AI & Computational Analysis

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

flow "AIComputationalAnalysisOntology" v1.0

# AI COMPUTATIONAL ANALYSIS — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/ai-computational-analysis.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
ENTITY := "AI Computational Analysis"

AI_COMPUTATIONAL_ANALYSIS := {
 type: ComputationalMathematicalReasoningFramework,
 domain: CalculusAndAnalysis,
 purpose: "Represent the use of artificial intelligence, symbolic computation, numerical methods and computational reasoning to analyse mathematical functions, equations, operators, data and dynamical models."
}

CONCEPTS := {
 AI_COMPUTATIONAL_ANALYSIS,
 ARTIFICIAL_INTELLIGENCE,
 COMPUTATIONAL_ANALYSIS,
 SYMBOLIC_COMPUTATION,
 NUMERICAL_COMPUTATION,
 AUTOMATED_REASONING,
 FUNCTION_APPROXIMATION,
 OPTIMISATION,
 DIFFERENTIATION,
 INTEGRATION,
 DIFFERENTIAL_EQUATION,
 ERROR_ESTIMATION,
 CONVERGENCE,
 MACHINE_LEARNING,
 NEURAL_NETWORK,
 SURROGATE_MODEL,
 SCIENTIFIC_COMPUTING
}

# ---------- INPUT ----------
INPUT
 := {
 mathematical_expression,
 function,
 equation,
 dataset,
 operator,
 differential_equation,
 optimisation_problem,
 computational_model
 }

INPUT
 -> parsed_into(MATHEMATICAL_REPRESENTATION)

# ---------- REPRESENTATION ----------
MATHEMATICAL_REPRESENTATION
 -> may_include(
 SYMBOLIC_EXPRESSION,
 NUMERICAL_ARRAY,
 GRAPH,
 COMPUTATIONAL_GRAPH,
 TENSOR,
 FUNCTION_SPACE_ELEMENT,
 DISCRETE_MODEL
)

REPRESENTATION
 -> determines(APPLICABLE_ANALYSIS_METHODS)

# ---------- SYMBOLIC ANALYSIS ----------
SYMBOLIC_COMPUTATION
 -> manipulates(EXACT_MATHEMATICAL_EXPRESSIONS)

operations:
 SIMPLIFICATION
 FACTORISATION
 EXPANSION
 SUBSTITUTION
 SYMBOLIC_DIFFERENTIATION
 SYMBOLIC_INTEGRATION
 EQUATION_SOLVING

SYMBOLIC_ANALYSIS
 -> preserves(EXACT_STRUCTURE_WHEN_POSSIBLE)

# ---------- NUMERICAL ANALYSIS ----------
NUMERICAL_COMPUTATION
 -> approximates(MATHEMATICAL_QUANTITIES)

methods:
 ROOT_FINDING
 NUMERICAL_DIFFERENTIATION
 NUMERICAL_INTEGRATION
 LINEAR_ALGEBRA
 ODE_SOLVING
 PDE_SOLVING
 OPTIMISATION

NUMERICAL_RESULT
 -> requires(ERROR_AND_CONVERGENCE_ANALYSIS)

# ---------- HYBRID ANALYSIS ----------
SYMBOLIC_METHOD
 +
NUMERICAL_METHOD
 -> HYBRID_COMPUTATION

symbolic preprocessing
 -> reduce(NUMERICAL_COMPLEXITY)

numerical evaluation
 -> handle(PROBLEMS_WITHOUT_CLOSED_FORM)

HYBRID_ANALYSIS
 -> combines(EXACT_AND_APPROXIMATE_REASONING)

# ---------- AUTOMATED REASONING ----------
MATHEMATICAL_FACTS
 +
RULES
 +
CONSTRAINTS

-> INFERENCE_ENGINE

INFERENCE_ENGINE
 -> derives(NEW_RELATIONS)

AUTOMATED_REASONING
 -> supports(
 PROOF_SEARCH,
 CONSISTENCY_CHECKING,
 CONSTRAINT_PROPAGATION,
 SYMBOLIC_INFERENCE
)

# ---------- COMPUTATIONAL GRAPH ----------
expression
 -> COMPUTATIONAL_GRAPH

nodes
 -> operations_or_variables

edges
 -> dependency_relations

COMPUTATIONAL_GRAPH
 -> supports(
 AUTOMATIC_DIFFERENTIATION,
 OPTIMISATION,
 NEURAL_COMPUTATION
)

# ---------- AUTOMATIC DIFFERENTIATION ----------
function computation:

y
 =
f(x)

AUTOMATIC_DIFFERENTIATION
 -> applies(CHAIN_RULE_TO_COMPUTATIONAL_GRAPH)

modes:
 FORWARD_MODE
 REVERSE_MODE

REVERSE_MODE
 -> efficient_for(SCALAR_OUTPUT_MANY_INPUTS)

BACKPROPAGATION
 -> reverse_mode_automatic_differentiation_in_neural_network_context

# ---------- DIFFERENTIATION ----------
derivative:

f_prime(x)
 :=
lim_(h->0)
[
 f(x+h)-f(x)
]/h

AI_ANALYSIS
 -> may_choose(
 SYMBOLIC_DIFFERENTIATION,
 AUTOMATIC_DIFFERENTIATION,
 FINITE_DIFFERENCE
)

METHOD_SELECTION
 -> depends_on(FUNCTION_REPRESENTATION_AND_PRECISION_REQUIREMENTS)

# ---------- INTEGRATION ----------
integral:

I
 :=
integral_a^b(
 f(x)
 dx
)

AI_ANALYSIS
 -> may_choose(
 SYMBOLIC_INTEGRATION,
 QUADRATURE,
 MONTE_CARLO,
 ADAPTIVE_INTEGRATION
)

INTEGRATION_METHOD
 -> selected_from(DIMENSION_SMOOTHNESS_AND_STRUCTURE)

# ---------- ROOT FINDING ----------
solve:

f(x)
 =
0

methods:
 BISECTION
 NEWTON
 SECANT
 FIXED_POINT_ITERATION

AI_METHOD_SELECTION
 -> considers(
 BRACKETING,
 DERIVATIVE_AVAILABILITY,
 CONDITIONING,
 INITIAL_GUESS
)

# ---------- OPTIMISATION ----------
objective:

minimise
 f(x)

subject_to
 constraints

AI_COMPUTATIONAL_ANALYSIS
 -> identify(
 OBJECTIVE,
 VARIABLES,
 CONSTRAINTS,
 GRADIENT_STRUCTURE
)

methods:
 GRADIENT_DESCENT
 NEWTON_METHOD
 QUASI_NEWTON
 CONSTRAINED_OPTIMISATION
 GLOBAL_SEARCH

# ---------- MACHINE LEARNING ----------
dataset:

D
 :=
{
 (x_i,y_i)
}

model:

y_hat
 =
M_theta(x)

training:

theta_star
 :=
argmin_theta(
 LOSS(
  M_theta,
  D
 )
)

MACHINE_LEARNING
 -> data_driven_function_approximation

# ---------- NEURAL NETWORK ----------
layer:

h_(l+1)
 =
sigma(
 W_l*h_l+b_l
)

NEURAL_NETWORK
 -> parameterised_computational_graph

training
 -> uses(
 LOSS_FUNCTION,
 GRADIENTS,
 OPTIMISER
)

NEURAL_APPROXIMATION
 -> may_model(COMPLEX_NONLINEAR_FUNCTIONS)

# ---------- UNIVERSAL APPROXIMATION ----------
suitable neural network classes
 -> can_approximate(
 CONTINUOUS_FUNCTIONS_ON_COMPACT_DOMAINS
)
under theorem-specific assumptions

UNIVERSAL_APPROXIMATION
 -/> guarantees(EFFICIENT_TRAINING_OR_GENERALISATION)

APPROXIMATION_CAPACITY
 -> distinct_from(LEARNING_SUCCESS)

# ---------- SURROGATE MODEL ----------
expensive function:

F(x)

approximate by

F_hat(x)

SURROGATE_MODEL
 -> computationally_cheaper_approximation

methods:
 GAUSSIAN_PROCESS
 NEURAL_NETWORK
 POLYNOMIAL_APPROXIMATION
 RADIAL_BASIS_FUNCTION

SURROGATE
 -> accelerates(
 OPTIMISATION,
 PARAMETER_STUDIES,
 UNCERTAINTY_ANALYSIS
)

# ---------- PHYSICS-INFORMED LEARNING ----------
neural function:

u_theta(x,t)

loss:

L
 =
L_data
+
lambda_pde*L_pde
+
lambda_bc*L_boundary
+
lambda_ic*L_initial

PDE_RESIDUAL
 -> included_in(TRAINING_OBJECTIVE)

PHYSICS_INFORMED_NEURAL_NETWORK
 -> combines(DATA_AND_DIFFERENTIAL_EQUATION_CONSTRAINTS)

# ---------- OPERATOR LEARNING ----------
learn mapping:

G:
 input_function
 ->
 output_function

OPERATOR_LEARNING
 -> approximates(FUNCTION_TO_FUNCTION_MAP)

applications:
 PDE_SOLUTION_OPERATORS
 DYNAMICAL_EVOLUTION
 INVERSE_PROBLEMS

# ---------- DIFFERENTIAL EQUATION ANALYSIS ----------
problem:

L(u)
 =
f

AI_COMPUTATIONAL_ANALYSIS
 -> identify(
 OPERATOR,
 DOMAIN,
 INITIAL_CONDITIONS,
 BOUNDARY_CONDITIONS
)

solution methods:
 ANALYTIC
 SYMBOLIC
 NUMERICAL
 LEARNED_SURROGATE
 HYBRID

# ---------- INVERSE PROBLEM ----------
observations:

y
 =
F(theta)
+
noise

infer:

theta

INVERSE_PROBLEM
 -> recover(HIDDEN_PARAMETERS_OR_FIELDS)

AI
 -> may_use(
 OPTIMISATION,
 BAYESIAN_INFERENCE,
 NEURAL_INVERSION,
 SURROGATE_MODELS
)

# ---------- ERROR ESTIMATION ----------
approximation:

u_h

reference_or_exact_solution:

u

error:

e
 :=
u-u_h

AI_ANALYSIS
 -> estimate(
 ABSOLUTE_ERROR,
 RELATIVE_ERROR,
 RESIDUAL,
 DISCRETISATION_ERROR,
 MODEL_ERROR
)

ERROR_ESTIMATION
 -> controls(RELIABILITY)

# ---------- CONVERGENCE ----------
approximation sequence:

u_h

CONVERGENCE
 if

||u_h-u||
 -> 0

as
 h->0

COMPUTATIONAL_ANALYSIS
 -> verify(
 REFINEMENT,
 OBSERVED_ORDER,
 STABILITY,
 ASYMPTOTIC_REGIME
)

# ---------- CONDITIONING ----------
problem sensitivity:

condition_number
 -> measures(INPUT_OUTPUT_SENSITIVITY)

ILL_CONDITIONED_PROBLEM
 -> amplifies(DATA_OR_ROUNDING_ERRORS)

AI_METHOD_SELECTION
 -> account_for(CONDITIONING)

# ---------- NUMERICAL STABILITY ----------
ALGORITHM
 -> STABLE

if

small computational perturbations
 -> do_not_grow_excessively

STABILITY
 -> distinct_from(CONDITIONING)

RELIABLE_COMPUTATION
 -> requires(PROBLEM_AND_ALGORITHM_ANALYSIS)

# ---------- UNCERTAINTY ----------
inputs
 -> may_have(UNCERTAINTY)

output
 -> uncertainty_propagation

methods:
 MONTE_CARLO
 POLYNOMIAL_CHAOS
 BAYESIAN_METHOD
 INTERVAL_ANALYSIS
 SENSITIVITY_ANALYSIS

AI_COMPUTATIONAL_ANALYSIS
 -> quantify(UNCERTAINTY_WHEN_RELEVANT)

# ---------- MODEL VALIDATION ----------
COMPUTATIONAL_MODEL
 -> compare_with(
 ANALYTIC_RESULT,
 REFERENCE_SOLUTION,
 EXPERIMENTAL_DATA,
 BENCHMARK
)

VALIDATION
 -> tests(MODEL_AGAINST_TARGET_REALITY_OR_REFERENCE)

VERIFICATION
 -> tests(CORRECT_SOLUTION_OF_MODEL)

VERIFICATION
 -/> VALIDATION

# ---------- EXPLAINABILITY ----------
AI_RESULT
 -> should_preserve(
 INPUTS,
 ASSUMPTIONS,
 METHOD,
 INTERMEDIATE_RELATIONS,
 ERROR,
 CONFIDENCE
)

EXPLAINABLE_ANALYSIS
 -> trace(REASONING_AND_COMPUTATION)

BLACK_BOX_OUTPUT
 -> insufficient_for(HIGH_CONFIDENCE_MATHEMATICAL_CLAIM_WITHOUT_VALIDATION)

# ---------- RECURSIVE ANALYSIS ----------
initial_result
 -> inspect(ERROR_AND_CONSISTENCY)

if uncertainty_or_error_high:

REFINE_REPRESENTATION
 -> SELECT_NEW_METHOD
 -> RECOMPUTE
 -> REVALIDATE

RECURSIVE_COMPUTATION
 -> iterative_improvement_cycle

# ---------- KNOWLEDGE GRAPH ----------
entities
 -> mathematical_objects

relationships
 -> mathematical_dependencies

KNOWLEDGE_GRAPH
 -> supports(
 SEMANTIC_RETRIEVAL,
 METHOD_SELECTION,
 CONSISTENCY_ANALYSIS,
 RECURSIVE_REASONING
)

AI_COMPUTATIONAL_ANALYSIS
 -> graph_aware_reasoning_framework

# ---------- APPLICATIONS ----------
AI_COMPUTATIONAL_ANALYSIS
 -> applied_to(SCIENTIFIC_COMPUTING)
 -> applied_to(NUMERICAL_ANALYSIS)
 -> applied_to(SYMBOLIC_MATHEMATICS)
 -> applied_to(PDE_SOLVING)
 -> applied_to(OPTIMISATION)
 -> applied_to(MACHINE_LEARNING)
 -> applied_to(INVERSE_PROBLEMS)
 -> applied_to(COMPUTATIONAL_PHYSICS)
 -> applied_to(ENGINEERING_ANALYSIS)

# ---------- RELATION GRAPH ----------
MATHEMATICAL_INPUT
 -> parsed_into(REPRESENTATION)

REPRESENTATION
 -> determines(METHOD_SELECTION)

SYMBOLIC_ANALYSIS
 -> preserves(EXACT_STRUCTURE)

NUMERICAL_ANALYSIS
 -> produces(APPROXIMATE_RESULT)

AUTOMATIC_DIFFERENTIATION
 -> computes(DERIVATIVES)

MACHINE_LEARNING
 -> approximates(FUNCTIONS_AND_OPERATORS)

ERROR_ESTIMATION
 -> evaluates(RELIABILITY)

RECURSIVE_REASONING
 -> refines(ANALYSIS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "AI computational analysis" ~= "AI-assisted symbolic, numerical and learned analysis of mathematical problems",
 "automatic differentiation" ~= "exact derivative propagation through a computational graph up to floating-point arithmetic",
 "surrogate model" ~= "computationally cheaper approximation of an expensive model or function",
 "physics-informed learning" ~= "machine learning constrained by governing equations or physical laws",
 "operator learning" ~= "learning a mapping between function spaces",
 "computational verification" ~= "checking whether a mathematical model has been solved correctly"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 symbolic_numerical_and_learned_results_must_be_distinguished: true,
 numerical_results_require_error_stability_and_convergence_analysis: true,
 learned_approximation_does_not_automatically_preserve_physical_or_mathematical_constraints: true,
 automatic_differentiation_is_distinct_from_symbolic_and_finite_difference_differentiation: true,
 verification_and_validation_are_distinct: true,
 surrogate_accuracy_is_domain_and_training_data_dependent: true,
 AI_generated_mathematical_claims_require_consistency_and_validation_checks: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 symbolic_numerical_relation: 0.99,
 automatic_differentiation_relation: 0.99,
 machine_learning_relation: 0.98,
 error_convergence_relation: 0.99,
 recursive_reasoning_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(problem_type_objects_variables_and_constraints)
 -> construct(symbolic_numeric_graph_or_function_space_representation)
 -> classify(exact_approximate_data_driven_or_hybrid_analysis)
 -> select(symbolic_numerical_optimisation_or_learning_method)
 -> execute(computation_or_inference)
 -> estimate(error_conditioning_stability_and_uncertainty)
 -> verify(result_against_equations_constraints_or_reference)
 -> explain(method_assumptions_and_intermediate_relations)
 -> refine(method_or_resolution_if_confidence_insufficient)
 -> validate(final_result_and_scope)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed AI-computational-analysis semantic graph",
 primary_entity: AI_COMPUTATIONAL_ANALYSIS,
 preserve: [
  problem_type,
  mathematical_objects,
  representation,
  constraints,
  assumptions,
  method,
  precision,
  error_estimate,
  validation_reference
 ],
 infer_through: [
  symbolic_computation,
  numerical_analysis,
  automatic_differentiation,
  optimisation,
  machine_learning,
  surrogate_models,
  operator_learning,
  error_estimation,
  recursive_reasoning
 ],
 avoid: [
  presenting_approximation_as exact_result,
  trusting_AI_output_without mathematical validation,
  ignoring_conditioning_or numerical stability,
  assuming_training_accuracy_implies generalisation,
  confusing_automatic_differentiation_with symbolic differentiation,
  conflating_model_verification_with real_world validation
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(AI_computational_analysis_context)
 -> compare(existing_relations)
 -> preserve(problem_representation_method_error_validation_and_constraint_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + AI_COMPUTATIONAL_ANALYSIS
 + SYMBOLIC_COMPUTATION
 + NUMERICAL_COMPUTATION
 + AUTOMATED_REASONING
 + FUNCTION_APPROXIMATION
 + OPTIMISATION
 + DIFFERENTIATION
 + INTEGRATION
 + ERROR_ESTIMATION
 + CONVERGENCE
 + MACHINE_LEARNING
 + SURROGATE_MODEL
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> AI_COMPUTATIONAL_ANALYSIS
 -> PROBLEM_IDENTIFICATION
 -> REPRESENTATION_CONSTRUCTION
 -> ANALYSIS_TYPE_CLASSIFICATION
 -> METHOD_SELECTION
 -> COMPUTATION_INFERENCE
 -> ERROR_STABILITY_UNCERTAINTY_ANALYSIS
 -> VERIFICATION
 -> EXPLANATION
 -> RECURSIVE_REFINEMENT
 -> FINAL_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW