## 5 Discussion and Future Research Directions
### Hybrid Physics-Data Models: A Bridge Between First-Principles Insight and Statistical Learning  
As highlighted throughout the quantitative (Section 3) and qualitative (Section 4) analyses, an increasingly visible research strand blends mechanistic insight with modern machine learning pipelines. Roughly 11% of the 354 primary studies selected in our SLR employ some form of hybrid physics-data modelling--by either (i) embedding governing equations or conservation laws in the loss function of a neural network, (ii) coupling a fast data-driven _surrogate_ with a high-fidelity simulator, or (iii) enforcing physically meaningful constraints (e.g., energy balance, cyclic symmetry, or resonance period) during feature engineering. Representative examples are summarised in Tables S2, S9, S12 and S13. Below, we distil the main patterns, benefits, and open questions that emerged from this sub-corpus. Taxonomy of hybrid approaches. 1. Physics-informed neural networks (PINNs). Recent studies [110,113] minimise a joint loss that combines data misfit with residuals of the underlying motion or wear equations. By forcing the latent representation to satisfy kinematic periodicity or impulse-response decay, PINNs raise diagnostic accuracy by 2-5 pp compared with purely data-driven CNN baselines while requiring only \(\sim\)50% of the original labelled samples. 2. Grey-box digital twins with learned surrogates. In compressor and pump studies (e.g., those by [PERSON] et al. [40] and [PERSON] et al. [39]), thermodynamic or CFD simulations generate synthetic fault trajectories that feed a lightweight surrogate network (LSTM, RBF, or autoencoder). Once calibrated, the surrogate runs three to four orders of magnitude faster than the full solver, enabling real-time residual generation for anomaly scoring. 3. Physics-guided feature engineering. Works on cavitation and bearing spalls (e.g., thos eby [PERSON] et al. [84] and [PERSON] et al. [37]) first isolate carrier or modulated frequencies predicted by fluid-structure interaction theory, then feed those physically meaningful sub-bands to classical classifiers (RF or SVM) or to compact CNNs. This grey-box preprocessing typically cuts feature dimensionality by 60-80% and improves class-imbalance robustness. Why does the hybrid route matter? * Label scarcity. Most industrial assets fail rarely; embedding conservation constraints or simulator priors substitutes for the thousands of fault cycles that deep learners usually demand. * Extrapolation and trust. When operating conditions drift outside the training envelope, hard-coded physical laws act as safety rails that curb unphysical predictions and ease regulatory acceptance--an advantage repeatedly stressed by domain experts during our validation workshops. * Computational frugality. Grey-box surrogates such as the self-adaptive RBF used by [PERSON] et al. [40] reduce inference latency to below 100 ms on edge-grade CPUs, satisfying the cycle times of most PLC-controlled plants. Frontier challenges. Despite these benefits, the reviewed papers reveal three recurring pain points:* Automatic weighting of physics vs. data losses. Nearly every PINN study tunes the balance term via grid search; adaptive schemes (e.g., dynamic gradient harmonisation) remain to be tested on industrial datasets. * Uncertainty quantification. Only two hybrid works--that of [PERSON] et al. [37] and the Bayesian ensemble of [PERSON] et al. [64]--provide predictive intervals. Extending probabilistic PINNs or ensemble surrogates to multi-fault settings is an open avenue. * Standardised benchmarks. Cross-study comparability is hampered by heterogeneous simulators and proprietary datasets. A curated, open repository pairing raw signals with physics metadata (steel grade, lubricant viscosity, and ambient pressure) would accelerate method transfer across domains. Research agenda and practical tips.