Robust Hybrid Physics-ML Framework

• The paper introduces a hybrid framework that integrates a mechanistic forward osmosis model with a residual Gaussian Process correction to achieve sub-percent prediction errors.
• The paper employs complete uncertainty quantification by analytically decomposing epistemic and aleatoric uncertainties, enhancing model interpretability and risk assessment.
• The paper demonstrates superior predictive performance with a MAPE of 0.26% and R² of 0.9990 using only 120 training samples, significantly outperforming traditional models.

A Robust Hybrid Physics–ML Framework combines the physical foundations of mechanistic modeling with the expressive power and uncertainty-awareness of modern machine learning, specifically to deliver accurate predictive surrogates even in data-scarce, high-variability applications. Such frameworks have emerged in response to the limitations of both pure first-principles and pure data-driven approaches. In forward osmosis (FO), for example, traditional transport models incorporate complex nonlinearities but suffer from empirical parameter ambiguities, while purely ML-based surrogates—though flexible—lack interpretability and rigorous uncertainty quantification. The recent "Hybrid Physics–ML Model for Forward Osmosis Flux with Complete Uncertainty Quantification" (Ratn et al., 11 Dec 2025) illustrates this paradigm by integrating a mechanistic FO model and a residual-correcting Gaussian Process Regression (GPR), along with a full decomposition of epistemic and aleatoric uncertainties.

1. Physical Model Backbone and Problem Formulation

The hybrid framework begins with a mechanistic model, in this case the classical Spiegler–Kedem solution–diffusion formulation of FO including both external (ECP) and internal concentration polarization (ICP). The steady-state water flux, $J_{w,\text{physical}}$, is defined via a transcendental nonlinear equation:

$$J_{w,\text{physical}} = A\, \frac{ \Pi_{D,b}\,\exp\left(-\frac{J_{w,\text{physical}}\,S}{D_{s}}\right) - \Pi_{F,b}\,\exp\left(\frac{J_{w,\text{physical}}}{k}\right) }{ 1 + \frac{B}{J_{w,\text{physical}}} \left[ \exp\left(\frac{J_{w,\text{physical}}}{k}\right) - \exp\left(-\frac{J_{w,\text{physical}}\,S}{D_{s}}\right) \right] }$$

where model parameters and features (active and support-layer membrane properties, geometric and operational variables) follow standard FO conventions. Brent's method is employed to numerically solve for $J_{w,\text{physical}}$, given a 10-dimensional input space covering membrane and process properties. This first-principles component guarantees physical consistency for bulk transport behavior.

2. Residual-Based Gaussian Process Correction

Instead of regressing the water flux $J_w$ end-to-end from inputs, the hybrid approach fits a GPR directly to the residual error:

$$r(\mathbf{z}) = J_{w,\text{actual}}(\mathbf{z}) - J_{w,\text{physical}}(\mathbf{z})$$

with $\mathbf{z}\in\mathbb{R}^{10}$ aggregating all process and membrane features. The GPR employs a Matérn $5/2$ kernel, recognizing the limited dataset regime (as few as 120 training points), which is highly advantageous for GPs. Hyperparameters are tuned via marginal likelihood maximization, and the GP posterior provides both a predictive mean $\hat{r}(\mathbf{z}_*)$ and variance $\sigma^2_{\text{model}}(\mathbf{z}_*)$ at any query point. The final surrogate is computed by

$$J_{w,\text{hybrid}}(\mathbf{z}) = J_{w,\text{physical}}(\mathbf{z}) + \hat{r}(\mathbf{z})$$

establishing an additive "correction" mechanism that is physically anchored but data-adaptive.

3. Complete Uncertainty Quantification

A central innovation of this framework is the analytic decomposition of predictive uncertainty into epistemic (model-driven) and aleatoric (input-propagated) components:

$$\sigma^2_{\text{total}} = \sigma^2_{\text{model}} + \sigma^2_{\text{input}}$$

• Epistemic: $\sigma^2_{\text{model}}(\mathbf{z}_*)$ is calculated from the GP's posterior variance.
• Aleatoric: $\sigma^2_{\text{input}}(\mathbf{z}_*)$ is propagated via the Delta method,

$$\sigma^2_{\text{input}}(\mathbf{z}) \approx \left[\nabla_{\mathbf{z}} J_{w,\text{hybrid}}(\mathbf{z})\right]^\top \Sigma_z \left[\nabla_{\mathbf{z}} J_{w,\text{hybrid}}(\mathbf{z})\right]$$

where $\Sigma_z$ encodes covariance of feature uncertainties and the full Jacobian combines derivatives of both the physical model and the GP correction. Validation against Monte Carlo simulations confirms the reliability of this analytic approach (relative errors <3%).

4. Model Training, Data Regime, and Predictive Performance

The model is trained with only 120 experimental points (augmented with 2854 independent test cases). Features are standardized, and deterministic splits avoid data leakage. On the test set, the hybrid delivers state-of-the-art accuracy: MAPE = 0.26%, $R^2 = 0.9990$. In comparison, a physics-free GPR baseline achieves MAPE = 0.35%, $R^2 = 0.9980$, and a pure ANN yields orders-of-magnitude worse error at MAPE ≈ 5.8%. The framework's data efficiency and superior generalization stem from its explicit exploitation of the mechanistic model structure.

5. Applications: Digital Twins, Design, and Process Optimization

The surrogate's reliability and decomposed uncertainty deliver three practical impacts:

• Physically consistent, interpretable predictions at sub-percent error level allow for robust digital twin integration.
• Quantitative risk assessment: Disentangled uncertainty enables formal safety factors and compliant process design.
• Bayesian optimization and real-time control: The framework's efficiency and reliability support model-based optimization loops, critical for process intensification and automated FO module control.

6. Limitations, Extensions, and Future Directions

• Delta method limitations: This first-order propagation may under-represent uncertainty for strongly nonlinear or non-Gaussian input distributions. Higher-order (unscented transform, polynomial chaos) approaches are proposed for future implementation.
• Dynamic operating conditions: Extending the current static framework to handle dynamic fouling, time-dependent parameter drift, or unsteady conditions remains an open research avenue.
• Multi-fidelity and deep-kernel GPs: Integration of data from diverse sources (lab, pilot, commercial scales) or leveraging deep GPs could further enhance model expressivity and transferability.

7. Impact and Generalization

This robust hybrid Physics–ML approach demonstrates a scalable blueprint for integrating first-principles models with advanced, uncertainty-aware probabilistic learning, particularly in domains constrained by scarce, high-quality ground truth. The explicit mathematical decomposition of uncertainty and physical-data residual correction advances both prediction fidelity and actionable risk quantification, defining new standards for digital process twins, model-based optimization, and regulatory compliance in engineering systems (Ratn et al., 11 Dec 2025).