ML-Enhanced Greybox Framework

Updated 3 January 2026

Machine-learning-enhanced greybox frameworks are hybrids that integrate interpretability of whitebox models with the adaptability of neural networks.
They embed neural, GP, and ensemble methods into simulation and optimization pipelines to improve efficiency and predictive accuracy.
Applications in software testing, physical simulations, and optimal control yield measurable gains in code coverage, runtime, and model generalization.

A machine-learning-enhanced greybox framework integrates data-driven models, particularly neural networks, into classical modeling or search pipelines, enabling principled exploitation of both domain knowledge and statistical learning. These frameworks have emerged across multiple domains—including software testing, physical systems modeling, simulation acceleration, and optimization—delivering marked increases in code coverage, generalizability, efficiency, and interpretability.

1. Core Principles and Taxonomy

Machine-learning-enhanced greybox frameworks achieve an overview between whitebox (first-principles, interpretable) and blackbox (data-driven, unconstrained) modeling. Architectures are constructed by embedding neural or statistical learning modules at critical points within a traditionally hand-engineered computation, such as simulation loops, controller policies, search heuristics, or physical equations.

Key variants include:

ML-accelerated input generation or mutation: Greybox fuzzing with LLM-based or bandit-driven mutators (Zhang et al., 2024, Patil et al., 2018, Karamcheti et al., 2018, Wüstholz et al., 2018).
Hybrid physical–neural simulation: DNNs replace sub-blocks of a physical system, sharing state variables and embedded in the solver loop (Agarwal et al., 2024).
Augmented physical models: Neural or GP surrogates learn unmodeled residuals or operators in PDE, ODE, or optimal control flows (Kag et al., 2024, Garg et al., 2021, Cantone et al., 30 Dec 2025).
Statistical surrogate-based optimization: Gaussian-process (GP) surrogates for constituent models, composite objectives, and multi-fidelity or partial observation cases (Hameed et al., 1 Sep 2025, Astudillo et al., 2022).
Ensemble (mixture-of-experts): Grey and blackbox components are interpolated or mixed, often with explicit regularization and gating for interpretability (Leoni et al., 2024).
Neural-symbolic explainable models: Symbolic rule-based layers enforced on top of neural representations for self-explaining predictions (Bennetot et al., 2022).

2. Framework Structures and Learning Integration

a. Model Formulation

Simulation/Physical Models: ML components approximate missing operators $\mathcal{N}$ in PDEs/ODEs (e.g., $u_t + \hat{\mathcal{N}} = 0$ ), or learn correction maps for model-form error as in greybox dynamical systems (Kag et al., 2024, Garg et al., 2021).
Optimization: GP or DNN surrogates interpolate expensive or partially known functions, with explicit consideration for partial or multi-fidelity information (Hameed et al., 1 Sep 2025, Astudillo et al., 2022).
Search/Mutation: Neural policies or bandits assign resource allocation (e.g., fuzzing "energy") or generate new candidate inputs, guided by reward/uncertainty or structural priors (Patil et al., 2018, Zhang et al., 2024, Karamcheti et al., 2018, Wüstholz et al., 2018).

b. ML Components and Training

Neural Networks: LSTMs, DNNs, or transformers operate on contextually relevant inputs (substrings, state windows, system parameters). Supervised or policy-gradient-based (REINFORCE) approaches are trained on coverage, residual reward, or predictive accuracy (Patil et al., 2018, Agarwal et al., 2024, Cantone et al., 30 Dec 2025).
Gaussian Processes: Used for direct correction of dynamics or residual forces, incorporating a kernel structure over system states (Garg et al., 2021).
Mixture-of-experts: Each expert is often a grey model or data-driven regressor; the gating function is optimized under loss and regularizations for smoothness and interpretability (Leoni et al., 2024).
Surrogate Models: Hierarchies of surrogates (low- and high-fidelity) are swapped based on trust-region agreement and predictive error, extending to TS, GP, and hybrids (Hameed et al., 1 Sep 2025).

3. Algorithmic Workflow and Integration

Physical–ML Composition: ML models may be embedded via residual connection, replacement of high-cost submodels, or parallel ensembles. In simulation, explicit state-sharing permits backpropagation of sensitivities for solver Jacobians (Newton–Raphson or DAEs), crucial for convergence guarantees and enforcement of global constraints (Agarwal et al., 2024).
Search Guidance: Contextual bandit or entropy-based ML models bias resource allocation (energy, test-case priority) or candidate execution selection within a feedback loop, maximizing coverage and bug-trigger rate in fuzzing (Patil et al., 2018, Karamcheti et al., 2018).
Greybox BO: Partial constituent observations are incorporated directly in GP posteriors; acquisition functions are designed to maximize expected improvement or knowledge gain with respect to composite objectives or at multiple fidelities (Astudillo et al., 2022).
Mixture-of-experts Training: Alternating minimization algorithms optimize local expert parameters and global mixture weights, with convex gating updates subject to smoothness penalties (Leoni et al., 2024).
Surrogate-based Optimization: Trust region subproblems are solved within ellipsoid constraints adaptive to Hessian or surrogate curvature, with step acceptance overseen via a filter and ratio-based update mechanism (Hameed et al., 1 Sep 2025).

4. Empirical Performance and Domain Applications

a. Software Testing (Fuzzing)

Framework/Paper	Coverage/Crashes vs Baseline	Key Mechanism
LLAMAFUZZ (Zhang et al., 2024)	$+41$ bugs, $+27.19\%$ branches	LLM-based mutation in async with AFL++
Contextual Bandit Fuzzing (Patil et al., 2018)	Matches/exceeds AFL on code coverage in several binaries	LSTM policy for energy allocation
ML-Guided Fuzzing (Karamcheti et al., 2018)	$+13$ bugs, $+23.7\%$ paths (avg, 3h)	Logistic regression and entropy selection
Greybox Input Learning (Wüstholz et al., 2018)	Up to $3\times$ more coverage, $+38\%$ more bugs	Analytic affine root-finding

b. Physics and Engineering Modeling

Greybox DHPM: Parametric operator networks generalize over unseen $f(x)$ , system parameters, and domain lengths, maintaining low $L_2$ errors ( $\approx1.9\times10^{-2}$ ) on PDE test cases, with no retraining for in-distribution variations (Kag et al., 2024).
Physics-Integrated Hybrid (GP): Residual-forces are learned via GPs and injected, improving predictions by an order of magnitude and generalizing robustly to new load scenarios (Garg et al., 2021).
DNN-Based Hybrid Simulation: DNN macromodels yield up to $38.4\%$ state-space and $19\%$ runtime reduction with voltage errors $<5.9\%$ in large power network cases (Agarwal et al., 2024).
Optimal Quantum Control Greybox: Transformer-augmented models achieve $>99\%$ single-qubit gate fidelity for weak noise and $>90\%$ in strong non-Markovian noise, outperforming whitebox and naive blackbox decouplings (Cantone et al., 30 Dec 2025).

c. Optimization

Trust-Region Filter with ML Surrogates: Spectral-Hessian adaptive variants reduce average blackbox evaluations and iterations by up to an order of magnitude in process design, while requiring minimal manual tuning (Hameed et al., 1 Sep 2025).
Greybox Bayesian Optimization: Composite and multi-fidelity objectives are efficiently optimized by exploiting intermediate outputs, partial observations, and GP posterior structure (Astudillo et al., 2022).

d. Model Interpretability, Language, and Symbolic Integration

Explainable Mixture-of-Experts: Grey–grey ensembles outperform hybrid or blackbox ensembles for time-varying system identification, with gating directly interpretable in domain terms (Leoni et al., 2024).
Neural-Symbolic XAI: Segmentation-attribute/KB-extraction paired with interpretable classifiers (logistic regression) yield state-of-the-art compositional image classifiers with perfect explanation faithfulness (Bennetot et al., 2022).
Greybox Active Learning of ERA: Structural knowledge of timings and regions is integated to efficiently infer DFA for timed languages; model output is dramatically more compact and interpretable than zone-automata-based learners (Majumdar et al., 2024).

5. Limitations and Open Challenges

Partial Greybox Replacement: Many greybox approaches currently replace only a subset of heuristics or physical submodels, leaving other system components hand-coded or suboptimally integrated (Patil et al., 2018, Agarwal et al., 2024).
Transfer and Generalization: Some frameworks tie generalization ability to training data domain coverage, with performance degrading on out-of-distribution inputs unless post-processing or symbolic regression is introduced (Kag et al., 2024, Cantone et al., 30 Dec 2025).
Interpretability vs. Performance: Blackbox neural components remain difficult to interpret unless paired with symbolic mixing or explicit knowledge bases (Leoni et al., 2024, Bennetot et al., 2022).
Scalability: Run-time complexity, memory overhead of gradient/backprop in simulation, and sensitivity to DNN errors outside training support remain limiting factors for very large-scale systems (Agarwal et al., 2024).
Hyperparameter Tuning and Surrogate Fidelity: Quality of surrogate-based optimization frameworks depends on adaptive switching and local fit accuracy, though recent spectral–Hessian methods have substantially reduced tuning effort (Hameed et al., 1 Sep 2025).
Optimality and Convergence: Policy-gradient approaches for energy allocation do not implement baseline correction or actor-critic stabilization, leaving variance and optimality improvements as future work (Patil et al., 2018).

6. Outlook and Future Directions

End-to-End Differentiable Hybrids: Embedding adaptive online-updated neural networks as internal nodes within physical solvers remains an open direction for maximizing transfer, sample efficiency, and physical constraint enforcement (Agarwal et al., 2024).
Surrogate-Driven and Multi-fidelity Extensions: Further work on dynamic fidelity-switching, cost-aware acquisition, and learning hybrid residuals for more complex objectives will improve process optimization and exploration efficiency (Hameed et al., 1 Sep 2025, Astudillo et al., 2022).
Compositional and Modular Self-Explanation: Machine-learning-enhanced greybox models will increasingly exploit explicit symbolic bases, interpretable gating, and knowledge graphs to ensure both transparency and adaptability (Bennetot et al., 2022, Leoni et al., 2024).
Automated Model Architecture Selection: Determining which submodules to treat as grey, black, or whitebox in large systems is a key open research area, with implications for data-efficiency and model-bias control (Gupta et al., 2019, Kag et al., 2024).
Integration in Active Learning and Synthesis: Methods such as greybox active automata learning exploit domain constraints to reduce sample/query complexity and yield highly interpretable models for system identification and verification (Majumdar et al., 2024).