Signe Riemer-Sørensen

Katarzyna Michałowska, Somdatta Goswami, George Em Karniadakis et al.

Deep neural networks are an attractive alternative for simulating complex dynamical systems, as in comparison to traditional scientific computing methods, they offer reduced computational costs during inference and can be trained directly from observational data. Existing methods, however, cannot extrapolate accurately and are prone to error accumulation in long-time integration. Herein, we address this issue by combining neural operators with recurrent neural networks, learning the operator mapping, while offering a recurrent structure to capture temporal dependencies. The integrated framework is shown to stabilize the solution and reduce error accumulation for both interpolation and extrapolation of the Korteweg-de Vries equation.

Katarzyna Michałowska, Somdatta Goswami, George Em Karniadakis et al.

Deep operator networks (DeepONets, DONs) offer a distinct advantage over traditional neural networks in their ability to be trained on multi-resolution data. This property becomes especially relevant in real-world scenarios where high-resolution measurements are difficult to obtain, while low-resolution data is more readily available. Nevertheless, DeepONets alone often struggle to capture and maintain dependencies over long sequences compared to other state-of-the-art algorithms. We propose a novel architecture, named DON-LSTM, which extends the DeepONet with a long short-term memory network (LSTM). Combining these two architectures, we equip the network with explicit mechanisms to leverage multi-resolution data, as well as capture temporal dependencies in long sequences. We test our method on long-time-evolution modeling of multiple non-linear systems and show that the proposed multi-resolution DON-LSTM achieves significantly lower generalization error and requires fewer high-resolution samples compared to its vanilla counterparts.

Sølve Eidnes, Alexander J. Stasik, Camilla Sterud et al.

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.

Aslak Djupskås, Alexander Johannes Stasik, Signe Riemer-Sørensen

Reliable uncertainty estimation is crucial for machine learning models, especially in safety-critical domains. While exact Bayesian inference offers a principled approach, it is often computationally infeasible for deep neural networks. Monte Carlo dropout (MCD) was proposed as an efficient approximation to Bayesian inference in deep learning by applying neuron dropout at inference time \citep{gal2016dropout}. Hence, the method generates multiple sub-models yielding a distribution of predictions to estimate uncertainty. We empirically investigate its ability to capture true uncertainty and compare to Gaussian Processes (GP) and Bayesian Neural Networks (BNN). We find that MCD struggles to accurately reflect the underlying true uncertainty, particularly failing to capture increased uncertainty in extrapolation and interpolation regions as observed in Bayesian models. The findings suggest that uncertainty estimates from MCD, as implemented and evaluated in these experiments, is not as reliable as those from traditional Bayesian approaches for capturing epistemic and aleatoric uncertainty.

Peter Collett, Alexander Johannes Stasik, Simone Casolo et al.

Accurate condition monitoring of industrial equipment requires inferring latent degradation parameters from indirect sensor measurements under uncertainty. While traditional Bayesian methods like Markov Chain Monte Carlo (MCMC) provide rigorous uncertainty quantification, their heavy computational bottlenecks render them impractical for real-time process control. To overcome this limitation, we propose an AI-driven framework utilizing Simulation-Based Inference (SBI) powered by amortized neural posterior estimation to diagnose complex failure modes in heat exchangers. By training neural density estimators on a simulated dataset, our approach learns a direct, likelihood-free mapping from thermal-fluid observations to the full posterior distribution of degradation parameters. We benchmark this framework against an MCMC baseline across various synthetic fouling and leakage scenarios, including challenging low-probability, sparse-event failures. The results show that SBI achieves comparable diagnostic accuracy and reliable uncertainty quantification, while accelerating inference time by a factor of82$\times$ compared to traditional sampling. The amortized nature of the neural network enables near-instantaneous inference, establishing SBI as a highly scalable, real-time alternative for probabilistic fault diagnosis and digital twin realization in complex engineering systems.

Katarzyna Michałowska, Helga Margrete Bodahl Holmestad, Signe Riemer-Sørensen

We propose a new method for inferring roads from GPS trajectories to map construction sites. This task presents a unique challenge due to the erratic and non-standard movement patterns of construction machinery, which significantly diverge from typical vehicular traffic on established roads. Our proposed method first identifies intersections in the road network that serve as critical decision points, and then connects them with edges to produce a graph, which can subsequently be used for planning and task-allocation. We demonstrate the approach by mapping roads at a real-life construction site in Norway. The method is validated on four increasingly complex segments of the map. In our tests, the method achieved perfect accuracy in detecting intersections and inferring roads in data with no or low noise, while its performance was reduced in areas with significant noise and consistently missing GPS updates.

Aleksandra Jekic, Afroditi Natsaridou, Signe Riemer-Sørensen et al.

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for which many properties and limitations remain unknown. PINNs are widely accepted as inferior to traditional methods for solving PDEs, such as the finite element method, both with regard to computation time and accuracy. However, PINNs are commonly claimed to show promise in solving inverse problems and handling noisy or incomplete data. We compare the performance of PINNs in solving inverse problems with that of a traditional approach using the finite element method combined with a numerical optimizer. The models are tested on a series of increasingly difficult fluid mechanics problems, with and without noise. We find that while PINNs may require less human effort and specialized knowledge, they are outperformed by the traditional approach. However, the difference appears to decrease with higher dimensions and more data. We identify common failures during training to be addressed if the performance of PINNs on noisy inverse problems is to become more competitive.

Pål V. Johnsen, Eivind Bøhn, Sølve Eidnes et al.

Time-series modeling 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 model 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 inference, 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, utilizing 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. This approach shows promise in developing automatic, adaptable forecasting models for decision-making and control systems in process industries and other complex systems.

Sigurd Holmsen, Sølve Eidnes, Signe Riemer-Sørensen

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.

Marius C. Landverk, Signe Riemer-Sørensen

Measuring model performance is a key issue for deep learning practitioners. However, we often lack the ability to explain why a specific architecture attains superior predictive accuracy for a given data set. Often, validation accuracy is used as a performance heuristic quantifying how well a network generalizes to unseen data, but it does not capture anything about the information flow in the model. Mutual information can be used as a measure of the quality of internal representations in deep learning models, and the information plane may provide insights into whether the model exploits the available information in the data. The information plane has previously been explored for fully connected neural networks and convolutional architectures. We present an architecture-agnostic method for tracking a network's internal representations during training, which are then used to create the mutual information plane. The method is exemplified for graph-based neural networks fitted on citation data. We compare how the inductive bias introduced in graph-based architectures changes the mutual information plane relative to a fully connected neural network.

Pål Vegard Johnsen, Inga Strümke, Signe Riemer-Sørensen et al.

Estimating feature importance is a significant aspect of explaining data-based models. Besides explaining the model itself, an equally relevant question is which features are important in the underlying data generating process. We present a Shapley value based framework for inferring the importance of individual features, including uncertainty in the estimator. We build upon the recently published feature importance measure of SAGE (Shapley additive global importance) and introduce sub-SAGE which can be estimated without resampling for tree-based models. We argue that the uncertainties can be estimated from bootstrapping and demonstrate the approach for tree ensemble methods. The framework is exemplified on synthetic data as well as high-dimensional genomics data.

Signe Riemer-Sørensen, Jie Wu, Halvor Lie et al.

Slender marine structures such as deep-water marine risers are subjected to currents and will normally experience Vortex Induced Vibrations (VIV), which can cause fast accumulation of fatigue damage. The ocean current is often three-dimensional (3D), i.e., the direction and magnitude of the current vary throughout the water column. Today, semi-empirical tools are used by the industry to predict VIV induced fatigue on risers. The load model and hydrodynamic parameters in present VIV prediction tools are developed based on two-dimensional (2D) flow conditions, as it is challenging to consider the effect of 3D flow along the risers. Accordingly, the current profiles must be purposely made 2D during the design process, which leads to significant uncertainty in the prediction results. Further, due to the limitations in the laboratory, VIV model tests are mostly carried out under 2D flow conditions and thus little experimental data exist to document VIV response of riser subjected to varying directions of the current. However, a few experiments have been conducted with 3D current. We have used results from one of these experiments to investigate how well 1) traditional and 2) an alternative method based on a data driven prediction can describe VIV in 3D currents. Data driven modelling is particularly suited for complicated problems with many parameters and non-linear relationships. We have applied a data clustering algorithm to the experimental 3D flow data in order to identify measurable parameters that can influence responses. The riser responses are grouped based on their statistical characteristics, which relate to the direction of the flow. Furthermore we fit a random forest regression model to the measured VIV response and compare its performance with the predictions of existing VIV prediction tools (VIVANA-FD).