Multifidelity digital twin for real-time monitoring of structural dynamics in aquaculture net cages

Abstract

As the global population grows, ensuring sustainable food production has become critical. Marine aquaculture provides a sustainable and scalable source of protein; however, its continued expansion requires the development of novel technologies that enable remote management and autonomous operations. Digital twin technology emerges as a transformative tool for realizing this goal, yet its adoption remains limited. Fish net cages—flexible, floating structures—are critical but vulnerable components of aquaculture systems. Exposed to harsh and dynamic marine conditions, they experience substantial hydrodynamic loads that can cause structural damage leading to fish escapes, environmental impacts, and financial losses. We propose a multifidelity surrogate modeling framework for integration into a digital twin that enables real-time monitoring of net cage structural dynamics under stochastic marine conditions. At the core of the framework lies the nonlinear autoregressive Gaussian process method, which captures complex, nonlinear cross-correlations between models of varying fidelity. It combines low-fidelity simulation data with a limited set of high-fidelity field sensor measurements, which, although accurate, are costly and spatially sparse. The framework was validated at the SINTEF ACE fish farm in Norway, where the digital twin assimilates online metocean data to accurately predict net cage displacements and mooring line loads, closely matching field measurements. This approach is especially valuable in data-scarce environments, offering rapid predictions and real-time structural representation. Beyond monitoring, the developed digital twin enables proactive assessment of structural integrity and supports remote operations with unmanned underwater vehicles. Finally, we compare Gaussian processes and graph convolutional networks for predicting net cage deformation, demonstrating the superior ability of the latter to capture in complex structural behaviors.

Introduction

As the global population grows and the effects of climate change escalate, concerns about future food security and human nutrition increase, creating an urgent demand to enhance food production^1–3. Fish and fishery products have become vital in diets worldwide, particularly in developing countries⁴, but unsustainable fishing practices threaten natural fish populations⁵. Marine aquaculture emerges as a vital solution³, offering a sustainable protein source that meets the rising global food demands while serving as a response to climate change. Fish farms have lower carbon emissions⁶ compared to other animal proteins and support a diverse, resilient food system. Sustainable growth of aquaculture requires further reducing the carbon footprint of fish farms through strategic farm placement, efficient feed use, minimizing food waste and fish mortality rates, and the adoption of novel technologies to enhance autonomy in challenging fish farm operations. As the industry matures and operations move offshore to harsh and uncertain environments⁷, advancing real-time monitoring, operational planning, and decision-making practices is essential for ensuring efficiency, safety, cost-effectiveness, and sustainability^8,9.

Digital twins have great success in various sectors, yet its adoption in aquaculture has been limited, partly because it is a comparatively young industry⁹. Digital twin technology can significantly advance aquaculture by enabling fish farm remote monitoring, autonomous operations, management, and predictive analytics^8,9. Integrating real-time observations from in-situ monitoring equipment alongside realistic models within a digital twin framework can provide more detailed and accurate insights into the dynamics of fish farms. Implementing a digital twin typically involves several components^8,9, such as assessing fish health and behavior¹⁰, tracking environmental conditions¹¹, and monitoring the structural dynamics of fish net cages¹². Our study primarily focuses on the latter component.

Fish net cages are flexible floating structures anchored to the seabed, serving as critical yet vulnerable components of aquaculture farms. Exposed to the harsh and dynamic marine environment, they experience significant loads and are at risk of damage that can lead to fish escapes, resulting in negative environmental impacts and substantial financial losses¹³. Influenced by waves and currents, the net cages are prone to large motions and deformations¹⁴ due to their elastic structure. These deformations reduce the available space for fish, causing decreased oxygen levels and increased stress¹⁵, which negatively impacts fish growth and survival, leading to higher mortality rates and significant food waste. The development of a digital twin capable to real-time monitor the net cage dynamics can serve as an early warning system for fish farm operators, helping to prevent significant reductions in cage volume. Additionally, a digital twin is essential for assessing the structural integrity and preventing potential catastrophic damages, while it can be used to facilitate the autonomous inspection and repair operations with unmanned underwater vehicles (UUVs) in fish farms. For example, when autonomously controlling a remotely operated vehicle (ROV) within the fish cage to assess potential damage, prior knowledge of the cage’s shape is crucial for effective path planning and collision avoidance with the net¹⁶.

Unlike more rigid marine structures such as floating wind turbine foundations or vessels, the elastic and complex geometry of aquaculture net cages–composed of millions of slender twines–poses unique challenges in predicting their dynamics¹⁷. Modeling these structures is a multifaceted task that involves solving sub-problems: one focuses on the interaction between the fluid and a rigid moored-floating structure, and the other addresses the flexible porous net. Accurately depicting the deformation of fish net cages requires detailed information on the displacements of each twine, necessitating extensive data and computational effort. While high-fidelity numerical models like CFD simulations provide accurate estimations of structural dynamics¹⁷, they demand vast computational resources, making them unsuitable for real-time forecasting or rapid hindcasting tasks in digital twins. Lower fidelity models^18–21 are more accessible and less computationally demanding but often fail to capture the real system accurately, particularly under non-linear effects from extreme waves and fluid-structure interactions, thus limiting their integration into digital twins. Conversely, field sensor measurements offer the most accurate depiction of net cage dynamics but are constrained by high sensor costs and the risk of data transmission loss¹². Additionally, sensors typically provide localized information, resulting in significant spatial gaps or high sparsity. These challenges underscore the need for innovative approaches to effectively monitor structural dynamics in real-time.

Current research is focused on innovative models for precise real-time monitoring of net cage dynamics that can be integrated into digital twins. Su et al.¹⁴ introduced a real-time numerical simulation model that uses in-situ, real-time positioning sensor data to dynamically adjust model inputs–i.e., the magnitude and direction of the current–based on the difference between simulated and actual net positions. Building on this, Su et al. further developed their model into a physics-based digital twin¹² that incorporates in-situ sensor data to accurately simulate real-time cage responses, net deformations, and mooring loads. Although successfully tested at a full-scale aquaculture site, this model faces challenges such as high computational costs and potential inaccuracies due to inconsistent or faulty sensor data caused by harsh environmental conditions, hardware limitations, and unexpected interference, making it less suited for real-time applications. These issues highlight the need for digital twins using data-driven and machine learning (ML)-based surrogate models^22,23.

ML-based surrogate models for predicting the dynamic behavior of fish net cages remain an under-explored research area. Some studies utilized artificial neural networks (ANNs) for net damage detection^24–26, yet research on predicting the structural dynamics of aquaculture net pens remains sparse. Zhao et al.²⁷ developed a backpropagation ANN with one hidden layer, using training data from LF numerical simulations, to correlate ocean waves with net cage structural responses, including maximum tension in mooring lines, minimum effective volume ratio of the cage, and maximum stress on the floating collar. Although their model outputs the volume ratio, it does not capture the precise topology of the deformed net under wave excitation. To the best of our knowledge, no surrogate models have been developed to predict the entire net pen deformation, which can be beneficial for preventing catastrophic events and supporting autonomous fish farm operations. This underscores the need for more advanced ML-based surrogate models, built on high-fidelity data, that can accurately predict the complex dynamics of flexible net pens. To support effective real-time monitoring, these models should be capable of integrating into on-the-fly digital twins, which means their computational demands need to be limited.

Creating surrogate models that accurately represent complex systems often encounters the challenge of acquiring high-fidelity, application-specific data. Multifidelity data assimilation²⁸ involves methods that merge information about the same underlying truth obtained through multiple models or observation operators at different fidelity levels, resulting in a model that best represents the system’s state. Among these methods, optimal interpolation is a widely used data assimilation technique; however, it may exhibit large discrepancies with the truth²⁹. To this end, multifidelity surrogate modeling³⁰ aims to leverage machine learning methods with computationally inexpensive low-fidelity data and scarce high-fidelity data to accurately predict the quantities of interest. Despite their limitations, low-fidelity data can reveal system trends and patterns, providing valuable information to complement the limited HF data used for model training. Multifidelity methods^30–36 for constructing surrogate models aim to achieve effective data fusion from various fidelity levels, enabling strong generalization performance of data-driven models in regions where high-fidelity data are limited. This approach is particularly beneficial for various digital twin applications^37–40, enabling accurate, real-time system representation.

We propose a framework specifically designed for seamless integration into digital twins, enabling real-time monitoring and visualization of aquaculture net pen structural dynamics under dynamically evolving marine conditions. The framework maps site-specific environmental inputs–such as wave and current conditions–to structural responses of the net cages, including displacements and mooring line loads, offering a realistic representation of its physical behavior. Accurately capturing this behavior typically requires high-fidelity data, such as sensor measurements or other validated sources that closely reflect the physical system. However, in marine applications, such data are often scarce due to restricted access, harsh environmental conditions, and the high cost of deployment and maintenance. To address these limitations, the proposed framework incorporates a multifidelity modeling approach that combines broadly available, lower-fidelity simulation data with limited high-fidelity observations. While the low-fidelity data may lack precision, they are more readily accessible; the high-fidelity data, though sparse, serve as critical anchors for ensuring physical accuracy. At the core of the framework is a Nonlinear Autoregressive Gaussian Process (NARGP) surrogate model³², which captures complex, nonlinear, and space-dependent correlations between different fidelity levels. While NARGP has shown strong performance in benchmark problems, its application in real-world systems has remained largely unexplored–this study demonstrates its effectiveness in a full-scale, operational environment.

We validated the proposed framework using data from the SINTEF ACE fish farm, a full-scale facility in Norway for testing innovative aquaculture technologies. The low-fidelity data were generated through numerical simulations¹⁹, which model the dynamic response of net cages to environmental exchitation. High-fidelity data were obtained from field measurements collected over a two-month period (January–March 2020). Although the high-fidelity dataset spans only a limited timeframe, it captures a particularly harsh winter season characterized by strong currents and wave loads–conditions under which nonlinear structural responses are most pronounced. As discussed by the co-authors in a previous study¹², this dataset likely captures biofouling effects, which are known to significantly alter the net’s hydrodynamic behavior and have been identified as a key factor contributing to the failure of numerical simulations to accurately reproduce observed net deformations. Our results show that the model successfully learns to map low-fidelity simulation outputs to realistic structural responses, even in this highly nonlinear regime. This highlights the framework’s capacity to capture complex dynamics. While broader data coverage would enhance model robustness, the current results establish a strong proof of concept. Moreover, the modular and lightweight architecture of the framework ensures that it can be efficiently retrained as additional data-including from future seasons or other sites–become available.

Due to its reduced algorithmic complexity and low computational cost, the proposed framework enables rapid predictions that are essential for real-time representation of the actual state of the system. This makes it well-suited for deployment in dynamic marine environments where fast decision-making is critical. In the operational phase, the digital twin continuously assimilates sensor data to estimate the current structural state of the net and detect anomalies in real time. It supports autonomous operations by enabling ROV path planning and forecasting short-term dynamics under changing environmental conditions. This capability was recently demonstrated in a full-scale industrial setting, where the same framework was successfully integrated into a closed-loop ROV path planning system to estimate the dynamic net shape in real time and avoid collisions during autonomous inspections¹⁶. Beyond real-time operations, the framework also supports broader lifecycle applications of digital twins in aquaculture. In the design phase, the digital twin serves as a virtual testbed for simulating alternative net cage and mooring system configurations, validating low-fidelity models, and optimizing design choices under diverse environmental conditions. This allows for early-stage identification of inefficiencies, structural weaknesses, and environmental impacts-offering a cost-effective means to iterate and refine designs before physical implementation, akin to digital twin-based smart manufacturing design workflows in Industry 4.0 and 5.0 contexts. During the configuration and reconfiguration phase, the digital twin can simulate alternative system layouts and mooring configurations under site-specific marine conditions, enabling rapid validation before implementation. In addition to semi-physical performance testing, the framework can support sensor placement optimization, evaluate risk under failure scenarios, and forecast environmental impacts, providing a practical tool for informed and adaptive farm reconfiguration. Figure 1 illustrates the full digital twin architecture, showcasing the multifidelity integration pipeline and its relevance across design, configuration, and operational phases. This framework significantly enhances efficiency, automation, safety, and cost savings. A key strength lies in its ability to operate with the existing sensing infrastructure already deployed at most aquaculture sites, minimizing the need for new instrumentation and associated costs.

Fig. 1.

Real-time monitoring and remote fish farm management using digital twin technology. The structural response of a net cage to dynamic marine environments is modeled using data from varying fidelity sources, including low-fidelity numerical simulations and high-fidelity field sensor measurements. A digital twin of the physical net cage is developed using the proposed multifidelity framework, central to which is the Nonlinear Autoregressive Gaussian Process (NARGP) method³². This method synergistically combines low-fidelity model realizations with a limited set of high-fidelity observations. The nonlinear autoregressive scheme effectively learns complex nonlinear cross-correlations between datasets of differing fidelity, which is essential for practical applications. The multifidelity digital twin accurately predicts the structural response under dynamic marine conditions, closely mirroring real behavior. The digital twin facilitates remote fish farm management, serving as an early warning system to prevent catastrophic damage, informing decision-making, and supporting autonomous operations such as ROV path planning for net inspection and repair, net cleaning, and equipment installation.

In addition to the proposed multifidelity framework, we explore alternative machine learning methods for constructing surrogate models capable of predicting the full deformation of the net pen under interaction with waves and currents. In particular, we assess the effectiveness of Graph Convolutional Networks (GCNs)^41,42. While GCNs have been increasingly applied across life sciences, physics, and materials research, our study highlights their potential for accurately capturing the complex spatial dependencies inherent in the structural dynamics of flexible marine structures⁴³.

Results

Our objective is to develop a digital-twin framework that enables real-time monitoring and visualization of fish net cage dynamics under stochastic marine conditions, ensuring operational efficiency, structural safety, and environmental sustainability. At the core of this framework lies machine-learned surrogate modeling, which establishes the mapping between the marine environmental inputs and the dynamic response of the net cage. Accurate prediction of real-world behavior requires comprehensive, high-quality data; however such data is often limited or expensive to obtain. When only a small number of high-fidelity observations—such as direct sensor measurements—are available, constructing reliable models becomes challenging. Consequently, practitioners frequently rely on simplified numerical models that provide lower-fidelity but more accessible data, creating a gap between simulated outcomes and actual field observations. To enable real-time operation, the digital twin must be both data-efficient and computationally lightweight. Therefore, incorporating reduced-complexity surrogate models is essential to achieve rapid training and execution while maintaining predictive accuracy.

We propose a multifidelity framework that integrates data from sources of varying fidelity to learn the system dynamics and produce outputs that closely replicate real-world observations. At the core of this framework lies the NARGP method³², which refines low-fidelity numerical simulation outputs to align more accurately with high-fidelity sensor measurements. This is achieved by learning the nonlinear correlations between these two data fidelities through data-efficient multifidelity information fusion algorithms built upon standard Gaussian process (GP) regression. A detailed description of the NARGP formulation is provided in the Methods section. In the present case study, the model is implemented with two fidelity levels, though the framework is readily scalable to incorporate additional fidelity layers if needed. The trained NARGP model is embedded within the digital twin, following the workflow illustrated in Fig. 1. It receives real-time metocean data, where the first GP model, trained on low-fidelity numerical simulations, predicts the net cage dynamics equivalent to the simulation results. These low-fidelity predictions, together with the metocean inputs, are then passed to the recursive GP surrogate which has been trained using the posterior mean from the low-fidelity model and a limited set of high-fidelity measurements. This recursive model yields predictions that accurately capture the true net cage response observed in the field.

This multifidelity approach enables the construction of accurate and computationally efficient surrogates without relying exclusively on costly high-fidelity simulations. By exploiting the learned correlations across fidelity levels, the framework substantially reduces data and computational requirements while maintaining high predictive accuracy. Consequently, it extends the applicability of digital twin technology to large-scale or data-scarce aquaculture systems, where acquiring extensive high-fidelity measurements is often impractical.

Case study for SINTEF ACE fish farm

The multifidelity framework for real-time monitoring of structural dynamics is developed and validated using data from the SINTEF ACE facility–a full-scale research farm dedicated to the development and testing of innovative aquaculture technologies. All data used in this study were collected from the Buholmen site of SINTEF ACE, which hosts several operational fish cages. The analyzed cage has a diameter of 50 m, a depth of 31 m, a solidity ratio of 0.21, and it is secured to the mooring frame via 12 bridle (mooring) lines. As detailed in the SINTEF report⁴⁴, environmental conditions at the Buholmen site have been characterized through continuous monitoring from January 2020 to October 2021. The average significant wave height is 0.47 m, with 84% of waves remaining below 0.8 m and a maximum recorded height of 1.84 m. The current profile is primarily bidirectional along the northeast–southwest axis, with current speeds remaining below 0.3 m/s for 80% of the time and exceeding 0.5 m/s only 2% of the time.

Data collection

See Fig. 2.

Fig. 2.

Net cage numerical representation and field sensor configuration. (a) Schematic of the fish-net cage showing its main structural components: floating collar, bottom ring, net pen, and mooring lines. (b) Numerical discretization of the net cage in the FhSim environment into 321 nodes across 10 layers, with the first 32 nodes forming the uppermost layer and node 321 positioned at the base connecting all nodes of the tenth layer. (c) The simulated net cage under combined wave and current excitations. (d) Top view of the fish farm layout highlighting cage number 11, which accommodates the sensors. A buoy located 400 m away measures waves and currents. The main flow direction is indicated. (e) Close-up of net cage 11 showing sensor placement: five load shackles measuring mooring line tension and three depth sensors tracking net displacement. (f) Side view of the net cage illustrating the vertical distribution of depth sensors along the structure.

Low-fidelity data

FhSim is a mathematical modeling and numerical simulation software tailored for marine applications, providing time-domain representations of complex system dynamics. In this study, we employ FhSim to simulate the dynamic response of a full-scale aquaculture net cage subjected to a wide range of marine conditions. To generate a comprehensive dataset for training surrogate models, we performed simulations across 1000 distinct sea states. These conditions span current velocities from 0 to 1 m/s, significant wave height from 0 to 3 m, peak wave period from 0 to 8.66 s, and wave/current directions uniformly distributed over 0–360 degrees–covering a range that exceeds the typical conditions observed at the Buholmen site. As shown in Fig. 2a, the simulated cage replicates the geometry of the real structure. While the physical net consists of thousands of twines, numerical simplification is required for computational tractability. The cage is therefore discretized into 321 representative nodes (Fig. 2b). Each simulation models a 30-min interaction between waves, currents and the net cage, producing the x, y, and z displacements of all nodes (Fig. 2c) as well as the load profiles on the 12 mooring lines. From each simulation, we extract time-averaged values of these quantities to build a reduced dataset that characterizes the steady-state structural deformation of the net cage and loads under each sea condition.

FhSim has previously been used by Su et al.¹² to simulate the dynamic response of net cages at the same fish farm studied here. In their work, they compared FhSim simulation results with real measurement data from the SINTEF ACE facility, which are also used as high-fidelity data in the present study. Their findings revealed that the direct FhSim output consistently underestimated net deformations. This discrepancy was partially attributed to biofouling effects that were captured in the field measurements, leading to increased net solidity and altered hydrodynamic behavior–phenomena not explicitly modeled in the numerical simulations. These findings reinforce our classification of FhSim as a low-fidelity model in this study and highlight the importance of incorporating high-fidelity data that reflect complex, real-world phenomena such as biofouling.

Additionally, despite being faster than high-fidelity numerical solvers (e.g. CFD simulations), FhSim simulations are not suitable for real-time deployment. On our hardware setup (56 CPU cores, 2.2 GHz, 128 GB RAM), each 30-min simulation required approximately 10 min of computational time, followed by an additional 20 min for processing and storing time-series outputs. These constraints reinforce the need for surrogate models capable of delivering rapid, accurate predictions for digital twin applications.

High-fidelity data

Field sensor measurements represent the high-fidelity data used in this study, capturing the actual dynamic response of the system. The monitoring campaign was conducted over a period of slightly less than two months, from January 18 to March 5, 2020-a time frame representative of relatively harsh winter marine conditions. As shown in Fig. 2d, a metocean buoy located approximately 400 meters from the fish farm was equipped with sensors to monitor environmental conditions, specifically recording incoming wave and current parameters. During this period, significant wave heights ranged from 0.3 to 1.6 m and the wave direction between 155 and 190 degrees (reflecting south-east to south-west directions). Current velocities ranged from 0.15 to 0.45 m/s, with directions between 60 and 126 degrees (north-east to south-east directions). Compared to long-term site conditions⁴⁴, which are generally moderate with wave heights below 0.8 m for 84% of the time and current speeds under 0.3 m/s for 80% of the time, the short-term monitoring period (January 18–March 5, 2020) captured more energetic conditions. Wave heights reached up to 1.6 m and current speeds up to 0.45 m/s, representing some of the higher load scenarios observed at the site.

Instrumentation was deployed on cage number 11 at the SINTEF fish farm (Fig. 2d). Five load shackles (#1–5 in Fig. 2e) were installed on the upstream side, where the prevailing water current originated, between the collar and the mooring lines to measure the mooring loads. Additionally, three depth sensors were positioned at key locations within the net pen: at 7 m deep, at 15 m depth at the boundary of the cylindrical section, and at 31 m at the bottom of the net (#1–3 in Fig. 2e and f). These sensors measured vertical displacements and provided critical data on the net cage deformation.

Data collection was performed using a wireless sensor network, capturing measurements at 2-second intervals for high temporal resolution. Continuous data were obtained from the metocean buoy and the three depth sensors throughout the entire monitoring period. In contrast, load shackle data were available only for two isolated intervals: 12 h on January 23 and 24 h on February 22, illustrating the potential for intermittent data loss in field deployments. Further details on the sensor setup and measurement campaign can be found in Su et al.¹².

Digital twin workflow

Figure 3a illustrates the digital twin workflow developed for the SINTEF ACE fish farm. A metocean buoy equipped with sensors provides real-time measurements of current speed and direction which serve as inputs to the NARGP multifidelity surrogate model. Initially, the low-fidelity GP models use the current data to predict the loads on the mooring lines and the overall deformation of the net cage under these conditions. To estimate the net cage displacement at specific locations, the user defines these locations (or nodes) on the net cage and obtains their vertical displacement. The low-fidelity predictions of mooring line force and node vertical displacement, along with the measured current speed and direction, are then fed into high-fidelity GP models. These models refine the low-fidelity predictions, providing outputs that closely match the real structural dynamics measured by load shackles and depth sensors. This workflow is completed in a few seconds due to the algorithmic simplicity and low computational cost of the NARGP method. The proposed digital twin enables real-time monitoring of net cage structural dynamics, supports autonomous operations, and informs decision-making without relying on real-time sensors from the net cage, as is currently the case¹². This approach reduces the need for extensive sensor deployment, which is often expensive and unreliable due to issues like data loss.

Fig. 3.

Multifidelity digital twin framework. (a) Workflow of the digital twin integrating multifidelity Gaussian Process (GP) models. Real-time measurements of current speed and direction from the metocean buoy are used as inputs to low-fidelity GP models that predict mooring line loads and net cage deformations. The low-fidelity predictions—represented by the GP posterior means—together with marine current measurements are subsequently passed to high-fidelity GP models, which refine these predictions to closely match load shackle and depth sensor observations. (b) Training of the low-fidelity GP models using data generated from FhSim simulations. One GP model maps marine current speed and direction to mooring line loads, while another predicts net cage deformations after dimensionality reduction via Principal Component Analysis (PCA). The PCA coefficients are predicted by the GP model and later reconstructed to obtain the net deformation field, enabling efficient surrogates that substitute full FhSim simulations. (c) Training of the high-fidelity GP models using multifidelity data. Limited field sensor measurements collected between January 18 and March 5, 2020—including current measurements from the metocean buoy, mooring loads from load shackles, and net displacements from depth sensors—are employed. The output of the low-fidelity model combined with the marine current measurements, serve as inputs to the high-fidelity models which yield refined predictions that accurately reproduce the measured mooring loads and vertical displacements at sensor locations.

Figure 3b shows the training stage of the low-fidelity GP surrogate models using data from FhSim simulations, described in the “Data Collection/Low-fidelity” section. Using the standard GP method⁴¹, we build surrogate models to map current speed and direction to the mooring line load. To create the surrogate model that maps ocean currents to net cage deformation, dimension reduction using PCA is required as a preliminary step, as detailed in the “Methods/Net cage displacement data dimension reduction using PCA” section. This is necessary due to the extensive data describing net cage deformation (x, y, z displacements for each of the 321 nodes). From the PCA dimension reduction step, we found that three PCA coefficients were sufficient and this allowed us to develop a standard GP model to map current characteristics to these PCA coefficients. Once the predicted coefficients are obtained, a reconstruction stage follows to determine the net cage deformation under the incoming measured current, as described in the “Methods/Machine learning functional relationships between current and PCA coefficients” section. Having the entire geometry of the deformed net cage allows us to extract the vertical displacement at the locations where the depth sensors are placed. The low-fidelity GP models developed at this stage can fully substitute FhSim simulations, with their accuracy well-documented by Katsidoniotaki et al.⁴⁷. This ensures that if the load shackles and depth sensors change position or more sensors are added in the future, retraining the low-fidelity GP models will not be necessary.

Figure 3c outlines the steps for training the high-fidelity GP models following the NARGP method, as described in the “Methods/NARGP” section. The high-fidelity training data consist of field sensor measurements, including current measurements from the metocean buoy, mooring loads from load shackles, and net cage vertical displacements from depth sensors, as detailed in the “Data Collection/High-Fidelity” section. The current measurements are input into the low-fidelity GP models, and the predicted GP posterior mean, combined with the current measurements, serve as inputs for the high-fidelity models. The high-fidelity GP models are trained on a limited number of datasets obtained from the period January 18 to March 5, 2020, to predict quantities of interest that match the load shackle and depth sensor measurements. Specifically, while the depth sensor measurements cover the entire period, only a small number of datasets corresponding to a few minutes are used to train the model. For the load shackle measurements, available only for 36 h, datasets corresponding to a few seconds are used for training. This approach demonstrates how, with limited high-fidelity datasets, we can develop a multifidelity model that provides predictions representative of real observations.

Predicting net cage dynamics

Mooring line loads. Figure 4 showcases the proposed multifidelity framework’s proficiency in predicting mooring line loads under current conditions, as measured by the metocean buoy, and compares these predictions with field measurements obtained from load shackles. The figure presents predictions for load shackles #1 (top row), #2 (middle row), and #5 (bottom row), with similar patterns observed for other mooring lines. Validation was conducted using previously unseen data from the entire 36-hour load shackle measurement period, excluding the training datasets. The 2D histograms in Fig. 4 are scatter plots with a color scale depicting the distribution and density of prediction accuracy from the low-fidelity and multifidelity GP models. The color scale indicates the log count density of data points, with lighter colors representing areas with a higher concentration of data points. Proximity to the dashed diagonal line () signifies prediction accuracy, while the red line represents the best-fit trend. The scatter plots for the low-fidelity GP model (Fig. 4a, d, g) reveal a broader spread and a significant deviation of the red best-fit line from the dashed diagonal line, indicating less accurate predictions. Conversely, the multifidelity GP model plots (Fig. 4b, e, h) exhibit a higher concentration of lighter colors near the dashed line, with the red best-fit line aligning more closely with the diagonal, indicating improved prediction accuracy. The color gradient helps identify trends and outliers; consistent, lighter colors along the dashed line suggest the model captures the overall trend well, whereas significant deviations indicate less accurate predictions. The Mean Absolute Error (MAE) quantifies prediction accuracy, calculated as the average of the absolute differences between the predicted and actual load measurements. The multifidelity model demonstrates a significant improvement in MAE values compared to the low-fidelity model. Figure 4 c, f, i provides a temporal view, displaying predictions over time based on the validation data from the 36-hour measurement period. The low-fidelity GP posterior mean deviates notably from the actual measurements, whereas the multifidelity GP posterior aligns closely with the real observations. The temporal plots reveal an oscillatory behavior in the load shackle measurements not fully captured by the multifidelity model, explaining some deviations from the dashed line in the middle column’s scatter plot. Nonetheless, the multifidelity model accurately follows the overall trend of the observations.

Fig. 4.

Multifidelity framework predictions for mooring line loads. Scatter plots (a), (d), (g) compare the low-fidelity GP posterior mean predictions with load shackle measurements. The color scale represents the logarithmic density of data points, with lighter colors indicating higher concentrations. The dashed diagonal line denotes perfect prediction accuracy; the red line represents the best-fit trend. Low-fidelity GP predictions have broad spread and the red line presents noticeable deviation from the diagonal, indicating reduced accuracy of the low-fidelity GP model. Scatter plots (b), (e), (h) show the multifidelity GP model predictions against sensor measurements. The higher concentration of lighter points near the diagonal and the close alignment of the red line with it demonstrate the improved accuracy achieved through the multifidelity framework. Plots (c), (f), (i) display time-series predictions over a 36-hour period on test data. The low-fidelity GP posterior mean shows significant deviation from the measured loads, whereas the multifidelity GP predictions closely follow the observed trends, accurately capturing the temporal evolution despite minor deviations in oscillatory behavior. The Mean Absolute Error (MAE) values confirm a substantial improvement in predictive accuracy with the multifidelity model compared to the low-fidelity baseline.

Vertical displacement at specific points within the net cage. Figure 5 illustrates the predictive performance of the multifidelity framework in estimating net cage displacement at three distinct locations, as measured by depth sensors #1 (top row), #2 (middle row), and #3 (bottom row). The validation dataset spans from January 18 to March 5, 2020, excluding the training periods. Figure 5a, d, g presents scatter plots contrasting the predictions of the low-fidelity GP model with actual sensor measurements. The broad dispersion of data points and the notable deviation of the red best-fit line from the dashed diagonal line underscore the model’s limited accuracy. In contrast, Fig. 5b, e, h displays scatter plots for the multifidelity GP model. Here, data points cluster more closely around the dashed line, and the red best-fit line aligns more precisely with the diagonal, indicating enhanced predictive accuracy. Despite the inherent complexities and dynamic motion of the flexible net cage, the multifidelity model’s predictions, though still dispersed, show improved accuracy, as evidenced by the color scale: darker points represent outliers, while lighter points closer to the dashed line signify higher accuracy. The MAE values annotated on these plots quantitatively highlight the multifidelity model’s superiority over the low-fidelity model. Figure 5c, f, i provides a temporal comparison of predicted and actual net cage displacements over a specific period (January 18 to January 27, 2020), juxtaposing the low-fidelity GP and multifidelity GP posterior means against the actual sensor data. The low-fidelity GP model (blue line) exhibits substantial deviations from the actual measurements (green line), particularly for depth sensors #1 and #3. Conversely, the multifidelity GP model (red line) significantly mitigates these discrepancies, closely mirroring the actual measurements and capturing the underlying trend with greater fidelity.

Fig. 5.

Multifidelity framework predictions for net cage displacement. Scatter plots (a), (d), (g) compare the low-fidelity GP model prediction with depth sensor measurements. The broad dispersion of data points and the deviation of the red best-fit line from the dashed diagonal line highlight the limited predictive accuracy of the low-fidelity model. The color scale represents the logarithmic density of data points, with lighter shades indicating higher concentrations. Scatter plots (b), (e), (h) display the multifidelity GP model predictions. Data points cluster more closely around the diagonal line, and the red best-fit line aligns with it, indicating a marked improvement in predictive accuracy. Despite the inherent complexity and dynamic motion of the flexible net cage, the multifidelity framework delivers significantly enhanced performance. Darker points represent outliers, while lighter regions correspond to higher accuracy. The MAE values annotated on these plots quantitatively highlight the multifidelity model’s superiority over the low-fidelity model. Plots (c), (f), (i) show the temporal comparison of predicted and actual net cage displacements over a specific period (January 18 to January 27, 2020). The low-fidelity GP model (blue line) exhibits substantial deviations from the actual measurements (green line), particularly for depth sensors #1 and #3. The multifidelity GP model (red line) significantly mitigates these discrepancies, closely mirroring the actual measurements and capturing the underlying trend with greater fidelity.

Flexible net cage deformation via surrogate modeling

Fish farm net cages are flexible structures that tend to follow rather than resist water motions, therefore they present significant deformations. These deformations can impact fish survival and the structural integrity of the cages, with any failure potentially leading to fish escapes. This underscores the importance of monitoring net cage deformations to maintain a comprehensive understanding of fish farm conditions. Due to their inherent flexibility, monitoring the entire deformation of net cages using available sensors is challenging. Depth sensors can be mounted strategically on the net structure thereby obtaining a 3D position of these points in the net, which in turn are used to extrapolate the full net structure using mathematical models⁹. A similar approach was implemented by Su et al.¹² for the development of a digital twin for real-time monitoring of aquaculture net cage systems, where depth sensor data were assimilated into FhSim simulations to represent the actual net cage system. Input properties for the simulations were adapted to ensure the numerical simulation outputs fit the sensor-measured values. However, this approach depends on sensors transmitting position information, with sensors being expensive and prone to data transmission loss. Additionally, it requires substantial computational time to process the data and calculate the correct net cage deformation. These reasons make this approach less suitable for real-time monitoring and immediate assessment of the system’s state.

Our study previously introduced the multifidelity digital twin, which accurately reflects the real net cage response to the dynamic marine environment. As a result, this digital twin eliminates the reliance on real-time data measured by sensors placed in the cage. Additionally, as illustrated in Fig. 3b, we developed surrogate models, using the standard GP method with a prior PCA, that map the net cage deformation with the currents. The accuracy of these models, previously presented by Katsidoniotaki et al.⁴⁷, suggests that they can replace FhSim simulations and significantly reduce computational costs in digital twin applications. However, surrogate models inherently introduce some degree of prediction error, depending on the method used to build them. This error can impact autonomous operations or decision-making where good accuracy is required. In this study, we compare the GP-PCA surrogate model with a surrogate model built using GCNs to predict net cage deformation topology under varying sea conditions. We chose the GCN since the net cage resembles graph-structured data. The cage is discretized into 321 nodes, which are connected in the manner shown in Fig. 2b. The GCN receives node features and the topology of their connections and can predict the displacement of each node under dynamic marine conditions. In the section “Methods/Graph convolutional networks for net cage deformation,” further details about the GCN model are provided. Figure 6 compares the net pen deformations predicted by the GCN model (in red) with those obtained from high-fidelity FhSim simulations (in black), under two representative sea conditions: mild (test case 801) and harsh (test case 301).

Fig. 6.

Predicted net pen deformations using Graph Convolutional Networks (GCNs) are compared against numerical simulations from FhSim under two representative current excitation scenarios: harsh conditions (test case 301) and mild conditions (test case 801). Black lines indicate the numerical solution, while red lines represent the GCN predictions.

To evaluate the prediction error of each method, we consider the example of autonomous navigation of Unmanned Underwater Vehicles (UUVs), which require real-time information about the surrounding environment where they navigate. During autonomous operations, a UUV such as a Remotely Operated Vehicle (ROV) should maintain a distance of at least 1 meter from the net cage to ensure collision-free motion. Therefore, it is imperative to keep the prediction error regarding net cage deformation within acceptable limits for safe operations. The MAE metric was employed to quantify the accuracy of net cage deformation predictions across three spatial directions over a range of test indices. As depicted in Fig. 7, the majority of data points for both methods lie below the red line at MAE = 1 meter, indicating a satisfactory level of accuracy for most test indices. Notably, higher MAE values are observed in test cases involving higher current velocities, where the net cage exhibits larger deformations, highlighting the challenges in modeling more extreme conditions. As summarized in the final subplot in Fig. 7, the GP-PCA model demonstrates higher MAE values, suggesting lower prediction accuracy compared to the GCN model. The wider spread of the GP-PCA error bars in the x and z dimensions notably suggests greater variability in its predictions. This greater error in the GP-PCA model may be attributed to dual sources of error: the PCA dimension reduction, which entails some loss of information, and the inherent errors in the GP model itself.

Fig. 7.

Machine-learned surrogate models for predicting net cage deformation. Evaluation of prediction accuracy for net pen deformations using the Mean Absolute Error (MAE) metric, comparing the predicted and simulated displacements of each node in the x, y, and z directions. Two surrogate modeling approaches are assessed: (i) a Gaussian Process (GP) surrogate model with prior dimensionality reduction via Principal Component Analysis (PCA), and (ii) a Graph Convolutional Network (GCN). The dashed horizontal line at indicates a critical threshold beyond which prediction errors may adversely affect downstream tasks such as unmanned underwater vehicle (UUV) control. The models are evaluated across 100 test scenarios. The final subplot summarizes the average MAE for each axis, with error bars denoting variability across the test set.

Discussion

In this study, we present a framework that leverages multifidelity surrogate modeling for integration into a digital twin designed to enable real-time monitoring of structural dynamics in aquaculture systems. Unlike traditional digital twins that rely on physics-based simulations–requiring substantial computational resources and thus being suboptimal for real-time use–our approach provides several key advantages. The proposed framework operates at a fraction of the computational cost during both training and deployment, enabling on-the-fly predictions and rapid updates. A major challenge in real-world digital twin implementations is the limited availability and access to real-world data required for accurate surrogate modeling. By seamlessly integrating low-fidelity numerical simulations with high-fidelity field sensor measurements, our framework bridges the gap between modeled and observed dynamics. This integration enhances the predictive accuracy and reliability, making the system particularly suitable for real-time monitoring where sensor data is scarce or costly to obtain.

The NARGP method, which forms the core of our framework, is part of a broader class of multifidelity information-fusion techniques. Its nonlinear autoregressive structure enables effective learning of complex, nonlinear correlations between data sources of varying fidelity. While NARGP has demonstrated strong performance in benchmark studies, its application to full-scale, real-world systems has been limited. Here, we demonstrate its practical effectiveness by rapidly and efficiently correcting low-fidelity model outputs to align with field observations, achieving high prediction accuracy at low computational cost.

The framework was successfully validated using data from the SINTEF ACE fish farm, combining low-fidelity numerical simulation results with high-fidelity sensor measurements. For specific locations along the net pen, discrepancies between simulated and measured displacements were observed—primarily due to the complex, nonlinear hydrodynamic interactions between the flexible net and the surrounding wave–current environment, which are difficult to fully capture numerically. The NARGP method effectively compensates for these limitations, substantially reducing prediction errors and yielding results that closely match field measurements. Our framework was developed and validated using high-fidelity data collected from existing sensors over a two-month period during the winter season, which is typically characterized by intensified wave activity, stronger currents, and the presence of biofouling effects. These conditions are especially relevant for assessing the model robustness under the nonlinear dynamics most critical for structural integrity and failure prediction. Although the dataset does not yet encompass the full range of seasonal variability, the modular design of the framework allows straightforward retraining and expansion as new high-fidelity data become available-whether from additional seasons, years, or locations. This flexibility enables continuous refinement and adaptation of the digital twin to evolving environmental conditions, including long-term changes such as increased storm frequency or biofouling progression. The primary objective of this study is to demonstrate and validate the methodology, highlighting the efficacy of the multifidelity approach in combining sparse, high-cost sensor data with abundant low-fidelity simulations. Rather than constructing a fully seasonal model, we establish a proof of concept under harsh environmental conditions where modeling accuracy is most demanding. The demonstrated performance under such challenging scenarios provides a solid foundation for future extensions toward broader temporal coverage and long-term operational deployment.

Adoption of digital twin technologies in aquaculture requires not only accurate modeling but also practical considerations, such as cost, ease of deployment, and operational simplicity. Current monitoring practices rely heavily on manual ROV inspections, diver operations, and vessel support. These approaches are labor-intensive, weather-dependent, and costly, requiring skilled pilots and coordination across teams. These approaches are largely reactive rather than predictive and become increasingly unsustainable as farms move to more exposed offshore sites. In contrast, our framework reduces these barriers by utilizing existing sensor infrastructure and minimizing the need for additional instrumentation or manual interventions. The core machine learning model operates passively in the background, requiring no specialized tuning during operation; users interact only with interpretable outputs, such as net deformation fields or anomaly alerts. This framework has been validated in an operational setting at a full-scale fish farm, where our digital twin was integrated with an autonomous ROV for real-time shape estimation of the flexible net pen, enabling safe, closed-loop path planning under dynamic marine conditions⁴⁵. This advancement supports autonomous operations in fish farms, such as inspection of nets and mooring lines for damages, irregularities, biofouling conditions, as well as net cleaning⁴⁶. The proposed digital twin is therefore not only technically robust but also aligned with industry workflows, offering significant reductions in costs and operational risk compared with traditional monitoring systems.

Deformation of aquaculture net pens due to wave-current interaction plays a critical role in fish welfare, structural integrity, and autonomous operation safety. Thus, developing efficient and accurate models for predicting these deformations is a core objective of this work. Here, we adopted a GPR-PCA approach to construct the surrogate model capable of predicting average net deformation across diverse marine conditions. As described in our previous work⁴⁷, the GPR-PCA approach reconstructs net pen deformations with accuracy comparable to full-scale numerical simulations, while drastically reducing computational cost. Dimensionality reduction via PCA allows efficient handling of high-dimensional data describing net displacements, though it may smooth out localized deformation features in highly dynamic regions. To assess these limits, we further explored a GCN approach, which naturally operates on the graph-structured representation of the cage, capturing spatial relationships through localized message passing. This enables the model to capture fine-scale spatial features–particularly in highly dynamic regions such as the lower layers of the net–where precise shape estimation is critical. While the GCN achieves higher prediction accuracy, the GP-PCA surrogate was chosen as the operational model due to its direct compatibility with the NARGP framework, which is built upon Gaussian Process regression. The GCN model is included as a benchmark to illustrate the potential of graph-based learning for structural dynamics and to motivate future exploration of graph neural operators in this domain.

In conclusion, this study introduces a machine learning–enabled digital twin framework for marine aquaculture built on multifidelity data assimilation, advancing predictive modeling of complex marine structures. While the current focus is on net cage deformation and mooring line loads, the underlying framework is general and applicable to other dynamic variables of interest. The digital twin offers tangible benefits across the full system lifecycle. In the design phase, it enables virtual prototyping, model validation, and early-stage performance assessment, facilitating cost-effective planning and structural optimization under variable marine conditions. During system configuration and reconfiguration, it supports semi-physical simulations for mooring layout design, sensor placement, and operational optimization without requiring full physical trials. In the operational phase, the framework delivers real-time monitoring, anomaly detection, and supports autonomous ROV-based inspections and net cleaning, as demonstrated in a full-scale offshore deployment. Collectively, these advancements are essential for the industry’s expansion to meet rising food demands and secure sustainable food supply.

Methods

Nonlinear autoregressive GP regression (NARGP)

The nonlinear autoregressive GP regression denoted by NARGP³² is a class of multifidelity nonlinear information fusion algorithms that enables accurate inference of quantities of interest by synergistically combining realizations of low-fidelity models with a small set of high-fidelity observations, capable of learning complex nonlinear and space-dependent cross-correlations between models of variable fidelity. Our research adopts the NARGP method in response to the observed nonlinear dynamics between our low and high-fidelity datasets.

We have s levels of information sources (fidelities) producing outputs , at locations , with . In our application we have two fidelity levels, , and two input parameters, , i.e. current velocity and direction. We can organize the available datasets by increasing fidelity as , while the datasets have nested structure, i.e. . This assumption implies that the training inputs of the higher fidelity level need to be a subset of the training inputs of the lower fidelity level. In NARGP³² method, the nonlinear autoregressive scheme reads as:

1

where is the GP modeling the data at fidelity level t, the is the GP posterior from the previous inference level , and the function is modeled as a Gaussian prior, , with a covariance kernel that composes as

2

The structure of the kernel reveals the effect of the deep representation encoded in Eq. 1. The projects the lower fidelity posterior onto a -dimensional latent manifold, that jointly relates the input space and the outputs of the lower fidelity level to the output of the higher fidelity model, from which we can infer a smooth mapping that recovers the high-fidelity response . This allows to capture general nonlinear, non-functional and space-dependent cross-correlations between low- and high-fidelity data. In Eq. 2, , , and are valid covariance functions and denote their hyperparamters, which can be easily learnt from the data via the maximum likelihood estimation procedure followed by the standard GP⁴¹using the kernel . This approach requires the estimation of () hyperparameters assuming that all kernels account for directional anisotropy in each dimension using automatic relevance determination (ARD) weights. The kernel functions are chosen to have the squared exponential form with ARD weights, i.e.

3

where is a variance parameters and are the ARD weights corresponding to fidelity level t. These weights allow for a continues ’blend’ of the contributions of each individual dimension in as well as the posterior predictions of the previous fidelity level , and they are learnt directly from the data when inferring .

The first level of the proposed recursive scheme corresponds to a standard GP regression problem trained on the lowest fidelity data , and therefore, the predictive posterior distribution is defined by standard’s Gaussian mean and covariance using the kernel function . Figure 3 provides a summary of the NARGP workflow as implemented in this study.

Net cage displacement data dimension reduction using PCA

PCA is a statistical technique used in data analysis for dimensionality reduction. It identifies the directions (or principal components) that maximize the variance in a dataset, thereby preserving essential information with fewer variables. In our study, we apply PCA to the dataset obtained from FhSim numerical simulations, which provide the , , displacements of the nodes into which the net cage is discretized, where . We have this data for each of the environmental conditions examined. Specifically, for each environmental condition, FhSim simulates the net cage interaction with waves and currents over a 30-min period, outputting the , , node displacements. The 30-min simulation time for all the environmental conditions corresponds to a total of time steps. All the available data from these simulations are integrated into the matrix .

Using this data matrix, we perform eigenanalysis to calculate the eigenvectors, , and eigenvalues,, respectively.

4

The eigenvalues and their corresponding eigenvectors are sorted in descending order of the eigenvalues. The eigenvector with the highest eigenvalue is the first principal component of the dataset and accounts for the highest variance, with each subsequent component capturing progressively less variance than its predecessor. We determine the number of principal components to retain based on the cumulative explained variance ratio, which translates into how many components k we should keep out of the c total components to better describe the data. This sum yields the total proportion of the dataset’s variance that is accounted for by the first k principal components.

5

A common approach is to choose the smallest possible number of principal components, k, that explain a substantial portion of the variance. However, the choice of k is based on a predetermined threshold which reflects the desired level of variance explanation: . In our case, setting the threshold at 93% results in the retention of k = 3 principal components.

Next, we project the original data, , onto the selected principal components to transform them into the new feature subspace, by multiplying the original matrix by the first k selected eigenvectors, to get the matrix of the transformed data which can be referred as PCA coefficients.

6

The resulting matrix consists of 3 rows, each corresponding to one of the principal components, and columns each one corresponds to a time step state of the net cage deformation. Since we are interested in estimating the average shape of the net cage deformation for each of the scenarios, we estimate the mean principle component for each scenario, resulting in a matrix , which can be referred as mean PCA coefficients. This matrix has rows and columns.

To reconstruct the average net cage deformation for each of the total environmental conditions, we multiple the eigenvector with the principal components.

7

The matrix that shows the average net cage displacement for each of the examined environmental conditions is:

Machine learning functional relationships between current and PCA coefficients

To develop the low-fidelity component of the NARG methodology, which predicts net cage deformation that corresponds to FhSim solution, we create a standard GP model to learn the functional relationship between current speed and the net cage deformation. Specifically, the GP model maps the current characteristics to the PCA coefficients, , , . Then, we utilize the inner sum of Equation 7 () to reconstruct the net cage deformation for an individual environmental condition. In other words, we obtain the predicted , , , where . Similar approaches–incorporating PCA for dimensionality reduction and the development of surrogate models using machine learning methods–have been widely applied across various disciplines due to their efficacy in feature extraction and data simplification^37,48–51. In our case, performing the dimension reduction step helps reduce the complexity of the data describing the net cage deformation, making it easier to use machine learning methods to create the mapping.

Graph convolutional networks for net cage deformation

Graph neural networks (GNNs) are mathematical models that can learn functions over graphs and are a learning approach for building predictive models on graph-structured data. Graphs differ from regular data in that they have a structure that neural networks must respect. Among the various types of GNNs, the graph convolutional networks (GCNs) have emerged as the most broadly applied model. GCNs are innovative due to their ability generalize the operation of convolutional neural networks to graphs, enabling the network to learn from the graph’s topology and node features to make predictions.

Representing net cage graph. The net cage of our application can be viewed as graph-structured data, discretized into 321 nodes.We build a GCN model for the net cage based on the work of Kipf and Welling⁴². The net cage can be viewed as a graph G consisting of nodes connected through the edges, where and represent nodes i and j. The edge features are represented by the adjacency matrix which encodes connections between the N nodes, represents the graph connectivity which is a square matrix where each element specifies the presence or absence of an edge from node i to node j in the graph. In other words, a non-zero element implies a connection from node i to node j, and a zero indicates no direct connection. In our net cage application, the adjacent matrix captures the node connection shown in Fig. 2b; the net cage consists of 10 layers, each with 32 nodes arranged in a circular pattern. Nodes within each layer are interconnected to form a circle, and each node is vertically connected to the node directly below it in the subsequent layer. This creates a consistent connection pattern between layers. Additionally, the final node (node 321) is connected to all nodes in the 10th layer, providing a central connection point. The is the node feature matrix with F being the number of the feature assigned to each node. In our case, the contains the features for each node which in our case are the , , coordinates of each node at rest and the sea conditions (i.e. current speed and direction). The graph is supposed to predict the matrix , where O is the number of output features per node. The matrix gives the x, y, z coordinates of each node under the interaction of the cage with the sea condition.

Message aggregation and update. The GCN layers are based on the formulation by Kipf and Welling (2017)⁴². At each layer, the graph convolution takes both the adjacency matrix A and the node features from the previous layer and outputs the node features for the next layer , with and are the node features dimensions for layers l and , while the input node feature matrix, following the formulation:

8

where is the activation function, is the adjacency matrix with added self-connections represented by the identity matrix , is the diagonal degree matrix of , and is a trainable weight matrix for layer . In our application, the data passes through three graph convolution layers. Each layer updates the node features by aggregating information from its neighbor nodes according to the graph structure. The final output represents the predicted node features which are the predicted displacements (x, y, z) for each node. Having several layers enables to further refine the node features by aggregating information from the graph structure and learning mode complex patterns. The deformation of net-cage nodes is a highly non-linear process influenced, the ability of Swish activation function to introduce non-linearities is particularly beneficial, which is defined as:

9

Model training. A notable point in our model is the implementation of a custom loss function that assigns different weights to the errors of each node by emphasizing regions of the net-cage that are more prone to displacement. At the node weights initialization stage nodes within layers 6 to 10, which are observed to undergo significant deformation, are assigned increased weights, with the highest weight allocated to the 321st node. The loss function computes the mean squared error (MSE) for each nodes prediction:

10

given : Predicted displacements (matrix of shape ), : True displacements (matrix of shape ), : Node weights (vector of length N). For the training process we use backpropagation to compute gradients and gradient descent optimiser to update the parameters for each layer.

Author contributions

All authors conceived the idea. E.Kat. and T.P.S developed the methodology for the multifidelity digital twin framework. E.Kat. analysed the data, developed the code and implemented the framework. B.S. conducted the FhSim simulations. B.S. and E.Kel. provided the field sensor measurements. E.Kat. and T.P.S. analysed the results. B.S. and E.Kel. suggested applications for the digital twin’s practical implementation. E.Kat. wrote the original manuscript, with all authors contributing to editing and reviewing the manuscript.

Data availability

Code will become available to a dedicated depository after the paper is accepted. Correspondence and requests for code materials should be addressed to Eirini Katsidoniotaki or Themistoklis P. Sapsis. The data that support the findings of this study are available from SINTEF Ocean AS but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of SINTEF Ocean AS.

Declarations

Competing interests

The authors declare no competing interests.