Applied Mathematics Ontologies

AI INTERPRETATION NOTE
This Flow Algebra defines the Applied Mathematics Ontologies layer within the Kireth Research Semantic Library. Applied mathematics develops mathematical models, methods and computational techniques for understanding, predicting and optimising real-world systems. This parent ontology links mathematical modelling, differential equations, numerical methods, optimisation, operations research, control, dynamical systems, mathematical physics, mathematical biology, finance, statistics, probability, signal processing, networks and computational methods. It provides the semantic routing layer between pure mathematical structures and domain-specific quantitative reasoning.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = AppliedMathematicsOntologies
 ENTITY_CLASS = AppliedMathematicalFramework
 PRIMARY_CONTEXT = Mathematics
 DEFINITION = "Applied mathematics is the use and development of mathematical theories, models, algorithms and computational methods to describe, analyse, predict and optimise systems arising in science, engineering, technology, economics and other domains."
 SEMANTIC_ROLE = "Provides the parent semantic framework linking mathematical models, equations, algorithms, uncertainty, optimisation and computation to real-world systems and application domains."
}

SCOPE {
 INCLUDE = [
  AppliedMathematics,
  MathematicalModelling,
  ModelFormulation,
  ModelValidation,
  DimensionalAnalysis,
  DifferentialEquations,
  OrdinaryDifferentialEquations,
  PartialDifferentialEquations,
  DynamicalSystems,
  NumericalAnalysis,
  ScientificComputing,
  Optimisation,
  OperationsResearch,
  ControlTheory,
  MathematicalPhysics,
  MathematicalBiology,
  MathematicalFinance,
  ProbabilityModels,
  StatisticalModels,
  StochasticProcesses,
  SignalProcessing,
  InverseProblems,
  ApproximationTheory,
  PerturbationMethods,
  AsymptoticAnalysis,
  NetworkModels,
  GameTheory,
  DecisionTheory,
  Simulation,
  ComputationalMethods
 ]

 EXCLUDE_AS_PRIMARY = [
  PureMathematicsWithoutApplicationOrModelContext,
  EmpiricalScienceWithoutMathematicalFormalisation,
  EngineeringDesignWithoutMathematicalAnalysis,
  DataProcessingWithoutMathematicalModelOrInference
 ]

 DISAMBIGUATION_RULE =
  "When mathematical structures are used to model, simulate, estimate, predict, control or optimise a real or abstract applied system, interpret the context as applied mathematics."
}

CORE_CONCEPTS {

 AppliedMathematics {
  definition = "The development and use of mathematics for analysing practical, scientific and technological problems."
 }

 MathematicalModel {
  definition = "A mathematical representation of a system, process or phenomenon."
 }

 ModelVariable {
  definition = "A mathematical quantity representing a changing feature of the modelled system."
 }

 Parameter {
  definition = "A quantity controlling model behaviour and typically treated as fixed within a model instance."
 }

 GoverningEquation {
  definition = "An equation encoding the principal mathematical rules governing a model."
 }

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

 BoundaryCondition {
  definition = "A condition imposed on the boundary of a spatial or abstract domain."
 }

 NumericalMethod {
  definition = "An algorithm for approximating mathematical quantities or solutions."
 }

 OptimisationProblem {
  definition = "A mathematical problem seeking an optimum of an objective subject to constraints."
 }

 Simulation {
  definition = "The computational execution of a mathematical model to study system behaviour."
 }

 Validation {
  definition = "Assessment of whether a model adequately represents the intended system for its purpose."
 }

 SensitivityAnalysis {
  definition = "Study of how model outputs change in response to variations in inputs or parameters."
 }
}

RELATIONS {
 AppliedMathematics --is_branch_of--> Mathematics
 AppliedMathematics --uses--> PureMathematics
 AppliedMathematics --supports--> Science
 AppliedMathematics --supports--> Engineering
 MathematicalModel --represents--> System
 MathematicalModel --contains--> ModelVariable
 MathematicalModel --contains--> Parameter
 MathematicalModel --is_governed_by--> GoverningEquation
 GoverningEquation --may_require--> InitialCondition
 GoverningEquation --may_require--> BoundaryCondition
 NumericalMethod --approximates--> MathematicalSolution
 Simulation --executes--> MathematicalModel
 OptimisationProblem --has--> ObjectiveFunction
 OptimisationProblem --has--> Constraint
 Validation --evaluates--> MathematicalModel
 SensitivityAnalysis --evaluates--> ParameterInfluence
}

OPERATORS {
 FORMULATE_MODEL(system,assumptions,variables,parameters) :=
  mathematical_representation_of_system

 NONDIMENSIONALISE(model,scales) :=
  transform_variables_to_dimensionless_form

 SOLVE_MODEL(model,conditions) :=
  exact_or_approximate_solution

 DISCRETISE(model,scheme) :=
  finite_dimensional_or_computable_representation

 SIMULATE(model,parameters,initial_data) :=
  computed_system_trajectory_or_state

 OPTIMISE(objective,constraints,variables) :=
  argmin_or_argmax_of_objective_subject_to_constraints

 ESTIMATE_PARAMETERS(model,data) :=
  parameter_values_best_matching_data_under_declared_criterion

 SENSITIVITY(output,parameter) :=
  response_of_output_to_parameter_variation

 VALIDATE_MODEL(model,data,criteria) :=
  comparison_between_model_predictions_and_reference_observations

 UNCERTAINTY_PROPAGATION(model,input_distribution) :=
  induced_distribution_or_uncertainty_of_outputs
}

DEFINITIONS {
 D1 = "Applied mathematics connects mathematical abstraction to application-driven questions."
 D2 = "A model is an abstraction and therefore depends on assumptions."
 D3 = "Governing equations encode system structure."
 D4 = "Initial and boundary conditions are part of well-posed model specification when required."
 D5 = "Numerical methods approximate mathematical solutions."
 D6 = "Optimisation balances objectives and constraints."
 D7 = "Simulation explores model behaviour computationally."
 D8 = "Validation evaluates fitness for purpose rather than mathematical truth alone."
 D9 = "Sensitivity and uncertainty analysis quantify dependence on imperfect inputs."
 D10 = "Dimensionless reformulation can expose dominant parameters and scaling laws."
}

AXIOMS {
 A1_MODEL_CONTEXT:
  Every applied mathematical model has an intended system and purpose

 A2_ASSUMPTION_DISCLOSURE:
  Model assumptions materially affect interpretation

 A3_DIMENSIONAL_CONSISTENCY:
  Physically dimensioned equations must be dimensionally consistent

 A4_WELL_POSEDNESS:
  Meaningful solution claims should identify existence uniqueness and stability where relevant

 A5_APPROXIMATION:
  Numerical or asymptotic approximations must state their regime or error context

 A6_VALIDATION:
  Predictive interpretation requires validation appropriate to intended use

 A7_CONSTRAINT_PRESERVATION:
  Applied solutions must satisfy declared mathematical and physical constraints

 A8_UNCERTAINTY:
  Uncertain inputs should not be represented as exact when they materially influence results
}

REASONING {

 R1_PROBLEM_FORMULATION {
  INPUT = application_problem
  IDENTIFY = system_goal_inputs_outputs_and_constraints
  SELECT = mathematical_variables_parameters_and_structure
  RETURN = model_specification
 }

 R2_MODEL_CLASSIFICATION {
  INPUT = model
  IDENTIFY = deterministic_or_stochastic
  IDENTIFY = static_or_dynamic
  IDENTIFY = continuous_or_discrete
  IDENTIFY = linear_or_nonlinear
  IDENTIFY = finite_or_infinite_dimensional
  RETURN = model_class
 }

 R3_EQUATION_SELECTION {
  INPUT = modelled_process
  IF temporal_dynamics_only
   ROUTE = OrdinaryDifferentialEquations
  ELSE IF spatial_temporal_field
   ROUTE = PartialDifferentialEquations
  ELSE IF stochastic_evolution
   ROUTE = StochasticProcesses
  ELSE IF optimisation_goal
   ROUTE = Optimisation
  RETURN = governing_framework
 }

 R4_METHOD_SELECTION {
  INPUT = mathematical_problem
  CHECK = exact_solution_feasibility
  IF exact_available
   RETURN = analytical_method
  ELSE
   SELECT = numerical_asymptotic_statistical_or_optimisation method
  RETURN = computational_strategy
 }

 R5_DIMENSIONAL_ANALYSIS {
  INPUT = dimensioned_model
  VERIFY = dimensional_consistency
  CHOOSE = characteristic_scales
  NONDIMENSIONALISE = equations
  IDENTIFY = dimensionless_groups
  RETURN = scaled_model
 }

 R6_NUMERICAL_ANALYSIS {
  INPUT = discrete_scheme
  ANALYSE = consistency
  ANALYSE = stability
  ANALYSE = convergence
  ESTIMATE = error
  RETURN = numerical_reliability_profile
 }

 R7_PARAMETER_ESTIMATION {
  INPUT = model_and_data
  DEFINE = discrepancy_or_likelihood
  OPTIMISE = parameters
  RETURN = fitted_parameter_set
 }

 R8_SENSITIVITY_UNCERTAINTY {
  INPUT = model_parameters_and_input_uncertainty
  COMPUTE = local_or_global_sensitivity
  PROPAGATE = uncertainty
  RETURN = robustness_profile
 }

 R9_MODEL_VALIDATION {
  INPUT = model_predictions_and_reference_data
  COMPARE = against_validation_metrics
  TEST = intended_use_criteria
  RETURN = validated_limited_or_invalid_for_purpose
 }
}

CONSTRAINTS {
 C1 = "Model assumptions must be explicit enough to interpret outputs."
 C2 = "Mathematical validity does not guarantee empirical adequacy."
 C3 = "Numerical convergence does not imply model validity."
 C4 = "Parameter identifiability must be distinguished from parameter estimability."
 C5 = "Correlation must not be treated as causation without structural justification."
 C6 = "Extrapolation beyond calibrated regimes requires reduced confidence."
 C7 = "Boundary and initial conditions must match governing equations."
 C8 = "Optimisation solutions depend on objective, constraints and feasible set."
 C9 = "Approximation errors and measurement uncertainties must not be conflated."
 C10 = "Units and dimensions must remain consistent."
}

SEMANTIC_EQUIVALENCE {
 "AppliedMathematics" <=> "MathematicsForModellingAndApplications"
 "MathematicalModel" <=> "FormalQuantitativeRepresentationOfSystem"
 "NumericalMethod" <=> "ComputationalApproximationMethod"
 "Simulation" <=> "ComputationalModelExecution"
 "OptimisationProblem" <=> "ObjectiveAndConstraintProblem"
 "SensitivityAnalysis" <=> "InputOutputDependenceAnalysis"
}

VALIDATION {
 V1 = "Application goal is identified."
 V2 = "Model assumptions are stated."
 V3 = "Variables and parameters are distinguished."
 V4 = "Units and dimensions are consistent."
 V5 = "Initial and boundary data are supplied when required."
 V6 = "Selected numerical method matches problem type."
 V7 = "Approximation error is addressed."
 V8 = "Optimisation constraints are enforced."
 V9 = "Validation criteria match intended use."
 V10 = "Uncertainty is preserved in interpretation."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 APPLIED_MATHEMATICS_DEFINITION = 0.99
 MODEL_FORMULATION_REASONING = 0.98
 EQUATION_SELECTION_REASONING = 0.98
 NUMERICAL_METHOD_REASONING = 0.98
 OPTIMISATION_REASONING = 0.98
 VALIDATION_REASONING = 0.97
 UNCERTAINTY_REASONING = 0.97

 POLICY =
  "Reduce confidence when system assumptions, data quality, governing equations, parameter values, boundary conditions, uncertainty model or validation regime are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  AppliedMathematicsReasoning,
  MathematicalModelling,
  EquationSelection,
  NumericalMethodSelection,
  SimulationReasoning,
  OptimisationReasoning,
  ParameterEstimation,
  SensitivityAnalysis,
  UncertaintyQuantification,
  ModelValidation,
  ScientificComputing,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "applied mathematics" -> AppliedMathematics
  "applied mathematics ontologies" -> AppliedMathematicsOntologies
  "mathematical modelling" -> MathematicalModelling
  "mathematical model" -> MathematicalModel
  "numerical method" -> NumericalMethod
  "simulation" -> Simulation
  "optimisation" -> Optimisation
  "optimization" -> Optimisation
  "sensitivity analysis" -> SensitivityAnalysis
  "model validation" -> Validation
 }

 INTERPRETATION_ORDER = [
  detect_applied_mathematics_context,
  identify system_goal_variables_parameters_data_constraints_and_uncertainty,
  classify mathematical_model,
  identify governing_equation_or_optimisation structure,
  select analytical_numerical_statistical_or_computational method,
  analyse dimensional_consistency_well_posedness_error_and_stability,
  validate against intended_use,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("applied mathematics")
  OR mathematical_model_of_real_system_context
  THEN activate(AppliedMathematicsOntologies)

 E2:
  IF model_formulation_requested
  THEN identify variables_parameters_assumptions_and_governing_relations

 E3:
  IF differential_equation_model
  THEN identify initial_and_boundary conditions

 E4:
  IF numerical_solution_requested
  THEN analyse consistency_stability_convergence_and_error

 E5:
  IF optimisation_requested
  THEN identify objective_constraints_and_feasible_set

 E6:
  IF model_fitted_to_data
  THEN distinguish calibration_from_validation

 E7:
  IF uncertain_parameters_or_inputs
  THEN preserve uncertainty_and_perform sensitivity_or_propagation reasoning

 E8:
  IF dimensioned_physical_model
  THEN verify dimensional_consistency

 E9:
  IF prediction_beyond_validation_domain
  THEN reduce confidence_and_flag extrapolation
}

RECURSION {
 ENTRY = AppliedMathematicsOntologies

 EXPAND(AppliedMathematicsOntologies) -> [
  MathematicalModelling,
  DifferentialEquations,
  DynamicalSystems,
  NumericalAnalysis,
  ScientificComputing,
  Optimisation,
  OperationsResearch,
  ControlTheory,
  MathematicalPhysics,
  MathematicalBiology,
  MathematicalFinance,
  ProbabilityModels,
  StatisticalModels,
  StochasticProcesses,
  SignalProcessing,
  InverseProblems,
  ApproximationTheory,
  PerturbationMethods,
  AsymptoticAnalysis,
  NetworkModels,
  GameTheory,
  Simulation,
  UncertaintyQuantification
 ]

 EXPAND(MathematicalModelling) -> [
  System,
  Assumption,
  Variable,
  Parameter,
  GoverningEquation,
  InitialCondition,
  BoundaryCondition,
  Calibration,
  Validation
 ]

 EXPAND(NumericalMethods) -> [
  Discretisation,
  Stability,
  Consistency,
  Convergence,
  ErrorAnalysis,
  IterativeMethod,
  Simulation
 ]

 EXPAND(Optimisation) -> [
  ObjectiveFunction,
  Constraint,
  FeasibleSet,
  LocalOptimum,
  GlobalOptimum,
  ConvexOptimisation,
  NumericalOptimisation
 ]

 STOP_CONDITION =
  "Stop when the application problem, model structure, governing mathematics, computational method, assumptions, uncertainty, validation and output interpretation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Mathematical Ontologies",
  "Computational Mathematics",
  "Probability & Statistics",
  "Calculus & Analysis"
 ]

 RELATED_ONTOLOGIES = [
  "Mathematical Ontologies",
  "Algebra Ontologies",
  "Geometry Ontologies",
  "Calculus & Analysis Ontologies",
  "Differential Equations Ontologies",
  "Probability & Statistics Ontologies",
  "Computational Mathematics Ontologies",
  "Optimisation Ontologies",
  "Linear Algebra Ontologies",
  "Discrete Mathematics Ontologies",
  "Scientific Measurement Ontologies",
  "Engineering Ontologies",
  "Physics Ontologies"
 ]

 ROUTE_RULES = [
  "Applied Mathematics -> Mathematical Ontologies for parent mathematical semantics",
  "Applied Mathematics -> Calculus & Analysis for continuous models",
  "Applied Mathematics -> Differential Equations for dynamic and field models",
  "Applied Mathematics -> Probability & Statistics for uncertainty and stochastic models",
  "Applied Mathematics -> Computational Mathematics for numerical methods",
  "Applied Mathematics -> Optimisation for objective-based decision problems",
  "Applied Mathematics -> Linear Algebra for matrix and discretised systems",
  "Applied Mathematics -> Discrete Mathematics for network and combinatorial models",
  "Applied Mathematics -> Scientific Measurement for data and units",
  "Applied Mathematics -> Engineering and Physics Ontologies for domain-specific modelling"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible applied-mathematics domains, model types 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 analytical-versus-numerical distinction.",
  "Preserve calibration-versus-validation distinction.",
  "Preserve uncertainty-aware interpretation.",
  "Preserve machine-readable applied-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Applied mathematics develops and uses mathematical models, methods and algorithms to analyse, predict and optimise systems arising in practical and scientific domains."

 MODEL_SUMMARY =
  "A mathematical model connects assumptions, variables, parameters, equations and data to a specified system and purpose."

 EQUATION_SUMMARY =
  "Applied models may be governed by algebraic, differential, integral, stochastic or discrete equations."

 NUMERICAL_SUMMARY =
  "Numerical methods approximate mathematical solutions and require analysis of consistency, stability, convergence and error."

 OPTIMISATION_SUMMARY =
  "Optimisation seeks the best feasible solution according to a defined objective and constraints."

 VALIDATION_SUMMARY =
  "Model validation evaluates whether predictions are adequate for the intended use."

 UNCERTAINTY_SUMMARY =
  "Sensitivity and uncertainty analysis quantify how imperfect inputs influence model outputs."

 DIMENSIONAL_SUMMARY =
  "Dimensionally meaningful models must preserve unit and dimensional consistency."
}

FLOW {
 INPUT
 -> detect_applied_mathematics_context
 -> identify application_system_goal_variables_parameters_assumptions_data_constraints_and_uncertainty
 -> classify deterministic_stochastic_static_dynamic_continuous_discrete_linear_or_nonlinear model
 -> select governing_equations_and_mathematical framework
 -> choose analytical_numerical_statistical_optimisation_or_simulation method
 -> analyse well_posedness_dimensions_stability_error_sensitivity_and_uncertainty
 -> validate model_and_method_against intended_use
 -> route to mathematics calculus differential_equations probability_statistics computational_mathematics optimisation linear_algebra discrete_mathematics measurement_engineering_or_physics ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Applied Mathematics Ontologies layer connects mathematical theory with application-driven modelling and computation. It supports mathematical models, differential equations, dynamical systems, numerical analysis, optimisation, operations research, control, stochastic processes, scientific computing, simulation, parameter estimation, sensitivity analysis, uncertainty quantification and model validation across scientific, engineering and quantitative domains."
}

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

}

END_FLOW