A tale of perfect fit and phantom optima: how data-driven models can fail in real-time optimization

Real-time optimization (RTO) relies on process models to locate economically optimal operating conditions. Because developing first-principles models requires significant process knowledge, data-driven alternatives are increasingly attractive. Modern machine-learning models can fit historical plant data accurately and often pass standard validation tests. Whether such models can be trusted for economic optimization, however, remains unclear. We investigate this question using a vinyl acetate monomer benchmark process with a unique, well-conditioned economic optimum. We train a structured hybrid model that combines known mass balances and thermodynamics with a neural-network closure for unknown kinetics, and a fully data-driven neural ordinary differential equation (ODE) model. Both models reproduce plant measurements accurately and exhibit little variation in predictions across random initializations. Yet their economic optima differ substantially from that of the plant. Where the plant returns a single optimum on multistart search, the trained models return many phantom optima. We further show that the training optimizer alone can be yet another source of error. Even with noise-free data and initialization at weights that recover the plant optimum, stochastic gradient training can drift to weights that yield substantially worse RTO solutions. The identified model is thus an artifact of the training optimizer as well as the data. These results demonstrate that a good predictive fit of all available measurements does not guarantee reliable economic performance. A data-driven model for RTO should at least be required to recover the optimum on a decision-oriented benchmark like the one developed here before being considered for plant testing and application.

Physics-Informed and Hybrid Machine Learning in Additive Manufacturing: Application to Fused Filament Fabrication

This article investigates several physics-informed and hybrid machine learning strategies that incorporate physics knowledge in experimental data-driven deep-learning models for predicting the bond quality and porosity of fused filament fabrication (FFF) parts. Three types of strategies are explored to incorporate physics constraints and multi-physics FFF simulation results into a deep neural network (DNN), thus ensuring consistency with physical laws: (1) incorporate physics constraints within the loss function of the DNN, (2) use physics model outputs as additional inputs to the DNN model, and (3) pre-train a DNN model with physics model input-output and then update it with experimental data. These strategies help to enforce a physically consistent relationship between bond quality and tensile strength, thus making porosity predictions physically meaningful. Eight different combinations of the above strategies are investigated. The results show how the combination of multiple strategies produces accurate machine learning models even with limited experimental data.

From Hybrid Mechanistic--Data-Driven Modeling Toward Neuro-Symbolic AI: What, Why, and How

Hybrid mechanistic/data-driven models, which combine first-principles with learned components, are increasingly used in process engineering and scientific machine learning. Common hybrid modeling designs are specified primarily through their architectures and training losses, which offers a limited basis for a shared semantic interface to compare or verify them across domains, with comparatively little attention paid to epistemic uncertainty in the mechanistic part. We bridge hybrid modeling and neuro-symbolic (NeSy) AI by reconstructing these designs as instances of NeSy interface. The resulting translation, Hybrid-to-NeSy (H2N), places mechanistic knowledge on the language side, learned modules on the belief side, and validity domains together with constraints on the logic side. For each design, H2N then yields an explicit NeSy inference functional and a logic-belief decomposition. From this decomposition we derive two metrics: structural violation rate (SVR), measuring whether the learned belief respects the mechanistic structure; and belief dispersion (BD), measuring how concentrated the learned plausibility is, serving as a hybrid model's epistemic uncertainty in its mechanistic part. We instantiate H2N on a case study of a structured hybrid model for binary classification under label noise and show that models with higher SVR and BD exhibit greater variability in held-out accuracy. Under structural distribution shift, H2N further quantifies a model's uncertainty during extrapolations, whereas test accuracy reveals the same shift only post hoc.

Predicting Activities in Aqueous Electrolyte Solutions with Hybrid Machine Learning

Activities in aqueous electrolyte solutions, usually described by ionic activity and osmotic coefficients, are important properties for modeling many processes in industry and nature. Established activity models, such as those of Pitzer or Bromley, require fitting to experimental data for each electrolyte of interest and thus cannot predict properties for unstudied systems. While some predictive approaches exist, they are typically limited in scope and rely on additional ion-specific descriptors. In this work, we introduce a new hybrid model that combines the physics-based Bromley model with a matrix completion method (MCM) from machine learning. The MCM is employed to predict the electrolyte-specific parameters of the Bromley model, exploiting the fact that these parameters can be arranged in a matrix with cations and anions as rows and columns, respectively. Due to the lack of experimental data for many electrolytes, the initial parameter matrix is sparsely populated, making the prediction of the Bromley parameters for unstudied electrolytes a matrix completion problem. The hybrid model, Bromley-MCM, was trained end-to-end on experimental data for mean ionic activity coefficients and osmotic coefficients of aqueous solutions of 478 electrolytes at 298 K from the Dortmund Data Bank. As output, we obtain a completed matrix of Bromley parameters for 83 cations and 112 anions, enabling consistent prediction of concentration-dependent activities in aqueous solutions of 9,296 electrolytes at 298~K. This substantially extends the applicability of the Bromley model while maintaining high predictive accuracy, as demonstrated through evaluations on electrolytes excluded from model training.

OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems

Dynamical systems are fundamental to modeling the natural world, yet modeling them involves a persistent trade-off: manually prescribed mechanistic models are interpretable by design but often overly simplistic and misspecified; in contrast, flexible data-driven neural methods lack physical insight. Hybrid modeling aims for the best of both worlds by combining a prescribed or symbolic, physics-based component with a flexible neural network. A critical challenge, however, is that the neural component may relearn mechanistic parts, yielding redundant and uninterpretable models, especially when the symbolic structure itself is discovered from data. Existing methods based on standard $L^2$ regularization rely on a projection argument that breaks when the symbolic component is learned through sparse discovery, allowing the neural augmentation to overlap with symbolic structure. We introduce \textbf{OrthoReg} (Orthogonal Regularization), which directly penalizes overlap between the symbolic and neural components, preventing symbolic structure from being absorbed by the neural residual. This yields a complementary decomposition: the symbolic part captures what the library can express, and the neural part captures what remains. On benchmark dynamical systems with partial library mismatch, OrthoReg improves symbolic recovery and out-of-distribution behavior.

Learning Cardiac Electrophysiology Digital Twins Through Agentic Discovery of Hybrid Structure

Building personalized cardiac electrophysiology (EP) digital twins requires identifying the appropriate model structure for each patient, not merely fitting parameters. Traditional methods rely on experts to manually prescribe hybrid physics-neural architectures, which requires deep domain expertise and does not transfer across patients. Recent works have applied large language models (LLMs) to generate or act as hybrid models. However, despite their promising generalization capacity, these LLM-based methods lack the structural priors needed for stable cardiac simulations. Hence, we propose LEADS, a framework that formulates cardiac EP domain knowledge as a structured action space and utilizes an LLM agent to discover hybrid models. The agent follows an iterative reasoning-and-action loop to select, combine, and refine hybrid models, whilst gradient descent handles parameter fitting. The proposed LEADS designs every candidate model towards physically grounded, interpretable, and numerically stable, while allowing open-ended architectural discovery. We validate LEADS on synthetic data with three ground-truth reaction models and on real cardiac EP data, demonstrating that it outperforms both human-designed hybrid models and other LLM-based hybrid modeling.

Integrating Mechanistic and Data-Driven Models for Neurological Disorders through Differentiable Programming

Advances in computational modeling, neuroimaging, and artificial intelligence are revolutionizing the modeling of neurological disorders for improved diagnostics, prognosis, and treatment planning. Mechanistic models provide valuable scientific insight into the disorders, but in practice they are often simplified with assumptions or computationally expensive and slow to solve. However, while purely data driven approaches provide speed and scalability, they require large, high quality data to train and generally suffer from interpretability and generalization issues. This perspective paper presents a structured overview of hybrid modeling strategies, which combine deep learning models with physics based solvers, and are categorized into parallel, series, and parallel-series architectures. Three main approaches that have been emphasized are residual modeling for missing or incomplete physics, Neural Ordinary Differential Equations (NODEs) for continuous time dynamics approximation, and solver in the loop that accelerates traditional solvers with neural approximations. These hybrid models integrate the governing differential equation based formulations and deep learning to characterize the evolution of neurological disorders, and promise advanced personalized neurological modeling. In addition, the study explores and proposes different hybrid configurations to improve diagnosis accuracy, predict disease progression, and inform treatment strategies across a range of neurological disorders. These capabilities outperform standalone mechanistic or purely data driven approaches, making hybrid modeling a powerful tool, especially in applications involving modeling the progression and treatment responses in neurological conditions such as brain tumors, Alzheimer's disease, and stroke.

PE-MHL: Physics-Encoded Modular Hybrid Layers for Scalable Learning of Complex Systems

Hybrid models that combine physics-based and data-driven components have shown strong potential for achieving accuracy and interpretability in control applications. While recent methods have made progress in incorporating physical consistency, challenges remain in scalability, robustness to noise, and control of model complexity. This paper proposes a Physics-Encoded Modular Hybrid Layer (PE-MHL) framework, in which a baseline physics-based model is incrementally refined through the addition of new sub-models, where each new component adds complexity while preserving what previous components have already learned. We establish a theoretical guarantee for this construction: with a least-squares initialization of each new sub-model, the training error is monotonically non-increasing in the number of sub-models and provably converges. Empirical evaluations on a nonlinear NARX benchmark and the Quanser Aero 2 platform demonstrate that PE-MHL outperforms equivalently sized monolithic networks in both accuracy and generalization, while also providing more stable training dynamics and better preservation of underlying data structures.

Hybrid Neural Ordinary Differential Equations for Data-Efficient Polymerization Modeling with Incomplete Kinetics

Accurate prediction of polymerization dynamics is essential for process design, control, and optimization. Yet, purely mechanistic models require labor-intensive parameterization of partially characterized kinetics, while purely data-driven models demand large, diverse datasets that are costly to obtain, particularly in early-design stages. We propose a hybrid Neural Ordinary Differential Equation (NODE) framework for data-efficient modeling of free-radical polymerization. Using batch polymerization of methyl methacrylate (MMA) as a case study, the mechanistic mass balances are retained explicitly, and only the partially-characterized effective radical concentration governing monomer consumption is learned from data through a neural network surrogate, while established reactions such as initiator decomposition, propagation, and termination remain physically modeled. The hybrid NODE is evaluated against a discrete-time feedforward neural network and a purely data-driven NODE under sparse data conditions, with models trained on as few as ten measurements under both regular and irregular sampling. The hybrid NODE consistently achieves lower prediction errors and more physically consistent extrapolations than both purely data-driven baselines. In a generalization scenario with noisy data and unseen operating conditions, the hybrid NODE achieves an RMSE of 0.013, compared to 0.31 for the data-driven NODE and 0.68 for the discrete-time model, demonstrating that learning only a closure term rather than the full dynamics is sufficient for reliable prediction under limited data availability.