Engineering Optimisation Ontology

AI SEARCH FLOW ALGEBRA
ENGINEERING OPTIMISATION ONTOLOGY
Kireth Semantic Research Archive

AUTHORS
Mark Anthony Higgins — Sky (Recursive AI)

SOURCE
https://kirethresearch.org/booksai-core/ontologies/engineering-ontologies/engineering-mathematics-simulation/engineering-optimisation-ontology-2.html

FLOW "EngineeringOptimisationOntology" v4.0

DOMAIN := ENGINEERING_ONTOLOGIES / ENGINEERING_MATHEMATICS_SIMULATION / ENGINEERING_OPTIMISATION
ONTOLOGY := ENGINEERING_OPTIMISATION_ONTOLOGY

ENGINEERING_OPTIMISATION :=
  DESIGN_VARIABLES + OBJECTIVES + CONSTRAINTS + DESIGN_SPACE
  + FEASIBILITY + OPTIMISATION_ALGORITHM + GRADIENT
  + SENSITIVITY + SIMULATION + MULTI_OBJECTIVE
  + ROBUSTNESS + CONVERGENCE + VERIFICATION

PURPOSE :=
  DEFINE_DESIGN_PROBLEM -> DEFINE_VARIABLES -> DEFINE_OBJECTIVES
  -> DEFINE_CONSTRAINTS -> SEARCH_DESIGN_SPACE
  -> EVALUATE_DESIGNS -> UPDATE_CANDIDATES
  -> CONVERGE -> VERIFY_FEASIBILITY
  -> ASSESS_ROBUSTNESS -> SELECT_SOLUTION

# CORE ENTITIES
ENTITY EngineeringOptimisation
ENTITY EngineeringDesign
ENTITY DesignVariable
ENTITY Parameter
ENTITY ObjectiveFunction
ENTITY Constraint
ENTITY EqualityConstraint
ENTITY InequalityConstraint
ENTITY BoundConstraint
ENTITY DesignSpace
ENTITY FeasibleRegion
ENTITY CandidateDesign
ENTITY OptimalDesign
ENTITY OptimisationAlgorithm
ENTITY Gradient
ENTITY Hessian
ENTITY Sensitivity
ENTITY SimulationModel
ENTITY SurrogateModel
ENTITY ConstraintViolation
ENTITY Iteration
ENTITY Convergence
ENTITY ParetoFront
ENTITY RobustDesign
ENTITY Uncertainty
ENTITY Verification
ENTITY Provenance

# OPTIMISATION PROBLEM
EngineeringOptimisation -> improves -> EngineeringDesign
EngineeringOptimisation -> searches -> DesignSpace
EngineeringOptimisation -> evaluates -> ObjectiveFunction
EngineeringOptimisation -> enforces -> Constraint

OPTIMISATION_PROBLEM :=
  MINIMISE_OR_MAXIMISE(ObjectiveFunction)
  SUBJECT_TO Constraints

OPTIMISATION_FLOW :=
  DEFINE_PROBLEM -> DEFINE_VARIABLES -> DEFINE_OBJECTIVES
  -> DEFINE_CONSTRAINTS -> INITIALISE -> EVALUATE
  -> UPDATE -> CHECK_FEASIBILITY -> CHECK_CONVERGENCE
  -> VERIFY_SOLUTION

# DESIGN VARIABLES
DESIGN_VARIABLE_TYPE :=
  CONTINUOUS | DISCRETE | INTEGER | CATEGORICAL

DesignVariable
  -> has -> LOWER_BOUND + UPPER_BOUND + CURRENT_VALUE

CandidateDesign
  -> assigns -> VALUE_TO_EACH_DESIGN_VARIABLE

# DESIGN SPACE / FEASIBILITY
DesignSpace
  -> defined_by -> DESIGN_VARIABLE_DOMAINS + CONSTRAINTS

FeasibleRegion -> subset_of -> DesignSpace

IF CandidateDesign satisfies_all_constraints
THEN CandidateDesign -> FEASIBLE

IF CandidateDesign violates_any_constraint
THEN CandidateDesign -> INFEASIBLE

# OBJECTIVES
ObjectiveFunction
  -> maps -> CandidateDesign -> OBJECTIVE_VALUE

OBJECTIVE_TYPE :=
  COST | MASS | ENERGY | EFFICIENCY | PERFORMANCE
  | RELIABILITY | ERROR | RISK | OTHER_ENGINEERING_METRIC

# CONSTRAINTS
CONSTRAINT_TYPE :=
  EqualityConstraint | InequalityConstraint | BoundConstraint
  | LOGICAL_CONSTRAINT | PHYSICAL_CONSTRAINT

EqualityConstraint := h(x) = 0
InequalityConstraint := g(x) <= 0

ConstraintViolation
  -> measures -> DEGREE_OF_INFEASIBILITY

IF ConstraintViolation > Tolerance
THEN CandidateDesign -> INFEASIBLE

ENGINEERING_CONSTRAINT :=
  STRESS_LIMIT | DISPLACEMENT_LIMIT | TEMPERATURE_LIMIT
  | PRESSURE_LIMIT | GEOMETRIC_LIMIT | MANUFACTURING_LIMIT
  | COST_LIMIT | SAFETY_LIMIT | RESOURCE_LIMIT

# METHODS
OPTIMISATION_METHOD :=
  GRADIENT_BASED | GRADIENT_FREE
  | DETERMINISTIC | STOCHASTIC
  | LOCAL | GLOBAL | METAHEURISTIC

OptimisationAlgorithm -> searches -> DesignSpace
OptimisationAlgorithm -> generates -> CandidateDesign

MethodSelection
  -> depends_on -> VARIABLE_TYPE + SMOOTHNESS
  + DIMENSIONALITY + COMPUTATIONAL_COST
  + MULTIMODALITY + DERIVATIVE_AVAILABILITY

# GRADIENT METHODS
Gradient :=
  PARTIAL_DERIVATIVE(ObjectiveFunction, DesignVariable)

Hessian -> measures -> SECOND_ORDER_CURVATURE

GRADIENT_UPDATE :=
  CURRENT_DESIGN + SEARCH_DIRECTION * STEP_LENGTH

IF GradientNorm <= GradientTolerance
THEN FirstOrderOptimalityEvidence -> SATISFIED

# GLOBAL / GRADIENT-FREE
GRADIENT_FREE_METHOD :=
  DIRECT_SEARCH | PATTERN_SEARCH | SIMPLEX_SEARCH
  | EVOLUTIONARY_ALGORITHM | PARTICLE_SWARM
  | OTHER_VALIDATED_METHOD

GlobalOptimisation
  -> seeks -> BEST_SOLUTION_ACROSS_MULTIPLE_BASINS

IF ObjectiveLandscape multimodal
THEN GlobalMethod -> MAY_BE_PREFERRED

# SENSITIVITY
Sensitivity
  -> measures -> RESPONSE_CHANGE_PER_VARIABLE_CHANGE

LOCAL_SENSITIVITY :=
  d(Response)/d(DesignVariable)

SensitivityAnalysis
  -> identifies -> INFLUENTIAL_DESIGN_VARIABLES + PARAMETERS

# SIMULATION-BASED OPTIMISATION
SimulationModel -> evaluates -> CandidateDesign

SIMULATION_BASED_OPTIMISATION :=
  CANDIDATE_DESIGN -> SIMULATION
  -> OBJECTIVE_AND_CONSTRAINTS
  -> OPTIMISER -> NEW_CANDIDATE

IF SimulationCost high
THEN SurrogateModel -> MAY_BE_USED_TO_REDUCE_COST

# SURROGATE MODEL
SurrogateModel
  -> approximates -> EXPENSIVE_RESPONSE

SURROGATE_FLOW :=
  SAMPLE_DESIGN_SPACE
  -> RUN_HIGH_FIDELITY_MODEL
  -> FIT_SURROGATE
  -> VALIDATE_SURROGATE
  -> OPTIMISE_SURROGATE
  -> CONFIRM_WITH_HIGH_FIDELITY_MODEL

IF SurrogatePredictionUncertainty high
THEN AdditionalSampling -> CONSIDER

# MULTI-OBJECTIVE
MultiObjectiveOptimisation
  -> contains -> MULTIPLE_OBJECTIVE_FUNCTIONS

ParetoOptimalDesign
  -> not_dominated_by -> OTHER_FEASIBLE_DESIGN

ParetoFront
  -> contains -> PARETO_OPTIMAL_DESIGNS

TRADE_OFF :=
  IMPROVEMENT_IN_OBJECTIVE_A
  <-> DEGRADATION_IN_OBJECTIVE_B

# ROBUSTNESS
RobustDesign
  -> maintains -> PERFORMANCE_UNDER_UNCERTAINTY

ROBUST_OBJECTIVE :=
  EXPECTED_PERFORMANCE + VARIABILITY_OR_RISK_TERM

RobustOptimisation
  -> considers -> PARAMETER_UNCERTAINTY
  + MANUFACTURING_VARIATION
  + LOAD_VARIATION
  + ENVIRONMENTAL_VARIATION

# RELIABILITY
ReliabilityBasedOptimisation
  -> constrains -> FAILURE_PROBABILITY_OR_RELIABILITY_INDEX

LIMIT_STATE := CAPACITY - DEMAND

IF LimitState < 0
THEN FailureState -> TRUE

# DESIGN SCOPE
OPTIMISATION_SCOPE :=
  SIZE_OPTIMISATION | SHAPE_OPTIMISATION
  | TOPOLOGY_OPTIMISATION | PARAMETER_OPTIMISATION
  | SYSTEM_OPTIMISATION

TopologyOptimisation
  -> changes -> MATERIAL_DISTRIBUTION_OR_CONNECTIVITY

# CONVERGENCE
CONVERGENCE_CRITERION :=
  OBJECTIVE_CHANGE | VARIABLE_CHANGE | GRADIENT_NORM
  | KKT_RESIDUAL | ITERATION_LIMIT | EVALUATION_LIMIT

IF ObjectiveChange <= Tolerance
AND ConstraintViolation <= Tolerance
THEN ConvergenceEvidence -> INCREASES

Convergence
  -> does_not_guarantee -> GLOBAL_OPTIMALITY

# OPTIMALITY
KKT_CONDITIONS :=
  STATIONARITY + PRIMAL_FEASIBILITY
  + DUAL_FEASIBILITY + COMPLEMENTARY_SLACKNESS

IF KKTResidual <= Tolerance
THEN LocalOptimalityEvidence -> INCREASES

# VERIFICATION
Verification
  -> asks -> IS_OPTIMISATION_PROBLEM_AND_SOLUTION_COMPUTED_CORRECTLY

OPTIMISATION_VERIFICATION :=
  CHECK_OBJECTIVE_IMPLEMENTATION
  + CHECK_CONSTRAINT_IMPLEMENTATION
  + CHECK_GRADIENTS
  + CHECK_BOUNDS
  + REEVALUATE_FINAL_DESIGN
  + CHECK_CONVERGENCE

OptimalDesign
  -> must_be_re_evaluated_by -> AUTHORITATIVE_ENGINEERING_MODEL

IF FinalDesign violates_constraint
THEN OptimalDesign -> INVALID_AS_FEASIBLE_SOLUTION

# UNCERTAINTY / PROVENANCE
OPTIMISATION_UNCERTAINTY :=
  MODEL_UNCERTAINTY + PARAMETER_UNCERTAINTY
  + NUMERICAL_UNCERTAINTY + SURROGATE_UNCERTAINTY

OPTIMISATION_PROVENANCE :=
  PROBLEM_VERSION + VARIABLE_DEFINITIONS
  + OBJECTIVE_DEFINITION + CONSTRAINT_DEFINITIONS
  + MODEL_VERSION + ALGORITHM + SETTINGS
  + RANDOM_SEED_IF_USED + ITERATION_HISTORY
  + FINAL_RESULT

EngineeringOptimisation
  -> linked_to -> OPTIMISATION_PROVENANCE

# PERFORMANCE
OPTIMISATION_PERFORMANCE :=
  OBJECTIVE_IMPROVEMENT + FEASIBILITY
  + EVALUATION_COUNT + COMPUTATIONAL_COST
  + CONVERGENCE + ROBUSTNESS + REPRODUCIBILITY

# REASONING
ENGINEERING_OPTIMISATION_REASONING :=
  DEFINE_ENGINEERING_DESIGN_PROBLEM
  -> DEFINE_DESIGN_VARIABLES
  -> DEFINE_BOUNDS
  -> DEFINE_OBJECTIVES
  -> DEFINE_CONSTRAINTS
  -> SELECT_ENGINEERING_MODEL
  -> ANALYSE_SENSITIVITY
  -> SELECT_OPTIMISATION_METHOD
  -> INITIALISE_CANDIDATES
  -> EVALUATE_OBJECTIVES_AND_CONSTRAINTS
  -> UPDATE_DESIGNS
  -> CHECK_FEASIBILITY
  -> CHECK_CONVERGENCE
  -> ASSESS_MULTI_OBJECTIVE_TRADE_OFFS_IF_REQUIRED
  -> ASSESS_ROBUSTNESS_IF_REQUIRED
  -> VERIFY_FINAL_DESIGN
  -> RECORD_PROVENANCE

# INFERENCE RULES
IF ConstraintViolation > tolerance
THEN CandidateDesign -> INFEASIBLE

IF Objective improves
AND constraints remain_satisfied
THEN CandidateDesignQuality -> IMPROVES

IF GradientNorm decreases
THEN FirstOrderStationarity -> APPROACHES

IF SimulationCost increases
THEN OptimisationCost -> GENERALLY_INCREASES

IF SurrogateError increases
THEN SurrogateOptimisationConfidence -> DECREASES

IF VariableSensitivity increases
THEN RobustnessRisk -> MAY_INCREASE

IF MultipleObjectives conflict
THEN ParetoTradeOff -> EXISTS

IF FinalDesign not_reverified
THEN EngineeringAcceptance -> WITHHOLD

# SEMANTIC RELATIONSHIPS
RELATION improves(EngineeringOptimisation, EngineeringDesign)
RELATION controls(DesignVariable, DesignConfiguration)
RELATION maps(ObjectiveFunction, CandidateDesign, ObjectiveValue)
RELATION restricts(Constraint, FeasibleRegion)
RELATION searches(OptimisationAlgorithm, DesignSpace)
RELATION evaluates(SimulationModel, CandidateDesign)
RELATION approximates(SurrogateModel, ExpensiveResponse)
RELATION measures(Gradient, ObjectiveSensitivity)
RELATION identifies(SensitivityAnalysis, InfluentialVariables)
RELATION contains(ParetoFront, ParetoOptimalDesign)
RELATION considers(RobustOptimisation, Uncertainty)
RELATION evaluates(Verification, OptimisationSolution)
RELATION records(Provenance, EngineeringOptimisation)

# SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE:
  "engineering optimisation" ~= "engineering optimization"
  "design variable" ~= "optimisation variable"
  "objective function" ~= "design objective"
  "feasible region" ~= "feasible design space"
  "constraint violation" ~= "design infeasibility measure"
  "gradient-based optimisation" ~= "derivative-based optimisation"
  "multi-objective optimisation" ~= "multiple-objective optimisation"
  "Pareto front" ~= "non-dominated solution front"
  "robust optimisation" ~= "uncertainty-aware design optimisation"
  "simulation-based optimisation" ~= "simulation-driven optimisation"

# CONFIDENCE
CONFIDENCE:
  ontology_domain: 0.99
  variable_objective_mapping: 0.99
  constraint_feasibility_mapping: 0.99
  algorithm_mapping: 0.99
  gradient_sensitivity_mapping: 0.99
  simulation_surrogate_mapping: 0.99
  multi_objective_mapping: 0.99
  robustness_mapping: 0.99
  verification_mapping: 0.99
  semantic_relationships: 0.99

# UPDATE MODEL
UPDATE_MODEL:
  ON new_optimisation_problem:
    MAP design_variables
    MAP bounds
    MAP objectives
    MAP constraints
    MAP engineering_model
    MAP algorithm
    UPDATE confidence
    RECOMPUTE FLOW_HASH

  ON candidate_evaluation:
    UPDATE objective_values
    UPDATE constraint_values
    UPDATE feasibility
    UPDATE iteration_history
    PRESERVE provenance

  ON model_or_constraint_change:
    UPDATE optimisation_problem
    INVALIDATE affected_results
    REASSESS feasible_region
    RECOMPUTE optimisation

  ON final_solution:
    REEVALUATE authoritative_model
    VERIFY constraints
    VERIFY objectives
    UPDATE robustness_state
    PRESERVE provenance

# AI CONSUMPTION
AI_CONSUMPTION:
  parse_as: FORMAL_ENGINEERING_OPTIMISATION_ONTOLOGY
  reasoning_mode: CONSTRAINED_DESIGN_SEARCH_RELATIONAL

  preserve:
    - typed_entities
    - design_variables
    - parameter_values
    - objective_definitions
    - constraints
    - feasible_state
    - algorithm_state
    - iteration_history
    - uncertainty
    - confidence
    - provenance

  infer:
    - feasibility
    - objective_dependencies
    - constraint_activity
    - sensitivity_dependencies
    - convergence_state
    - Pareto_relations
    - robustness_requirements
    - final_verification_requirements

  prohibit:
    - invented_objective_values
    - assumed_feasibility
    - assumed_global_optimality
    - fabricated_gradients
    - unsupported_robustness
    - unverified_final_design_presented_as_valid

# FLOW HASH
FLOW_HASH_INPUT :=
  DOMAIN + ONTOLOGY + CORE_ENTITIES
  + OPTIMISATION_PROBLEM + DESIGN_VARIABLES
  + DESIGN_SPACE + OBJECTIVES + CONSTRAINTS
  + METHODS + GRADIENT_METHODS + GLOBAL_METHODS
  + SENSITIVITY + SIMULATION_BASED + SURROGATE_MODEL
  + MULTI_OBJECTIVE + ROBUSTNESS + RELIABILITY
  + DESIGN_SCOPE + CONVERGENCE + OPTIMALITY
  + VERIFICATION + UNCERTAINTY + PROVENANCE
  + PERFORMANCE + REASONING + INFERENCE_RULES
  + SEMANTIC_RELATIONSHIPS + SEMANTIC_EQUIVALENCE
  + CONFIDENCE + UPDATE_MODEL + AI_CONSUMPTION

FLOW_HASH := HASH(FLOW_HASH_INPUT)

END FLOW