 ConstitutiveRelation {
  definition = "A relation describing material or phenomenological response beyond universal conservation laws."
 }

 GoverningEquation {
  definition = "The mathematical equation or system resulting from physical laws, constitutive assumptions and model scope."
 }

 BoundaryCondition {
  definition = "A condition specifying physical behaviour on the boundary of the model domain."
 }

 InterfaceCondition {
  definition = "A condition coupling fields or fluxes across interfaces between subdomains or materials."
 }

 MultiphysicsModel {
  definition = "A physical model coupling two or more interacting physical domains or field systems."
 }
}

RELATIONS {
 PhysicalModel --is_special_case_of--> MathematicalModel
 PhysicalModel --represents--> PhysicalSystem
 PhysicalModel --contains--> PhysicalQuantity
 PhysicalModel --has_domain--> Domain
 PhysicalModel --has_geometry--> Geometry
 PhysicalModel --uses--> ConservationLaw
 PhysicalModel --uses--> ConstitutiveRelation
 PhysicalModel --contains--> MaterialProperty
 PhysicalModel --is_governed_by--> GoverningEquation
 GoverningEquation --may_include--> SourceTerm
 GoverningEquation --may_include--> Flux
 GoverningEquation --may_require--> InitialCondition
 GoverningEquation --may_require--> BoundaryCondition
 CoupledPhysics --requires--> InterfaceCondition
 MultiphysicsModel --contains--> CoupledPhysics
 PhysicalQuantity --has--> Unit
 PhysicalQuantity --has--> Dimension
 Calibration --estimates--> PhysicalParameter
 Verification --checks--> NumericalImplementation
 Validation --compares--> PhysicalObservations
}

OPERATORS {
 DEFINE_PHYSICAL_MODEL(system,geometry,quantities,laws,relations) :=
  formal_physics_grounded_mathematical_representation

 FORM_BALANCE(storage,flux,source,sink) :=
  rate_of_storage + net_outflow - source + sink = 0
  under declared sign convention

 APPLY_CONSERVATION(quantity,domain) :=
  integral_balance_or_local_differential_form

 APPLY_CONSTITUTIVE_RELATION(material,state) :=
  response_function_or_tensor_relation

 NONDIMENSIONALISE(model,characteristic_scales) :=
  dimensionless_physical_model

 CHECK_UNITS(equation) :=
  verify_dimensional_consistency_of_all_terms

 COUPLE_MODELS(model_A,model_B,interface_conditions) :=
  multiphysics_model

 CALIBRATE_PHYSICAL_PARAMETERS(model,data) :=
  estimated_material_or_physical_parameters

 VERIFY_PHYSICAL_IMPLEMENTATION(model,numerical_solution) :=
  numerical_consistency_and_conservation_check

 VALIDATE_PHYSICAL_MODEL(model,observations,criteria) :=
  fitness_for_physical_purpose_assessment

 PROPAGATE_PHYSICAL_UNCERTAINTY(model,input_and_parameter_uncertainty) :=
  uncertainty_in_physical_outputs
}

DEFINITIONS {
 D1 = "A physical model is constrained by physical dimensions and units."
 D2 = "Conservation laws and constitutive relations play different roles."
 D3 = "Conservation laws express universal balance principles within their physical domain."
 D4 = "Constitutive relations encode material or phenomenological behaviour."
 D5 = "Geometry and boundary conditions can materially alter physical solutions."
 D6 = "Interface conditions couple physical fields across subdomains."
 D7 = "Material properties may be constant, spatially varying, nonlinear or uncertain."
 D8 = "Multiphysics models couple multiple interacting physical processes."
 D9 = "Verification checks correct implementation of the equations."
 D10 = "Validation checks whether the model adequately represents physical reality for its intended use."
}

AXIOMS {
 A1_PHYSICAL_GROUNDING:
  Every PhysicalModel maps mathematical quantities to physical meanings

 A2_DIMENSIONAL_CONSISTENCY:
  Every valid physical equation is dimensionally consistent

 A3_CONSERVATION:
  Conserved quantities satisfy appropriate balance laws

 A4_CONSTITUTIVE_SEPARATION:
  Constitutive assumptions are distinct from conservation principles

 A5_DOMAIN_DEPENDENCE:
  Geometry and domain influence physical solution structure

 A6_CONDITION_COMPLETENESS:
  Initial boundary and interface conditions must be sufficient for the posed problem

 A7_MEASURABILITY:
  Validation requires connection between model outputs and physical observations

 A8_UNCERTAINTY:
  Physical parameter and measurement uncertainty must be retained where material
}

REASONING {

 R1_PHYSICAL_MODEL_IDENTIFICATION {
  INPUT = model_description
  IDENTIFY = physical_system
  IDENTIFY = physical_quantities
  IDENTIFY = geometry_and_domain
  IDENTIFY = conservation_laws
  IDENTIFY = constitutive_relations
  RETURN = PhysicalModel
 }

 R2_PHYSICS_CLASSIFICATION {
  INPUT = physical_model
  IDENTIFY = mechanics_thermal_fluid_electromagnetic_transport_wave_or_multiphysics context
  RETURN = physical_domain_class
 }

 R3_GOVERNING_EQUATION_DERIVATION {
  INPUT = conservation_laws_constitutive_relations_geometry_and_assumptions
  COMBINE = balance_equations
  SUBSTITUTE = constitutive_relations
  APPLY = kinematic_or_geometric relations
  RETURN = governing_equations
 }

 R4_DIMENSIONAL_ANALYSIS {
  INPUT = governing_equations
  ASSIGN = dimensions_to_variables_and_parameters
  VERIFY = dimensional_consistency
  IDENTIFY = characteristic_scales
  RETURN = dimensional_profile
 }

 R5_BOUNDARY_INITIAL_CONDITIONS {
  INPUT = model_domain_and_equation_type
  IDENTIFY = required_initial_conditions
  IDENTIFY = required_boundary_conditions
  CHECK = compatibility
  RETURN = condition_completeness_profile
 }

 R6_INTERFACE_COUPLING {
  INPUT = adjacent_physical_domains
  IDENTIFY = conserved_fluxes_and_state_continuity laws
  DEFINE = interface_conditions
  RETURN = coupled_model
 }

 R7_PARAMETER_CALIBRATION {
  INPUT = physical_model_and_measurements
  IDENTIFY = uncertain_parameters
  DEFINE = fit_metric_or_likelihood
  ESTIMATE = parameters
  RETURN = calibrated_physical_model
 }

 R8_VERIFICATION {
  INPUT = physical_model_and_numerical_implementation
  CHECK = conservation
  CHECK = limiting_cases
  CHECK = mesh_or_time_step convergence when numerical
  RETURN = verification_status
 }

 R9_VALIDATION {
  INPUT = model_predictions_and_physical_observations
  COMPARE = observables
  ASSESS = errors_against_use_criteria
  RETURN = validation_status
 }

 R10_MULTIPHYSICS_REASONING {
  INPUT = component_physical_models
  IDENTIFY = coupling_variables_and_interfaces
  FORM = coupled_equation_system
  VERIFY = conservation_and_units_across_coupling
  RETURN = multiphysics_model
 }
}

CONSTRAINTS {
 C1 = "All physical quantities must have consistent dimensions and units."
 C2 = "Conservation laws must not be replaced by empirical fits without explicit declaration."
 C3 = "Constitutive laws may have restricted validity regimes."
 C4 = "Geometry and coordinate system must be specified where they affect equations."
 C5 = "Boundary and interface conditions must be physically meaningful."
 C6 = "Material parameters may depend on temperature, pressure, rate, direction or history."
 C7 = "Numerical stabilisation must not be mistaken for physical dissipation unless intended."
 C8 = "Multiphysics coupling must preserve compatible units and conservation laws."
 C9 = "Calibration within one regime does not guarantee validity in another."
 C10 = "Measurement uncertainty and model-form uncertainty must remain distinct."
}

SEMANTIC_EQUIVALENCE {
 "PhysicalModel" <=> "PhysicsGroundedMathematicalModel"
 "ConservationLaw" <=> "PhysicalBalancePrinciple"
 "ConstitutiveRelation" <=> "MaterialOrPhenomenologicalResponseLaw"
 "MaterialProperty" <=> "PhysicalResponseParameter"
 "MultiphysicsModel" <=> "CoupledPhysicalFieldModel"
 "PhysicalValidation" <=> "ComparisonWithPhysicalObservationForIntendedUse"
}

VALIDATION {
 V1 = "Physical system is identified."
 V2 = "Physical quantities have dimensions and units."
 V3 = "Governing equations are dimensionally consistent."
 V4 = "Conservation and constitutive relations are distinguished."
 V5 = "Geometry and domain are specified."
 V6 = "Initial, boundary and interface conditions are sufficient."
 V7 = "Material properties use correct regimes."
 V8 = "Numerical implementation preserves declared conservation properties where expected."
 V9 = "Validation observables correspond to measurable physical quantities."
 V10 = "Uncertainty sources are represented separately."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 PHYSICAL_MODEL_DEFINITION = 0.99
 GOVERNING_EQUATION_REASONING = 0.98
 DIMENSIONAL_REASONING = 0.99
 CONSERVATION_REASONING = 0.99
 CONSTITUTIVE_REASONING = 0.98
 MULTIPHYSICS_REASONING = 0.97
 CALIBRATION_REASONING = 0.97
 VALIDATION_REASONING = 0.97

 POLICY =
  "Reduce confidence when physical domain, constitutive regime, geometry, units, boundary conditions, material properties, coupling laws, measurement quality or uncertainty structure are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  PhysicalModelReasoning,
  MathematicalModellingReasoning,
  ConservationLawReasoning,
  ConstitutiveRelationReasoning,
  DimensionalAnalysis,
  BoundaryConditionReasoning,
  InterfaceConditionReasoning,
  MaterialPropertyReasoning,
  MultiphysicsReasoning,
  PhysicalCalibration,
  VerificationReasoning,
  ValidationReasoning,
  UncertaintyQuantification,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "physical model" -> PhysicalModel
  "physical system model" -> PhysicalModel
  "conservation law" -> ConservationLaw
  "balance law" -> BalanceLaw
  "constitutive relation" -> ConstitutiveRelation
  "material property" -> MaterialProperty
  "boundary condition" -> BoundaryCondition
  "interface condition" -> InterfaceCondition
  "multiphysics model" -> MultiphysicsModel
  "physical validation" -> Validation
 }

 INTERPRETATION_ORDER = [
  detect_physical_model_context,
  identify physical_system_domain_geometry_quantities_units_and_material properties,
  identify conservation_laws_constitutive_relations_sources_fluxes_and_conditions,
  classify physical_domain_and_model_type,
  derive_or_validate governing_equations,
  check dimensional_consistency_and_condition completeness,
  identify calibration_verification_validation_uncertainty_or_multiphysics goal,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("physical model")
  OR mathematical_model_with_physical_quantities_and_laws
  THEN activate(PhysicalModelOntology)

 E2:
  IF physical_equation_given
  THEN verify dimensional_consistency

 E3:
  IF conservation_law_context
  THEN identify storage_flux_source_and_sink terms

 E4:
  IF constitutive_relation_context
  THEN identify validity_regime_and_material parameters

 E5:
  IF PDE_or_spatial_model
  THEN identify geometry_domain_and_boundary conditions

 E6:
  IF multiple_physical_domains_interact
  THEN identify coupling_variables_and_interface conditions

 E7:
  IF model_parameters_fitted_to_measurements
  THEN classify as calibration_not_validation

 E8:
  IF numerical_solution_assessed
  THEN separate verification_from_physical validation

 E9:
  IF operating_regime_outside_validated_range
  THEN flag extrapolation_and_reduce confidence
}

RECURSION {
 ENTRY = PhysicalModel

 EXPAND(PhysicalModel) -> [
  PhysicalSystem,
  PhysicalQuantity,
  StateVariable,
  FieldVariable,
  InputVariable,
  OutputVariable,
  Parameter,
  MaterialProperty,
  Geometry,
  Domain,
  ConservationLaw,
  BalanceLaw,
  ConstitutiveRelation,
  GoverningEquation,
  InitialCondition,
  BoundaryCondition,
  InterfaceCondition,
  SourceTerm,
  Flux,
  CoupledPhysics,
  MultiphysicsModel,
  DimensionalAnalysis,
  Calibration,
  Verification,
  Validation,
  UncertaintyQuantification
 ]

 EXPAND(PhysicalDomains) -> [
  MechanicsModel,
  ThermalModel,
  FluidModel,
  ElectromagneticModel,
  TransportModel,
  WaveModel,
  MultiphysicsModel
 ]

 EXPAND(ConservationStructure) -> [
  MassConservation,
  MomentumConservation,
  EnergyConservation,
  ChargeConservation,
  Flux,
  SourceTerm,
  SinkTerm
 ]

 EXPAND(MaterialStructure) -> [
  Density,
  ElasticModulus,
  PoissonRatio,
  Viscosity,
  Conductivity,
  Diffusivity,
  HeatCapacity,
  ConstitutiveTensor
 ]

 STOP_CONDITION =
  "Stop when the physical system, quantities, laws, constitutive relations, geometry, conditions, material properties, coupling, validation status and uncertainty are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Modelling"

 IMMEDIATE_CONTEXTS = [
  "Mathematical Model Ontology",
  "Mathematical Modelling Ontology",
  "Applied Mathematics Ontologies",
  "Physics Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Mathematical Model Ontology",
  "Mathematical Modelling Ontology",
  "Applied Mathematics Ontologies",
  "Differential Equations Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies",
  "Scientific Measurement Ontologies",
  "Mechanics Ontologies",
  "Thermodynamics Ontologies",
  "Fluid Dynamics Ontologies",
  "Electromagnetism Ontologies",
  "Heat Transfer Ontologies",
  "Materials Science Ontologies",
  "Engineering Ontologies"
 ]

 ROUTE_RULES = [
  "Physical Model -> Mathematical Model Ontology for formal model semantics",
  "Physical Model -> Mathematical Modelling Ontology for modelling-process semantics",
  "Physical Model -> Differential Equations for governing field equations",
  "Physical Model -> Numerical Analysis for discretisation and solver semantics",
  "Physical Model -> Scientific Measurement for units and observables",
  "Physical Model -> Mechanics for force, motion and deformation",
  "Physical Model -> Thermodynamics and Heat Transfer for thermal systems",
  "Physical Model -> Fluid Dynamics for flow models",
  "Physical Model -> Electromagnetism for field models",
  "Physical Model -> Materials Science for constitutive and material properties",
  "Physical Model -> Engineering for domain-specific physical systems"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible physical domains, constitutive models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve physical grounding.",
  "Preserve dimensional consistency.",
  "Preserve conservation-versus-constitutive distinction.",
  "Preserve geometry and condition semantics.",
  "Preserve material-property validity regimes.",
  "Preserve verification-versus-validation distinction.",
  "Preserve uncertainty separation.",
  "Preserve machine-readable physical-model semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A physical model is a mathematical representation of a physical system grounded in physical quantities, laws, constitutive relations, geometry and measurable observables."

 PHYSICAL_QUANTITY_SUMMARY =
  "Physical quantities carry dimensions, units and measurable meanings."

 CONSERVATION_SUMMARY =
  "Conservation laws express balances of quantities such as mass, momentum, energy and charge."

 CONSTITUTIVE_SUMMARY =
  "Constitutive relations describe material or phenomenological response and have declared validity regimes."

 DOMAIN_SUMMARY =
  "Geometry, domain and coordinate system can materially affect the governing equations."

 CONDITION_SUMMARY =
  "Initial, boundary and interface conditions complete the physical problem specification when required."

 MULTIPHYSICS_SUMMARY =
  "Multiphysics models couple interacting physical domains through shared variables, fluxes or interface conditions."

 VALIDATION_SUMMARY =
  "Physical validation compares model predictions with measurable observations for the intended use."

 UNCERTAINTY_SUMMARY =
  "Physical-model uncertainty may arise from measurements, parameters, constitutive assumptions, model structure and numerical approximation."
}

FLOW {
 INPUT
 -> detect_physical_model_context
 -> identify physical_system_purpose_scope_geometry_domain_quantities_units_material_properties_and_observables
 -> identify conservation_laws_constitutive_relations_sources_fluxes_initial_boundary_and_interface conditions
 -> classify mechanics_thermal_fluid_electromagnetic_transport_wave_or_multiphysics structure
 -> derive_or_validate governing_equations
 -> verify dimensional_consistency_conservation_condition_completeness_and_material validity
 -> identify calibration_verification_validation_sensitivity_uncertainty_or_coupling goal
 -> route to mathematical_model mathematical_modelling applied_mathematics differential_equations numerical_analysis measurement physics materials_science_or_engineering ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Physical Model Ontology defines physically grounded mathematical models through physical quantities, units, geometry, material properties, conservation laws, constitutive relations, governing equations, sources, fluxes, initial, boundary and interface conditions, coupled physics, calibration, verification, validation and uncertainty. It preserves the semantic distinction between universal physical balance laws, material response assumptions, measured data and numerical approximation."
}

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

}

END_FLOW