This article presents a unified framework for constrained optimization solved by using damped dynamical systems on the Stiefel manifold, combining variational principles, projected-gradient methodologies, and asymptotic stability theory. For smooth objective functions defined on the Stiefel manifold, first-order optimality conditions are derived using both intrinsic tangent-space projections and classical Lagrange multiplier formulations, which naturally lead to second-order damped dynamical systems whose equilibrium points coincide with the Karush–Kuhn–Tucker solutions of the constrained optimization problem. Two complementary formulations are studied in detail: a Lagrange-based approach in which constraint satisfaction is enforced through dynamically evolving multipliers, and a projected-gradient formulation in which the dynamics evolve intrinsically on the tangent bundle of the Stiefel manifold. It is shown that both formulations admit identical stationary solutions, and explicit analytical relationships between the Lagrangian and projected dynamics are established. The proposed framework is applied to two canonical problems, namely the linear eigenvalue problem for computing invariant subspaces associated with the smallest eigenvalues of a symmetric positive definite matrix and the orthogonal Procrustes problem formulated in the Frobenius norm, for which explicit expressions for the gradients, multiplier dynamics, and reduced systems are derived. A rigorous asymptotic stability analysis is carried out by linearizing the resulting first-order systems and characterizing the spectra of the associated reduced Jacobian operators acting on the tangent space of the constraint manifold, leading to sufficient conditions for asymptotic stability that clarify the role of damping parameters in guaranteeing convergence. Numerical implementations are developed by discretizing the proposed second-order dynamical systems using a symplectic Euler scheme that preserves the qualitative stability properties of the continuous-time models, resulting in algorithms that rely on standard linear algebra operations, including Sylvester equation solvers and thin singular value decompositions, and that enable a direct comparison between intrinsic and extrinsic approaches to optimization on the Stiefel manifold.

Covering Tour Problem (CTP) is a combinatorial optimization problem in which the objective is to identify a minimum-cost tour that satisfies the coverage of a certain subset of nodes in a graph. The Covering Tour Problem with Varying Coverage (CTP-VC) is an extension of this problem in which the coverage radius is dependent on the amount of time spent at each node. In this paper, we propose a novel approach to address the CTP-VC using a Deep Reinforcement Learning Hyperheuristic (DRLH). This study includes experiments on the existing Adaptive Metaheuristic to solve CTP-VC, to enhance its solution quality. Further, new heuristics and three selection methods, namely Uniform Random Selection (URS), adaptive Metaheuristic (AMH), and the proposed DRLH are introduced. We detail the computational setup, including the instance sets utilized, the training process for the DRLH agent, and the validation procedures for model selection. Through extensive experimentation and analysis, we evaluate the performance of different selection methods, assess the solution quality of the DRLH approach, investigate the robustness of selection methods, examine heuristic selection frequency, and analyze solution convergence. Our results demonstrate the efficacy of the DRLH approach in tackling the CTP-VC, offering promising insights for future research in the interface of combinatorial optimization and reinforcement learning methodologies.

Underwater acoustics is an important component of fisheries management. Echograms, generated from acoustic backscatter data collected via echosounders, provide information on the distribution and abundance of marine life. Acoustic target classification (ATC) categorizes backscatters by assigning them into specific groups, such as fish or seabed. Traditionally, ATC requires extensive manual annotation, which is both time-consuming and prone to inconsistency. Automated approaches often rely on labeled data, but obtaining precise annotations remains challenging. This study introduces a method combining self-supervised learning (SSL) and clustering to perform high-resolution analysis of acoustic data. SSL features are extracted from echograms and clustered to classify pixels in a binary task. To address class imbalance, an over-clustering strategy is applied, followed by a greedy cluster selection algorithm guided by labeled data to maximize the F1 score. Aggregated probabilities for each pixel are calculated based on the selected clusters and thresholded to assign class labels. This feature-based approach is compared to a version using untreated data instead of features, and to a fully supervised method. Results show that the feature-based approach outperforms the raw data-based version and achieves performance close to the supervised model. Self-supervised data representations enable the training of simple yet effective downstream classification models.

The main object of our study is travelling waves in vast neuronal ensembles modelled using neural field equations. We obtained conditions that guarantee the existence of travelling wave solutions and their continuous dependence under the transition from sigmoidal neuronal activation functions to the Heaviside activation function. We, thus, filled the gap between the continuous and the discontinuous approaches to the formalization of the neuronal activation process in studies of travelling waves. We provided conditions for admissibility to operate with simple closed-form expressions for travelling waves, as well as to significantly simplify their numerical investigation. This opens the possibilities of linking characteristics of cortical travelling waves, e.g., the wave shape and the wave speed, to the physiological parameters of the neural medium, e.g., the lengths and the strengths of neuronal connections and the neuronal activation thresholds, in the framework of the neural field theory.

This paper examines optimal portfolio selection using quantile-based risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We address the case of a singular covariance matrix of asset returns, which may arise due to potential multicollinearity and strong correlations. This leads to an optimization problem with infinitely many solutions. An analytical form for a general solution is derived, along with a unique solution that minimizes the L2-norm. We show that the general solution reduces to the standard optimal portfolio for VaR and CVaR when the covariance matrix is non-singular. We also provide a brief discussion of the efficient frontier in this context. Finally, we present a real-data example based on the weekly log returns of assets included in the S&P 500 index.

We will report on two studies we have done related to Lagrangian transport of plastic particle in marine waters. In the first study, we investigate the particle dynamics of tyre-wear microplastics that come from road traffic across two major bridges in Byfjorden, namely the Nordhordland Bridge and the Askøy Bridge. We employ a Lagrangian particle tracking framework, OpenDrift, with background horizontal velocities from Bergen Ocean Model (BOM), paired with a vertical sinking velocity obtained from Stokes law to track individual particle paths along the flow field until they reach the seafloor. The sinking velocity is picked from a distribution that is designed based on results from point source experiments, enabling us to cover the particle dynamics for a spectrum of sinking velocities. The basis of this study lies in using the variability in local currents, by conducting multiple experiments with distinct initial locations and release times to understand the similarities and differences in the footprint. In the particle simulation, the horizontal velocity experienced by individual particles depends on release time which is related to when in the tidal cycle the particle is released. We seek insights to discover potential aggregation zones and their corresponding gradients along the bottom of the fjord. We plan to shed light on ‘how particle dynamics change when we vary the sinking velocity’. These results could be applicable in identifying the mechanisms behind particle transport in fjords and can assist in designing sampling campaigns. In the second, we assess the amount of transboundary plastic coming along the coast of western Norway, employing a nested modelling approach. We utilize emissions data of buoyant plastics from major European rivers (Meijer et al., 2021), as an input to our Lagrangian particle tracking model simulated using OpenDrift. The background currents are provided by the nested model which includes surface currents from three grids: A 4km model of the North Atlantic - Nordic4K (Lein et al., 2013), An 800m model covering Norway’s coastline - Norkyst800 (Albertsen et al., 2011), and 160m hydrodynamical model - NorFjords160 (Dalsøren et al., 2020). As particles transit through these nested grids, we precisely track the plastic pathways into the western Norwegian fjords around the city of Bergen. Employing this nested grid setup addresses problems with boundary conditions and mass balance. We present the estimates for the fraction of plastic moving into the fjord with focus on relative influence of wind and ocean currents on the transboundary movement of plastic. This study sheds light on processes responsible for near and far field transport, providing valuable insights for agencies working on trans-national pollution laws and implementing ocean clean-up strategies.

Marine plastic debris (MPD) is a form of transboundary pollution, with many common types having far-field transport potential. Such movement can undermine efforts to tackle marine plastic pollution and disincentivize taking action. Norway's extended coastline, which borders a major transport pathway out of the North Sea - the Norwegian Coastal Current (NCC) - makes it susceptible to transboundary fluxes of MPD from other countries. The complex network of fjords along the coast can act as natural traps for marine litter transported via the NCC. In this study, we investigate the origins of transboundary plastic reaching Western Norway using a nested modelling approach. Buoyant plastic emissions from major European rivers serve as input for a Lagrangian particle tracking model. Background currents are provided by three nested ocean models with increasing resolution: a 4 km model of the North-East Atlantic, an 800 m model of the Norwegian coast, and a 160 m hydrodynamic model for western Norway. As particles move through these grids, we track their pathways to the Norwegian Exclusive Economic Zone (EEZ) and into western Norwegian fjords. Between 15%-88% of released plastic mass enters the Norwegian EEZ from the countries bordering North Sea, of which 1%-35% enters fjords along the west coast. Five major rivers are identified as significant contributors to the simulated plastic load (both in terms of mass and number of particles) in these fjords, where concentrations can be an order of magnitude higher than in surrounding waters. The study highlights key processes in both near- and far-field plastic transport and offers insight for agencies working on international pollution policy and cleanup efforts.

Acoustic surveys play a pivotal role in fisheries management. During the surveys, acoustic signals are sent into the water and the strength of the reflection, so-called backscatter, is recorded. The collected data are typically annotated manually, a process that is both labor-intensive and time-consuming, to support acoustic target classification (ATC). The primary objective of this study is to develop an annotation-free deep learning model that extracts acoustic features and improves the representation of acoustic data. For this purpose, we adopt a self-supervised method inspired by the Self DIstillation with NO Labels (DINO) model. Extracting useful acoustic features is an intricate task due to the inherent variability and complexity in biological targets, as well as environmental and technical factors influencing sound interactions. The proposed model is trained with three sampling methods: random sampling, which ignores class imbalance present in the acoustic survey data; class-balanced sampling, which ensures equal representation of known categories; and intensity-based sampling, which selects data to capture backscatter variations. The quality of extracted features is then evaluated and compared. We show that the extracted features lead to improvement, in comparison to using the untreated data, in the discriminative power of several machine learning methods (k-nearest neighbor (kNN), linear regression, multinomial logistic regression) for ATC. The improvement was measured through higher accuracy in kNN (77.55% vs. 71.93%), Macro AUC in logistic regression (0.92 vs. 0.80), and R2 in linear regression (0.69 vs. 0.45) when comparing extracted features to the untreated data. Our findings highlight the advantage of applying emerging self-supervised techniques in fisheries acoustics. This study thus contributes to the ongoing efforts to improve the efficiency of acoustic surveys in fisheries management.

Abstract This paper introduces a novel methodology for characterizing parameter equivalence between Rectified Linear Unit (ReLU) Neural Networks (NNs) and 1D Finite Element Models (FEMs). By expressing the FEM solution equation in terms of neural networks and employing a hybrid scheme involving partially trained nested neural networks, we bridge the gap between these two computational paradigms. Our approach offers a promising avenue to leverage the flexibility of neural networks while retaining the established principles of FEM. Through extensive numerical experiments, we demonstrate the efficacy of our scheme in function interpolation and solving partial differential equations (PDEs). In function interpolation tasks, our hybrid scheme consistently outperforms fully trainable ReLU NNs, achieving significantly lower error factors across various scenarios. Similarly, in solving PDEs, our approach exhibits superior accuracy compared to fully trainable ReLU NNs, as evidenced by lower L2-error factors. We further extend the application of our hybrid scheme to diverse PDEs including the heat, Poisson, and wave equations, showcasing its versatility and effectiveness across different problem domains. This work not only presents a significant advancement in computational methods but also lays the groundwork for redefining the FEM algorithm within an algorithmic framework. These findings have implications for a wide range of scientific and engineering disciplines, and we believe they warrant consideration for publication in high-impact journals.

In this paper, we present a novel variant of the Covering Tour Problem (CTP), called the Covering Tour Problem with Varying Coverage (CTP-VC). We consider a simple graph G=(V,E), with a measure of importance assigned to each node in V. A vehicle with limited battery capacity visits the nodes of the graph and has the ability to stay in each node for a certain period of time, which determines the coverage radius at the node. We refer to this feature as stay-dependent varying coverage or, in short, varying coverage. The objective is to maximize a scalarization of the weighted coverage of the nodes and the negation of the cost of moving and staying at the nodes. This problem arises in the monitoring of marine environments, where pollutants can be measured at locations far from the source due to ocean currents. To solve the CTP-VC, we propose a mathematical formulation and a heuristic approach, given that the problem is NP-hard. Depending on the availability of solutions yielded by an exact solver, we evaluate our heuristic approach against the exact solver or a constructive heuristic on various instance sets and show how varying coverage improves performance. Additionally, we use an offshore CO2 storage site in the Gulf of Mexico as a case study to demonstrate the problem's applicability. Our results demonstrate that the proposed heuristic approach is an efficient and practical solution to the problem of stay-dependent varying coverage. We conduct numerous experiments and provide managerial insights.

In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz’ problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.

Acoustic surveys provide important data for fisheries management. During the surveys, ship-mounted echo sounders send acoustic signals into the water and measure the strength of the reflection, so-called backscatter. Acoustic target classification (ATC) aims to identify backscatter signals by categorizing them into specific groups, e.g. sandeel, mackerel, and background (as bottom and plankton). Convolutional neural networks typically perform well for ATC but fail in cases where the background class is similar to the foreground class. In this study, we discuss how to address the challenge of class imbalance in the sampling of training and validation data for deep convolutional neural networks. The proposed strategy seeks to equally sample areas containing all different classes while prioritizing background data that have similar characteristics to the foreground class. We investigate the performance of the proposed sampling methodology for ATC using a previously published deep convolutional neural network architecture on sandeel data. Our results demonstrate that utilizing this approach enables accurate target classification even when dealing with imbalanced data. This is particularly relevant for pixel-wise semantic segmentation tasks conducted on extensive datasets. The proposed methodology utilizes state-of-the-art deep learning techniques and ensures a systematic approach to data balancing, avoiding ad hoc methods.

Optimised marine monitoring strategies for CCS can be developed through several computational approaches. Many of these methods exist already including simulating hypothetical release events through models of hydrodynamics and biogeochemistry, quantifying highly sensitive criteria to distinguish from anomalous background conditions as an anomaly indicator, and the use of complex algorithms or machine learning to increases the certainty of detecting anomalies whilst utilising the most cost effective mobile or stationary monitoring platform distributions. However, one of the biggest challenges is the accessibility of these approaches, including closed source programs and inaccessible data, along with the need for programming skills and high-powered supercomputers. We integrate a number of these approaches and make them accessible through the ACTOM toolbox, the end-product of which is to aid users in defining a monitoring plan that will satisfy local stakeholders The toolbox uses pre-existing hydrodynamic and marine biogeochemical data that can be readily attained from existing high-resolution models and simulations, combined with available in-situ measurements, allowing the methodologies to be applied coherently to multiple offshore storage sites and to answer specific stakeholder led questions around required sensitivity and assurance levels. The ACTOM toolbox is designed to provide value over a range of field cases with diverse subsurface geology and environmental marine characteristics. Here we present the technical details of the ACTOM toolbox, and its components. The preparation steps for various sites, and how the toolbox can address regional concerns and be tuned to local characteristics is presented in another contribution to GHGT-16.

The ACTOM toolbox is designed to provide value over a range of field cases with diverse subsurface geology and environmental marine characteristics. Whereas the end-product of the toolbox is to aid users in defining a monitoring plan that will satisfy local stakeholders, one intermediate outcome from our study is to understand how data quality and availability as well as site variability will affect the development of the monitoring plan. For this assessment we collect and use available data from three representative sites as input to the toolbox: the Gulf of Mexico and northern and southern North Sea to visualize how the output from the semi- automated planning toolbox works in practice. In an ideal setting, hydrodynamic and marine biogeochemical data can be readily attained from calibrated high-resolution models and simulations, combined with available in-situ measurements. The process of identifying data availability, and later gathering and pre-processing the data, will have to be site specific. The technical details of the toolbox, and its components will be presented in another contribution to GHGT-16, here we will demonstrate the preparation steps for the three sites, and how the toolbox can address regional concerns and be tuned to local characteristics.

Carbon capture and storage is key for mitigating greenhouse gas emissions, and offshore geological formations provide vast CO2 storage potential. Monitoring of sub-seabed CO2 storage sites requires that anomalies signifying a loss of containment be detected, and if attributed to storage, quantified and their impact assessed. However, monitoring at or above the seabed is only useful if one can reliably differentiate abnormal signals from natural variability. Baseline acquisition is the default option for describing the natural state, however we argue that a comprehensive baseline assessment is likely expensive and time-bound, given the multi-decadal nature of CCS operations and the dynamic heterogeneity of the marine environment. We present an outline of the elements comprising an efficient marine environmental baseline to support offshore monitoring. We demonstrate that many of these elements can be derived from pre-existing and ongoing sources, not necessarily related to CCS project development. We argue that a sufficient baseline can be achieved by identifying key emergent properties of the system rather than assembling an extensive description of the physical, chemical and biological states. Further, that contemporary comparisons between impacted and non-impacted sites are likely to be as valuable as before and after comparisons. However, as these emergent properties may be nuanced between sites and seasons and comparative studies need to be validated by the careful choice of reference site, a site-specific understanding of the scales of heterogeneity will be an invaluable component of a baseline.

We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of $2\times 2$ -stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.

Carbon capture and storage (CCS) is a key technology to reduce carbon dioxide (CO2) emissions from industrial processes in a feasible, substantial, and timely manner. For geological CO2 storage to be safe, reliable, and accepted by society, robust strategies for CO2 leakage detection, quantification and management are crucial. The STEMM-CCS (Strategies for Environmental Monitoring of Marine Carbon Capture and Storage) project aimed to provide techniques and understanding to enable and inform cost-effective monitoring of CCS sites in the marine environment. A controlled CO2 release experiment was carried out in the central North Sea, designed to mimic an unintended emission of CO2 from a subsurface CO2 storage site to the seafloor. A total of 675 kg of CO2 were released into the shallow sediments (∼3 m below seafloor), at flow rates between 6 and 143 kg/d. A combination of novel techniques, adapted versions of existing techniques, and well-proven standard techniques were used to detect, characterise and quantify gaseous and dissolved CO2 in the sediments and the overlying seawater. This paper provides an overview of this ambitious field experiment. We describe the preparatory work prior to the release experiment, the experimental layout and procedures, the methods tested, and summarise the main results and the lessons learnt.

Over the last few years a number of computational approaches have been developed that enable the optimisation of marine monitoring strategies for CCS. These include methods to: simulate and quantify hypothetical release events, identify highly sensitive criteria by which to distinguish anomalous biogeochemistry which may indicate a seep and define minimum deployments of monitoring platforms which guarantee an acceptable chance of detection. In order for these tools to be operationally useful it is necessary to place them in an accessible and integrated digital format that allows the methodologies to be applied coherently to multiple offshore storage sites and to answer specific stakeholder led questions around required sensitivity, cost and assurance levels. In this work we describe progress towards achieving this integration as well as the site specific data necessary to generate bespoke monitoring guidance.

We present a new data-driven model to reconstruct nonlinear flow from spatially sparse observations. The proposed model is a version of a Conditional Variational Auto-Encoder (CVAE), which allows for probabilistic reconstruction and thus uncertainty quantification of the prediction. We show that in our model, conditioning on measurements from the complete flow data leads to a CVAE where only the decoder depends on the measurements. For this reason, we call the model semi-conditional variational autoencoder. The method, reconstructions, and associated uncertainty estimates are illustrated on the velocity data from simulations of 2D flow around a cylinder and bottom currents from a simulation of the southern North Sea by the Bergen Ocean Model. The reconstruction errors are compared to those of the Gappy proper orthogonal decomposition method.