Sølve Eidnes

Elena Celledoni, Sølve Eidnes, Brynjulf Owren et al.

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.

Sølve Eidnes, Brynjulf Owren, Torbjørn Ringholm

A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches, thereby also including finite element approaches. The schemes are then extended to accommodate $r$-, $h$- and $p$-adaptivity. The method is applied to the Korteweg-de Vries equation and the Sine-Gordon equation and results from numerical experiments are presented.

Elena Celledoni, Sølve Eidnes, Alexander Schmeding

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.

Håkon Noren, Sølve Eidnes, Elena Celledoni

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.

Elena Celledoni, Sølve Eidnes, Markus Eslitzbichler et al.

In this paper we are concerned with the approach to shape analysis based on the so called Square Root Velocity Transform (SRVT). We propose a generalisation of the SRVT from Euclidean spaces to shape spaces of curves on Lie groups and on homogeneous manifolds. The main idea behind our approach is to exploit the geometry of the natural Lie group actions on these spaces.

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.

Sølve Eidnes, Kjetil Olsen Lye

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.

Sunniva Meltzer, Sølve Eidnes, Alexander Johannes Stasik

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.

Sølve Eidnes, Torbjørn Ringholm

Energy preserving numerical methods for a certain class of PDEs are derived, applying the partition of unity method. The methods are extended to also be applicable in combination with moving mesh methods by the rezoning approach. These energy preserving moving mesh methods are then applied to the Benjamin--Bona--Mahony equation, resulting in schemes that exactly preserve an approximation to one of the Hamiltonians of the system. Numerical experiments that demonstrate the advantages of the methods are presented.

Eivind Bøhn, Sølve Eidnes, Kjell Rune Jonassen

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 ($\mathrm{NO_3^-}$) reduction to molecular nitrogen ($\mathrm{N_2}$) in the biogeochemical process known as \textit{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.

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.

Elena Celledoni, Sølve Eidnes, Brynjulf Owren et al.

This paper concerns an extension of discrete gradient methods to finite-dimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative ordinary differential equations. This includes Riemannian gradient flow systems which occur naturally in optimization problems. The Itoh--Abe discrete gradient is formulated and applied to gradient systems, yielding a derivative-free optimization algorithm. The algorithm is tested on two eigenvalue problems and two problems from manifold valued imaging: InSAR denoising and DTI denoising.