Adaptive LLM mutation operators in AdaEvolve can be guided by uncertainty estimates from reduced-order models to focus evolutionary search on structurally informative regions.

In evolutionary optimization of physics-informed models (e.g., PDE-constrained inverse design, multiphysics parameter search), an AdaEvolve variant that biases LLM-generated mutation/crossover proposals toward regions of high predictive uncertainty in a reduced-order model (ROM) surrogate — as estimated via ensemble disagreement or Gaussian-process posterior variance — will reach a target fitness threshold (e.g., within 5% of best-known optimum) using at least 30% fewer full-fidelity simulation evaluations than: (a) vanilla AdaEvolve with unguided/random LLM mutations, and (b) standard uncertainty-agnostic evolutionary strategies (CMA-ES, NSGA-II) of matched simulation budget, on a benchmark suite of ≥3 physics problems (e.g., airfoil shape optimization, heat-diffusion inverse parameter fitting, structural topology optimization).

Minimum viable test (MVT): single benchmark problem (2D heat-diffusion inverse parameter estimation, 8-parameter design space), comparing 3 conditions × 10 random seeds: (1) AdaEvolve + uncertainty-guided LLM mutation (proposed) (2) AdaEvolve + unguided LLM mutation (ablation baseline) (3) CMA-ES with matched full-fidelity simulation budget (external baseline) Fixed compute budget: 500 full-fidelity solver calls per run. Success measured by best fitness achieved at budget exhaustion and by number of evaluations to reach 95%-of-optimal fitness (determined via long-run reference solve). Full validation extends to 3 benchmark problems with 20 seeds each and adds NSGA-II as a 4th baseline plus the "fake uncertainty" ablation.

# Pseudocode: Uncertainty-Guided AdaEvolve

initialize population P (size N) randomly in design space
train ROM_ensemble on initial low-fidelity samples (5 NNs or GP)

for generation in range(G):
    fitness = evaluate_full_fidelity(P)  # expensive solver calls (budget-tracked)
    uncertainty_map = ROM_ensemble.predict_variance(candidate_grid)
    high_uncertainty_regions = top_k(uncertainty_map, k=fraction * grid_size)

    # LLM mutation conditioning
    for parent in select_parents(P, fitness):
        prompt = build_prompt(
            parent_params=parent,
            fitness_history=fitness_log,
            focus_zones=high_uncertainty_regions,  # numeric uncertainty scores injected
            instruction="mutate parameters toward high-uncertainty, high-potential regions"
        )
        child = LLM.generate_mutation(prompt)  # structured numeric output
        offspring.append(child)

    P = select_survivors(P + offspring, fitness)
    if generation % retrain_interval == 0:
        ROM_ensemble.update(new_full_fidelity_samples)

    if budget_exhausted(full_fidelity_calls):
        break

return best_candidate(P)

# Ablation variants:
# - fake_uncertainty: shuffle high_uncertainty_regions labels before prompt injection
# - no_LLM heuristic: replace LLM.generate_mutation with Gaussian perturbation biased toward focus_zones