Hybrid Physics–ML Risk Framework

• Hybrid Physics–ML risk assessment frameworks are integrated systems that combine physical modeling and machine learning to quantify risks under uncertainty.
• They employ feature-level fusion, model blending, and physics-informed loss functions to enhance prediction accuracy and maintain physical consistency.
• Applications span infrastructure, safety-critical controls, and energy systems, with robust uncertainty quantification validated through rigorous cross-validation and ensemble methods.

A hybrid Physics–Machine Learning (Physics–ML) risk assessment framework combines explicit physical knowledge with statistical or deep learning models to provide robust, scalable, and interpretable risk quantification under uncertainty. These frameworks leverage the complementary strengths of physics-based modeling (structural fidelity, extrapolation, physical consistency) and data-driven learning (pattern discovery, flexibility, scalability), enabling high-confidence risk assessments in engineered, natural, and socio-economic domains characterized by limited data, complex interactions, or rare events.

1. Foundational Principles of Hybrid Physics–ML Risk Assessment

Hybrid Physics–ML frameworks are structured integrations of physical models—either first-principles equations, empirical laws, or mechanistic surrogates—with machine learning architectures capable of regression, classification, probabilistic estimation, or control. Core principles include:

• Feature-level fusion: Physics-based quantities (e.g., calibrated health parameters, de-noised virtual sensors) are input to data-driven models, enriching feature representations for prognostic or risk models (Chao et al., 2020).
• Model-level integration: Machine learning predictions are convexly blended with explicit physical models, often via weighted ensembles where the physical component regularizes or extrapolates ML predictions in under-sampled regimes (Kriuk, 2 Oct 2025).
• Physics-informed loss and constraints: Learning is guided by physics-based constraints (e.g., satisfaction of governing PDEs, monotonicity, conservation laws) imposed as soft or hard losses in the ML objective (Wang et al., 2023, Hoshino et al., 25 Mar 2024, Sreenath et al., 26 Aug 2025).
• Uncertainty quantification and calibratable generalization: By leveraging physical structure alongside data, these frameworks quantify and propagate both epistemic and aleatoric uncertainty even in regions where training data is scarce (Kriuk, 2 Oct 2025, Wang et al., 2023).

2. Mathematical and Algorithmic Structures

Hybrid frameworks manifest in diverse algorithmic forms. The following summarizes typical approaches:

Structure	Physics Component	ML Component
Feature Augmentation	Calibrated health parameters, virtual sensors (Chao et al., 2020)	FNN, CNN, ensemble regressors
Model Blending	Physical sensitivity laws (Kriuk, 2 Oct 2025)	Random Forest, GBM, Elastic Net
PINNs (Physics-Informed NNs)	Governing-risk PDE (Wang et al., 2023, Hoshino et al., 25 Mar 2024, Wang et al., 11 Jul 2024)	Fully connected DNN, Q-networks
Generalized Additive Models	Basis from physics of hazard (Sreenath et al., 26 Aug 2025)	Additive NN pathways with constrained weights

Physics-constrained neural surrogates: PINNs enforce partial differential equations for risk probabilities, enabling direct estimation of long-term risk and its gradients. For example, the steady-state or finite-horizon reach-avoid probability $F(x,T)$ for an SDE:

$$\frac{\partial F}{\partial T}(x,T) - f(x,u) \nabla_x F(x,T) - \frac{1}{2} \mathrm{tr}(\sigma^2 \nabla^2_x F(x,T)) = 0$$

is encoded as a loss term in the DNN's training objective, in combination with empirical MC data (Wang et al., 2023, Wang et al., 11 Jul 2024).

Convex hybrid ensemble: Under extrapolation or future climate scenarios, the prediction model is a convex combination of the ML estimate $f_{ML}(x, \Delta T)$ and a physics-based response $f_{phys}(T_0, \Delta T)$:

$$f_{hybrid}(x, \Delta T) = w f_{ML}(x, \Delta T) + (1-w) f_{phys}(T_0, \Delta T), \quad w \in [0,1]$$

as in permafrost projection frameworks, where $f_{phys}$ encodes a physically justified permafrost decline rate (Kriuk, 2 Oct 2025).

3. Application Domains and Pipeline Architectures

Critical Infrastructure and Geophysical Risk

• Permafrost infrastructure risk: 2.9 million gridded observations across Arctic Russia are used to train an ML ensemble (RF+HGB+EN), whose predictions are hybridized with a physics-based permafrost sensitivity model ($-10$ pp/°C). Rigorous spatial-temporal cross-validation and scenario perturbation (e.g., RCP8.5 warming) yield actionable risk maps. Risk classes are assigned via fixed quantiles of loss magnitude and baseline exposure, and model uncertainty is mapped via the ensemble spread (Kriuk, 2 Oct 2025).
• Seismic ground motion: Generalized additive ML models, reflecting key seismological principles (magnitude scaling, geometric spreading, attenuation), are trained with hazard- and bin-balanced loss (HazBinLoss) to maintain physical interpretability and prevent underestimation in rare, high-impact regions (Sreenath et al., 26 Aug 2025).

Safety-Critical Stochastic Control

• Risk PDE surrogates: PIPE frameworks employ PINNs to learn risk probability maps and their gradients, given scarce empirical data and explicit physics constraints. Training minimises both a data loss and a PDE residual, producing surrogates that are sample-efficient, can generalize across state–time–parameter domains, and support real-time evaluation and gradient-based safe control (Wang et al., 2023, Wang et al., 11 Jul 2024).

Power Systems and Operational Risk

• Layered architectural frameworks: Hybrid ML surrogates replace computationally intractable physics-based SCED solvers in large-scale Monte Carlo pipelines, accelerating near real-time operational risk assessment while preserving fidelity to physical system constraints (Stover et al., 2022).
• Physical interpretability quantification: Hybrid models for grid security are quantitatively evaluated for physical interpretability, trainability, inference complexity, scalability, and generalizability. Physics are embedded through sparsity patterns, hard constraints, or global system embeddings (Li et al., 2022).
• Warm starting contingency analyses: Conditional Gaussian random field architectures use physical network topologies to initialize post-contingency states, greatly accelerating security evaluations under N−x or cyberattack scenarios (Li et al., 2022).

Prognostics and Remaining Useful Life (RUL)

• Feature-augmented deep learning: Calibrated health parameters inferred via physics-based models are concatenated with sensor and operating condition data, enabling deep networks to perform enhanced lifetime prediction. Hybrid models demonstrably extend RUL prediction horizons and outperform pure ML under domain shift or data scarcity (Chao et al., 2020).

4. Uncertainty Quantification, Validation, and Generalization

Hybrid frameworks offer institutional rigor in uncertainty quantification and model validation:

• Calibration under spatiotemporal partition: Ensemble approaches validated via spatial–temporal cross-validation, blocking by geography and historical timeline, prevent metric inflation due to spatial or temporal autocorrelation (Kriuk, 2 Oct 2025).
• Epistemic uncertainty mapping: Spread among ML ensemble outputs, or explicitly propagated uncertainty in probabilistic predictions, is retained in final risk quantification, providing spatially resolved measures of confidence (Kriuk, 2 Oct 2025).
• Provable generalization bounds: Theoretical guarantees show that under bounded PDE residuals and small boundary mismatch, PINN surrogates yield uniformly bounded error across domains. Empirically, this delivers $5-10\times$ improvement in sample efficiency and 2–3 orders of magnitude improvement in gradient estimation accuracy relative to direct MC (Wang et al., 2023, Wang et al., 11 Jul 2024).
• Adaptation to parameter shift: By embedding system (or scenario) parameters in surrogate inputs, hybrid models support real-time parametric adaptation, retraining with incremental MC data or controlling scenario (Wang et al., 2023, Wang et al., 11 Jul 2024).

5. Evaluation Attributes, Interpretability, and Loss Strategies

Rigorous frameworks explicitly evaluate proposed hybrid architectures via layered attributes (Li et al., 2022):

• Physical interpretability: Assessed on axes of scale (global/partial) and precision (exact/approximate) of physics integration.
• Scalability and trainability: Quantified in terms of parameter count, computational cost, and generalization to unseen system topologies or domains.
• Prediction reliability and application performance: Out-of-sample error, recall, variance, and operational utility are measured.
• Interpretability by design: GAM-style, basis-driven layering and monotonicity constraints ensure that subcomponents of the ML are physically meaningful and inspectable, with explicit contribution analysis—contrasting with SHAP-style post-hoc explanations, which may underrepresent critical physical terms (Sreenath et al., 26 Aug 2025).

Loss design adapts to risk domain:

• Asymmetric and hazard-aware losses: Emphasize accuracy near safety thresholds or rare, high-impact events, using parameterized penalties for under/overestimation in safe/unsafe domains (e.g., HAL in JITRALF (Stover et al., 2022); HazBinLoss in seismic GMM (Sreenath et al., 26 Aug 2025)).
• Physics residuals as regularizers: Imposed as differentiable losses, encouraging the network to satisfy governing (S)PDEs everywhere in the training domain (Wang et al., 2023, Hoshino et al., 25 Mar 2024).

6. Operationalization and Deployment

Hybrid physics–ML frameworks are implemented as open-source toolkits, supporting:

• End-to-end risk pipelines: Data ingestion, feature extraction, hybrid model evaluation (including uncertainty quantification and scenario specification), risk categorization and mapping (Kriuk, 2 Oct 2025, Stover et al., 2022).
• Real-time inference: Surrogate-based architectures achieve orders-of-magnitude speed-up over repeated large-scale physics-based simulations, reducing per-scenario computation to milliseconds without loss of fidelity (Stover et al., 2022).
• Integration with engineering practice: Probabilistic outputs and uncertainty quantification feed directly into design codes, adaptation planning, and field prioritization, supporting risk-informed decision making (Kriuk, 2 Oct 2025).

7. Limitations and Research Directions

Identified limits and future avenues include:

• Dependence on accurate physics: PINN and physics-constrained surrogates require high-fidelity physical models or equations; misspecification or large unmodeled effects can bias results (Hoshino et al., 25 Mar 2024).
• Sample complexity and high-dimensionality: Nonlinear high-dimensional domains demand efficient PDE sampling strategies and larger models, which can challenge tractability (Wang et al., 2023, Wang et al., 11 Jul 2024).
• Extensions and robustness: Research directions involve hybridization with robust optimization, adversarial training, and extension to broader risk and control measures (e.g., CVaR, full risk distributions), as well as multi-modality (incorporating cyber, economic, and physical dynamics) (Li et al., 2022, Hoshino et al., 25 Mar 2024).
• Validation and conservative usage: Conservative buffer regions, dense boundary sampling, and thorough independent validation are recommended for deployment in safety-critical settings (Hoshino et al., 25 Mar 2024).
• Adaptability: Parameter-dependent surrogate architectures allow online updating as regime or parameter values shift, with minimal retraining (Wang et al., 2023, Wang et al., 11 Jul 2024).

Hybrid Physics–ML risk assessment frameworks thus provide a mathematically principled, empirically validated, and operationally scalable approach to risk quantification and decision-support in complex, uncertain, and safety-critical domains, combining the explanatory power and extrapolation capability of physics with the pattern-recognition and flexibility of modern machine learning (Kriuk, 2 Oct 2025, Wang et al., 2023, Chao et al., 2020, Hoshino et al., 25 Mar 2024, Li et al., 2022, Stover et al., 2022, Sreenath et al., 26 Aug 2025, Wang et al., 11 Jul 2024).