BACO Framework

Extracted documentation from src/mdotoolbox/frameworks/baco.py.

Overview

Key Innovations

• GP surrogates for subsystem objectives (J_i) and constraints (g_i)
  • Acquisition function optimization (Expected Improvement)
  • Latin Hypercube Sampling for efficient DoE initialization
  • Reduced subsystem evaluations through surrogate modeling

Mathematical Formulation

System Level: (same as CO) min_{z,x_bar,y_bar} f(z, x_bar, y_bar) s.t. g_sys(z, x_bar, y_bar) >= 0 J_i <= epsilon for all i

Subsystem Level i: (Bayesian optimization)
    - Build GP surrogates: mu_{J_i}(z_under_i, x_under_i), mu_{g_i}(z_under_i, x_under_i)
    - Optimize acquisition: max alpha_{EI}[mu_{J_i}] s.t. mu_{g_i} >= 0
    - Update DoE with new evaluation
    - Retrain surrogates

Advantages

• Reduced subsystem calls (5-10x fewer than CO)
  • Better for expensive analysis codes
  • Uncertainty quantification via GP variance
  • Adaptive sampling focuses on promising reg_ions

Configuration

• SMTGPConfig: GP hyperparameters, kernel settings
  • DoE size: Initial sampling (typically 5-20 points per subsystem)
  • Acquisition function: Expected Improvement (log_ei)

BACOSubsystem

Attributes

problem: Problem Local subsystem problem definition z_idxs: ndarray Indices of shared design variables x_idxs: ndarray Indices of local design variables y_idxs: ndarray Indices of subsystem outputs y_coupled_idxs: List[ndarray] Coupled variable indices from other subsystems surrogate_config: SMTGPConfig GP configuration (kernel, hyperparameters) doe_J_i: DoE Design of Experiments for discrepancy J_i doe_g_i: DoE Design of Experiments for constraints g_i gp_J_i: GP model Trained surrogate for J_i objective gp_g_i: dict Trained surrogates for each constraint {name: GP} z_under_i, x_under_i, y_i: ndarray Current subsystem variable values best_J_i, best_h: float Best achieved discrepancy and constraint violation

Workflow: 1. initialize_doe(): LHS sampling to create initial DoE 2. build_surrogate_J_i(), build_surrogate_g_i(): Train GP surrogates 3. solve_acquisition(): Optimize acquisition function 4. Update DoE and retrain (repeat)

initialize_doe

Notes

• Uses Latin Hypercube for space-filling design
  • Samples from subsystem variable bounds [z_under_i, x_under_i]
  • Evaluates actual discipline for each sample
  • Forms baseline for GP training

Example: >>> subsystem.initialize_doe( ... n_samples=20, ... z_bar=np.array([1.0, 2.0]), ... x_bar=np.array([0.5]), ... y_bar=np.array([1.0, 1.5]) ... )

build_surrogate_J_i

Notes

• Must call initialize_doe() first
  • GP trained on all DoE samples
  • Hyperparameters optimized during training
  • Model ready for predict_values() calls

Example: >>> subsystem.initialize_doe(n_samples=20, z_bar, x_bar, y_bar) >>> subsystem.build_surrogate_J_i() >>> # Now can predict: subsystem.gp_J_i.predict_values(X_new)

build_surrogate_g_i

Notes

• Must call initialize_doe() first
  • One GP per constraint in self.problem.constraints
  • Each GP predicts g_i(z_under_i, x_under_i, y_coupled)
  • Used to enforce constraints in acquisition optimization

Example: >>> subsystem.build_surrogate_g_i() >>> # Predict constraint 'stress' at new point >>> X_test = np.array([[1.0, 2.0, 3.0]]) >>> g_pred = subsystem.gp_g_i['stress'].predict_values(X_test)

solve_acquisition

Notes

• Acquisition maximized = minimize negative acquisition
  • GP mean predictions used for constraint handling
  • True function evaluated at optimal acquisition point
  • DoE automatically updated for next GP training

Example: >>> from mdotoolbox.surrogates import log_ei >>> subsystem.solve_acquisition( ... z_bar=np.array([1.0, 2.0]), ... x_bar=np.array([0.5]), ... y_bar=np.array([1.0, 1.5]), ... optimizer='cobyqa', ... acq_func=log_ei, ... n_multistart=20 ... )