Reliable monitoring and control of lithium-ion batteries require models that capture electrochemical dynamics while remaining computationally efficient. Classical equivalent circuit models can be efficient but neglect detailed physics, whereas purely data-driven neural networks rely heavily on large datasets and lack interpretability, resulting in poor generalization beyond the training domain. Recent rapid advances in physics-informed machine learning (PIML) offer promising pathway to bridge this gap by integrating physical structure with learning. This review summarizes representative PIML approaches for lithium-ion batteries, including physics-guided, physics-informed, and physics-encoded methods. By comparing their capabilities and limitations in terms of data efficiency, physical consistency, and deployment potential, particular attention is given to the physics-encoded architectures, which enforce key constraints by design and show potential for robust and reliable battery modeling. Overall, this review provides a timely synthesis of recent progress in PIML for lithium-ion batteries modeling and highlights key opportunities for future research and BMS deployment.

When learning dynamical systems from data, embedding physical structure can constrain the solution space and improve generalization, but many physics-informed models assume access to the full system state. This limits their use in partially observed settings, where some state variables are completely unobserved and must be inferred without direct supervision. Here, we present neural Hamiltonian ordinary differential equations (NHODE), a framework that combines Hamiltonian neural networks (HNNs) with neural ordinary differential equations (neural ODEs) to learn partially observed dynamical systems from data. The Hamiltonian structure enforces energy conservation by construction, while the neural ODE framework enables a flexible training procedure that allows the loss to be defined only on observed variables. We also incorporate additional physical constraints through symmetry-aware coordinate transformations and separable energy formulations. The framework is evaluated on systems of increasing complexity, from linear and nonlinear mass-spring systems to the chaotic three-body problem. Across all examples, increasing the amount of embedded physical structure improves the accuracy and long-horizon stability of the predictions. Even in the most challenging regimes, the NHODE framework captures both observed and latent dynamics, whereas purely data-driven baselines become unstable.

Accurate estimation of key process variables is essential for effective monitoring and control in wastewater treatment plants (WWTPs), particularly when direct measurements are limited by noise, delays, or operational constraints. This study investigates machine learning–based prediction of nitrate concentration at the outlet of a pilot-scale denitrification reactor at the Veas wastewater treatment facility in Norway. Using high-resolution multivariate sensor data, the task is formulated as a data-driven state estimation problem in which outlet nitrate concentration is inferred from twelve process measurements capturing chemical, hydraulic, pressure, dosing, and thermal dynamics. A full year of operational data from October 2023 to September 2024 is used strictly for model training, while data from October 2024 to March 2025—representing future, unseen operating conditions—is used exclusively for evaluation. This provides a real-situation temporal test rather than a conventional cross-validation or holdout split, enabling a more realistic assessment of model generalization under operational variability. A broad range of modeling approaches is evaluated, including shallow regressors such as Elastic Net, Random Forest Regression, and Gradient Boosted Decision Trees. Traditional deep learning architectures for sequential data are also explored, including Recurrent Neural Networks (RNNs), Long Short-Term Memory networks (LSTMs), Gated Recurrent Units (GRUs), Temporal Convolutional Networks (TCNs), Deep AutoRegressive Networks (DeepAR), and Neural Basis Expansion Analysis for Time Series (N-BEATS). Finally, transformer-based modeling is examined using the Temporal Fusion Transformer (TFT), adapted in this study for regression by aligning the target with the input window. Across all experiments, the TFT achieves the strongest predictive accuracy, with a test RMSE of 1.7707, outperforming the best shallow model (RMSE 2.6018) and best traditional deep learning model (RMSE 2.8320). The results confirm that TFT models nonlinear and temporal dynamics more effectively than other approaches, enabling reliable nitrate state estimation in real WWTP operations.

Time series modelling in process industries faces the challenge of dealing with complex, multi-faceted and evolving data characteristics. Conventional single-model approaches often struggle to capture the interplay of diverse dynamics, resulting in suboptimal forecasts. Addressing this, we introduce the Recency-Weighted Temporally-Segmented (ReWTS, pronounced 'roots') ensemble model, a novel chunk-based approach for multi-step forecasting. The key characteristics of the ReWTS method are twofold: 1) It facilitates specialization of models into different dynamics by segmenting the training data into 'chunks' of data and training one model per chunk; 2) During forecasting, an optimization procedure assesses each model on the recent past and selects the active models, such that the appropriate mixture of previously learned dynamics can be recalled to forecast the future. This method not only captures the nuances of each period, but also adapts more effectively to changes over time compared to conventional 'global' models trained on all data in one go. We present a comparative analysis, using two years of data from a wastewater treatment plant and a drinking water treatment plant in Norway, demonstrating the ReWTS ensemble's superiority. It consistently outperforms the global model in terms of mean squared forecasting error across various model architectures by 10-70% on both datasets, notably exhibiting greater resilience to outliers. We further explore the generalizability of ReWTS by applying it to four publicly available datasets. The results indicate that ReWTS is particularly valuable in non-stationary, heterogeneous environments with frequent concept drift, as exemplified by the process industry datasets. This approach shows promise in developing automatic, adaptable forecasting models for decision-making and control systems in process industries and other complex systems.

Identifying the underlying dynamics of physical systems can be challenging when only provided with observational data. In this work, we consider systems that can be modelled as first-order ordinary differential equations. By assuming a certain pseudo-Hamiltonian formulation, we are able to learn the analytic terms of internal dynamics even if the model is trained on data where the system is affected by unknown damping and external disturbances. In cases where it is difficult to find analytic terms for the disturbances, a hybrid model that uses a neural network to learn these can still accurately identify the dynamics of the system as if under ideal conditions. This makes the models applicable in some situations where other system identification models fail. Furthermore, we propose to use a fourth-order symmetric integration scheme in the loss function and avoid actual integration in the training, and demonstrate on varied examples how this leads to increased performance on noisy data.

Wastewater treatment plants are increasingly recognized as promising candidates for machine learning applications, due to their societal importance and high availability of data. However, their varied designs, operational conditions, and influent characteristics hinder straightforward automation. In this study, we use data from a pilot reactor at the Veas treatment facility in Norway to explore how machine learning can be used to optimize biological nitrate (NO_3^-) reduction to molecular nitrogen (N_2) in the biogeochemical process known as denitrification. Rather than focusing solely on predictive accuracy, our approach prioritizes understanding the foundational requirements for effective data-driven modelling of wastewater treatment. Specifically, we aim to identify which process parameters are most critical, the necessary data quantity and quality, how to structure data effectively, and what properties are required by the models. We find that nonlinear models perform best on the training and validation data sets, indicating nonlinear relationships to be learned, but linear models transfer better to the unseen test data, which comes later in time. The variable measuring the water temperature has a particularly detrimental effect on the models, owing to a significant change in distributions between training and test data. We therefore conclude that multiple years of data is necessary to learn robust machine learning models. By addressing foundational elements, particularly in the context of the climatic variability faced by northern regions, this work lays the groundwork for a more structured and tailored approach to machine learning for wastewater treatment. We share publicly both the data and code used to produce the results in the paper.

. Discrete gradient methods are a class of numerical integrators producing solutions with exact preservation of first integrals of ordinary differential equations. In this paper, we apply order theory combined with the symmetrized Itoh-Abe discrete gradient and finite differences to construct an integral-preserving fourth-order method that is derivative-free. The numerical scheme is implicit and a convergence result for Newton's iterations is provided, taking into account how the error due to the finite difference approximations affects the convergence rate. Numerical experiments verify the order and show that the derivative-free method is significantly faster than obtaining derivatives by automatic differentiation. Finally, an experiment using topographic data as the potential function of a Hamiltonian oscillator demonstrates how this method allows the simulation of discrete-time dynamics from a Hamiltonian that is a combination of data and analytical expressions.

Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for learning dynamical systems that can be modelled by ordinary differential equations. In this paper, we extend the method to partial differential equations. The resulting model is comprised of up to three neural networks, modelling terms representing conservation, dissipation and external forces, and discrete convolution operators that can either be learned or be given as input. We demonstrate numerically the superior performance of PHNN compared to a baseline model that models the full dynamics by a single neural network. Moreover, since the PHNN model consists of three parts with different physical interpretations, these can be studied separately to gain insight into the system, and the learned model is applicable also if external forces are removed or changed.

Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems, both energy conserving and not energy conserving. We introduce a pseudo-Hamiltonian formulation that is a generalization of the Hamiltonian formulation via the port-Hamiltonian formulation, and show that pseudo-Hamiltonian neural network models can be used to learn external forces acting on a system. We argue that this property is particularly useful when the external forces are state dependent, in which case it is the pseudo-Hamiltonian structure that facilitates the separation of internal and external forces. Numerical results are provided for a forced and damped mass-spring system and a tank system of higher complexity, and a symmetric fourth-order integration scheme is introduced for improved training on sparse and noisy data.

Offshore slender marine structures experience complex and combined load conditions from waves, current and vessel motions that may result in both wave frequency and vortex shedding response patterns. Field measurements often consist of records of environmental conditions and riser responses, typically with 30-minute intervals. These data can be represented in a high-dimensional parameter space. However, it is difficult to visualize and understand the structural responses, as they are affected by many of these parameters. It becomes easier to identify trends and key parameters if the measurements with the same characteristics can be grouped together. Cluster analysis is an unsupervised learning method, which groups the data based on their relative distance, density of the data space, intervals, or statistical distributions. In the present study, a Gaussian mixture model guided by domain knowledge has been applied to analyze field measurements. Using the 242 measurement events of the Helland-Hansen riser, it is demonstrated that riser responses can be grouped into 12 clusters by the identification of key environmental parameters. This results in an improved understanding of complex structure responses. Furthermore, the cluster results are valuable for evaluating the riser response prediction accuracy.

We introduce the mean inverse integrator (MII), a novel approach to increase the accuracy when training neural networks to approximate vector fields of dynamical systems from noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge-Kutta methods. We show that the class of mono-implicit Runge-Kutta methods (MIRK) has particular advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained when inserting the training data in the MIRK formulae, unlocking symmetric and high-order integrators that would otherwise be implicit for initial value problems. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with experiments using data from several (chaotic) Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, supporting the favorable performance of MII.

Digitisation in the mining and metal processing industries plays a key role in their modernisation. Production processes are more and more supported by a variety of sensors that produce large amounts of data that meant to provide insights into the performance of production infrastructures. In the metal processing industry vibration sensors are essential in the monitoring of the production infrastructure. In this position paper we present the installation of vibration sensors in a real industrial environment and discuss the data quality issues we encountered while using such sensors.

The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive specific results for the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented, and their applications are demonstrated on the Hénon–Heiles system and a Lotka–Volterra system, and on both the training and integration of a pendulum system learned from data by a neural network.

Production of high silicon alloys must adhere to a range of expectations and demands. In addition to meeting the customer expectations on quality, the production process itself should be operated such that the raw material and energy consumption is minimized per metric ton produced. Elkem is working on upgrading the decision support systems for plant operators to meet the above-mentioned expectations. In this work, new sensor technology coupled with data-based models has been applied to develop an online decision support system (aka hybrid twin) for production of FeSi75. Infrared camera technology has been applied to the tapping, refining/alloying and casting processes and some preliminary results will be presented, focusing on determining the amount of slag produced in the furnace and tapped with the liquid alloy. In parallell, a data-driven model has been applied to furnace operational parameters in order to predict the amount of slag Based on initial results, application of IR technology combined with image analysis seems to be a viable path forward in order to establish more precise decision support models that will improve product quality and increase yield.

Conventional vortex-induced vibration (VIV) prediction tools are semi-empirical, in other words, based on several empirical parameters extracted from model tests in laboratory. Generally, the lab tests are costly, include small scale test conditions and with a limited test matrix. The extracted empirical databases are not directly applicable to full-scale VIV predictions of various slender marine structures. Therefore, large safety factors have been used by industry for VIV prediction in the past decades. To reduce the uncertainty (e.g. over-conservatism) related to semi-empirical VIV prediction tools, the NLPQL algorithm for parameter optimization of a semi-empirical time-domain prediction tool has been investigated. This methodology was demonstrated on pure cross-flow VIV prediction in an earlier study. It was shown that by setting appropriate constraints and cost functions of the optimization algorithm, this method is feasible to improve the VIV prediction accuracy. In this study, the NLPQL optimization algorithm was applied for combined cross-flow and in-line VIV predictions using time domain numerical model. Selected cases from field measurements representing multi-fidelity data were used to validate and verify the method.

Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems. In this work, we stress-test the method on both simple mass-spring systems and more complex and realistic systems with several internal and external forces, including a system with multiple connected tanks. We quantify performance under various conditions and show that imposing different assumptions greatly affect the performance during training presenting advantages and limitations of the method. We demonstrate that port-Hamiltonian neural networks can be extended to larger dimensions with state-dependent ports. We consider learning on systems with known and unknown external forces and show how it can be used to detect deviations in a system and still provide a valid model when the deviations are removed. Finally, we propose a symmetric high-order integrator for improved training on sparse and noisy data.

Kahan's method and a two-step generalization of the discrete gradient method are both linearly implicit methods that can preserve a modified energy for Hamiltonian systems with a cubic Hamiltonian. These methods are here investigated and compared. The schemes are applied to the Korteweg-de Vries equation and the Camassa-Holm equation, and the numerical results are presented and analysed.

We present linearly implicit methods that preserve discrete approximations to local and global energy conservation laws for multi-symplectic PDEs with cubic invariants. The methods are tested on the one-dimensional Korteweg-de Vries equation and the two-dimensional Zakharov-Kuznetsov equation; the numerical simulations confirm the conservative properties of the methods, and demonstrate their good stability properties and superior running speed when compared to fully implicit schemes.

The energy preserving discrete gradient methods are generalized to finite-dimensional Riemannian manifolds by definition of a discrete approximation to the Riemannian gradient, a retraction, and a coordinate center function. The resulting schemes are intrinsic and do not depend on a particular choice of coordinates, nor on embedding of the manifold in a Euclidean space. Generalizations of well-known discrete gradient methods, such as the average vector field method and the Itoh--Abe method are obtained. It is shown how methods of higher order can be constructed via a collocation-like approach. Local and global error bounds are derived in terms of the Riemannian distance function and the Levi-Civita connection. Some numerical results on spin system problems are presented.

Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. One computationally efficient approach to shape analysis is based on the Square Root Velocity Transform (SRVT). In this paper we propose a generalised SRVT framework for shapes on homogeneous manifolds. The method opens up for a variety of possibilities based on different choices of Lie group action and giving rise to different Riemannian metrics.