Using The Cross-Validation Method to Predict Computer Model Outputs Using The Stochastic Gaussian Process Regression Model

Gaussian process regression with skewed errors

The Gaussian process regression (GPR) model, which is a powerful machine learning tool for probabilistic prediction, is introduced into slope displacement prediction. Using this model, the displacements of the slope of the permanent ship lock of the Three Gorges Project, the Wolongsi slope, and the high slope of Longtan hydropower station were predicted. In addition, the predictive uncertainty index (PUI) for describing the uncertainty of the predicted results was proposed, and the corresponding classification of the PUI was established. The study results demonstrate that the GPR model can self-adaptively acquire model parameter values and has satisfactory adaptability for predicting nonlinear time series of slope displacement. The proposed PUI and its classification based on the GPR model enable quantitative uncertainty analysis and, in turn, reliability evaluation of the predicted results. The GPR model provides a new approach to displacement prediction and safety management in slope engineering.

Simulating root length density dynamics of sunflower in saline soils based on machine learning

Machine learning (ML) force fields are revolutionizing molecular dynamics (MD) simulations as they bypass the computational cost associated with ab initio methods but do not sacrifice accuracy in the process. In this work, the GPyTorch library is used to create Gaussian process regression (GPR) models that are interfaced with the next-generation ML force field FFLUX. These models predict atomic properties of different molecular configurations that appear in a progressing MD simulation. An improved kernel function is utilized to correctly capture the periodicity of the input descriptors. The first FFLUX molecular simulations of ammonia, methanol, and malondialdehyde with the updated kernel are performed. Geometry optimizations with the GPR models result in highly accurate final structures with a maximum root-mean-squared deviation of 0.064 Å and sub-kJ mol-1 total energy predictions. Additionally, the models are tested in 298 K MD simulations with FFLUX to benchmark for robustness. The resulting energy and force predictions throughout the simulation are in excellent agreement with ab initio data for ammonia and methanol but decrease in quality for malondialdehyde due to the increased system complexity. GPR model improvements are discussed, which will ensure the future scalability to larger systems.

The increasing effects of climate change require the accurate measurement and prediction of carbon emissions from building construction operations. This study aims to assess the performance of Gaussian Process Regression (GPR) and Support Vector Machine (SVM) in predicting emissions and determine which model is best suited for improving sustainable construction practices. The GPR and SVM models were trained and tested using empirical datasets with varying parameter values. Model performance is evaluated using correlation coefficient (R), root mean square error (RMSE), and mean absolute error (MAE). The analysis reveals that GPR consistently performs better than SVM across diverse datasets, especially in intricate and varied data situations. The superior performance of GPR highlights its robustness and versatility in handling the complexities of construction-related emissions data. The results indicate that GPR has the potential to assist significantly many stakeholders in the construction sector, including planners and environmental managers, in making better-informed decisions that are in line with carbon reduction objectives. Further studies could examine the incorporation of further predictive variables into the GPR model, such as economic indicators and the effects of material innovation.

Gaussian process regression to determine water content of methane: Application to methane transport modeling

The control of underactuated balance robots is aimed at performing both the external (actuated) subsystem trajectory tracking and internal (unactuated) subsystem balancing tasks. In this article, we propose a learning-based control design for underactuated balance robots. The key idea integrates a model predictive control method to design the desired internal subsystem trajectory and perform the external subsystem tracking task, while an inverse dynamics controller is used to stabilize the internal subsystem to its desired trajectory. The control design is based on Gaussian process (GP) regression models that are learned from experiments without requiring <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge about the robot dynamics or the demonstration of successful stabilization. GP regression models also provide estimates of modeling uncertainties of the robotic systems, and these estimations are used to enhance control robustness to modeling errors. The learning-based control design is analyzed with guaranteed stability and performance. The proposed design is demonstrated by experiments on a Furuta pendulum and an autonomous bikebot.

The growth of rice is a sequence of three different phenological phases. This sequence of change in rice phenology implies that the condition of the plant during the vegetative phase relates directly to the health of leaves functioning during the reproductive and ripening phases. As such, accurate monitoring is important towards understanding rice growth dynamics. Leaf Area Index (LAI) is an important indicator of rice yields and the availability of this information during key phenological phases can support more informed farming decisions. Satellite remote sensing has been adopted as a proxy to field measurements of LAI and with the launch of freely available high resolution Satellite images such as Sentinel-2, it is imperative that accurate retrieval methods are adopted towards monitoring LAI at irrigated rice fields. Here, we evaluate the potential of a hybrid radiative transfer model (i.e., PROSAIL - Gaussian Process Regression (GPR), for estimating the phenological dynamics of irrigated rice LAI using imager derived from the Sentinel-2 multispectral instrument. LAI field measurements were obtained from an experimental rice field in Nasarawa state, Nigeria during the dry season. We used the PROSAIL radiative transfer model to create a look up table (LUT) that was subsequently used to train a GPR model. Afterwards, we evaluated the potential of the hybrid modelling approach by assessing the overall model accuracy and the extent to which LAI was able to accurately predict LAI during key rice phenological phases. We compared the predicted hybrid GPR LAI values with LAI values generated from the SNAP toolbox, based on a hybrid Artificial Neural Network (ANN) modelling approach. Our results show that the overall predictive accuracy of the hybrid GPR model (R2 = 0.82, RMSE = 1.65) was more accurate than that of the hybrid ANN model (R2 = 0.66, RMSE = 3.89) for retrieving LAI values from Sentinel-2 imagery. Both models underestimated LAI values during the reproductive and ripening phases . However, the accuracy during the phenological phases were more significant when using the hybrid GPR model (P < 0.05). During the different phenological phases, the hybrid GPR model predicted LAI more accurately during the reproductive (R2 = 0.7) and ripening (R2 = 0.59) phases compared to the hybrid ANN reproductive and ripening phases. When monitoring LAI phenological profiles of both hybrid models, the hybrid GPR and ANN models underestimated LAI during the reproductive and ripening phases. However, the ANN model underestimations were statistically significantly greater than those for the hybrid GPR model (P < 0.05). Our results highlight the potential of hybrid GPR models for estimating the phenological dynamics of irrigated rice LAI from Sentinel-2 data. They provided more accurate estimation of LAI patterns from varying nitrogen and water applications than hybrid ANN models.

Due to the recent advances in digitisation of the manufacturing industry and the generation of manufacturing data, there is increasing interest to integrate machine learning on the shop floor to improve efficiency and quality control. Ultrasonic welding is an emerging joining process used in various manufacturing industries, and is an energy efficient, cost-effective method of joining similar or dissimilar materials. However, the quality of the joint achievable is heavily dependent on process input parameters. In this study, a Gaussian Process Regression (GPR) model is developed to map the relationship between process parameters and joint performance for ultrasonically welded aluminium joints, with a view to improving quality control in a manufacturing setting. Initially, a 33 full factorial design of experiments is conducted to investigate the influential parameters, then a GPR model is trained on the experimental data. In-process sensor data is also used to infer process performance. To assess the prediction performance of the model, ten unseen parameter combinations are predicted and compared to their respective experimental result. The model demonstrates a high level of accuracy producing a Pearson’s correlation coefficient of 0.982 between the predicted and actual results for all data. The mean relative predictive error for unseen data is 7.35%.KeywordsUltrasonic weldingMachine learningGaussian process regressionQuality control

Forecasting short-term solar radiation is crucial for many solar energy applications. Additionally, solar energy has a lower environmental impact than conventional sources like fossil fuels and can be used for investment purposes through the construction of large solar farm facilities. To test, evaluate, and compare various solar radiation models, short-term observations of meteorological, astronomical, computational, and geographical data were collected at two distinct locations from 2012 to 2015. In this study, seven machine learning models were employed: multi-layer perceptron (MLP), feedforward backpropagation algorithm (FFBP), autoregressive integrated moving average (ARIMA), linear regression (LR), radial basis function neural network (RBFNN), random forest (RF), and Gaussian process regression (GPR) models. These models were used to forecast hourly global solar radiation (GSR) using the aforementioned data as model input. The performance of the selected models' forecast accuracy was thoroughly examined by assessing it for a typical day, for four seasons, and under three sky conditions. The RF model can forecast GSR with satisfactory accuracy, and MLP and GPR models provide better accuracy than LR, FFBP, RBF, and ARIMA models. For example, the R2 value range of RF are 0.9621 for Tetuan site and 0.9534 for Tangier site, respectively. Meanwhile, RF, MLP, and GPR models under-forecast few high radiation values on clear days, which may due to the differences in training and testing data ranges and distributions of the sky conditions. Finally, the obtained result of this study indicate that the proposed RF model is a reliable alternative for short-term global solar radiation forecasting due to its high forecast accuracy.

Over the last decade, innovations in satellite remote sensing (RS) and data science have widened the scope and relevance of agricultural monitoring and management applications at farm and territorial scales. Recently launched and upcoming hyperspectral satellite missions (e.g., ASI-PRISMA, DLR-ENMAP, Planet-Tanager, ESA-CHIME, Kuva-Hyperfield) provide high spectral resolution (&lt; 10 nm) across the 400-2500 nm range, opening new frontiers for assessing biophysical and biochemical functional traits of agroecosystems, while advancements in machine learning (ML) and artificial intelligence (AI) allow the efficient exploitation of the information content of high-dimensionality spectral datasets. Here we summarize the results and lessons learned from three experiments in different European agricultural systems (croplands and grasslands), analysing how the synergy between hyperspectral imaging spectroscopy (field and satellite), ML, and foundation models could support adaptive agroecosystem management through the retrieval of vegetation and soil properties related to the nutrient cycle. The three case studies are:(1) Assessment of biomass and nutritional quality of Alpine pastures: Gaussian Process Regression (GPR) models were calibrated on 250 vegetation samples and field spectra collected in 2024 and 2025 from semi-natural pastures in Valtournanche (Aosta, Italy) and Val Camonica (Brescia, Italy). PRISMA-derived maps for biomass and leaf-level protein, and fiber content showed good accuracy against in-situ data (LAI [-] R2 = 0.71, RMSE = 0.89; Biomass [g · m-2] R2 = 0.67, RMSE = 178.71; DM [%] R2 = 0.65, RMSE = 2.70; CP [%] R2 = 0.58, RMSE = 0.52; ADF [%] R2 = 0.45, RMSE = 2.42; NDF [%] R2 = 0.50, RMSE = 0.61), allowing mapping of pasture metabolizable energy in support of grazing management.(2) Monitoring of nutritional status of paddy fields: GPR models were developed on 200 vegetation samples and field spectra collected in 2024 and 2025 in several fields located in the Ferrara region (Italy). These models, demonstrated on PRISMA and EnMAP time-series, effectively monitored crop development across a temporally and spatially independent test set (LAI [-] R2 = 0.83, RMSE = 0.30; Fresh Biomass [g · m-2] R2 = 0.72, RMSE = 627.67; LCC [μg · cm-2] R2 = 0.58, RMSE = 3.40; LNC [μg · cm-2] R2 = 0.34, RMSE = 22.51; CNC [g · m-2] R2 = 0.56, RMSE = 0.77).(3) Retrieval of Soil Organic Carbon (SOC) and soil Nitrogen (N) on arable lands: A transformer-based, sensor-agnostic deep learning architecture was fine-tuned on open global spectral libraries. When applied to EMIT and Tanager-1 imagery over the Po Plain (Italy) and Northern Netherlands regions, the model yielded high accuracy (SOC [%] R2 = 0.61, MAE = 0.37; N [%] R2 = 0.68, MAE = 0.12) against 289 independent field observations.Our findings demonstrate that satellite hyperspectral spectroscopy can complement operational multi-spectral missions, adding key information about agroecosystems nutritional status. Furthermore, we show that the use of pre-trained ML and AI models on global spectral libraries and field reflectance data allows accurate retrieval even in the absence of ground truth acquisition synchronous to the satellite overpass, offering a potential scalable solution for agroecosystems management and monitoring at landscape scale.

Active learning sensitivity analysis of γ'(L12) precipitate morphology of ternary co-based superalloys

Predicting binary solution phase behavior of polymers has remained a challenge since the early theory of Flory-Huggins, hindering the processing, synthesis, and design of polymeric materials. Herein, we take a complementary data-driven approach by building a machine learning framework to make fast and accurate predictions of polymer solution cloud point temperatures. Using polystyrene, both upper and lower critical solution temperatures are predicted within experimental uncertainty (1-2 °C) with a deep neural network, Gaussian process regression (GPR) model, and a combination of polymer, solvent, and state features. The GPR model also enables intelligent exploration of solution phase space, where as little as 25 cloud points are required to make predictions within 2 °C for polystyrene of arbitrary molecular weight in cyclohexane. This study demonstrates the effectiveness of machine learning for the prediction of liquid-liquid equilibrium of polymer solutions and establishes a framework to incorporate other polymers and complex macromolecular architectures.

Knitting is an effective technique for producing complex three‐dimensional surfaces owing to the inherent flexibility of interlooped yarns and recent advances in manufacturing, providing better control of local stitch patterns. Fully yarn‐level modeling of large‐scale knitted membranes is not feasible. Therefore, we use a two‐scale homogenization approach and model the membrane as a Kirchhoff‐Love shell on the macroscale and as Euler‐Bernoulli rods on the microscale. The governing equations for both the shell and the rod are discretized with cubic B‐spline basis functions. For homogenization, we consider only the in‐plane response of the membrane. The solution of the nonlinear microscale problem requires a significant amount of time due to the large deformations and the enforcement of contact constraints, rendering conventional online computational homogenization approaches infeasible. To sidestep this problem, we use a pretrained statistical Gaussian process regression (GPR) model to map the macroscale deformations to macroscale stresses. During the offline learning phase, the GPR model is trained by solving the microscale problem for a sufficiently rich set of deformation states obtained by either uniform or Sobol sampling. The trained GPR model encodes the nonlinearities and anisotropies present in the microscale and serves as a material model for the membrane response of the macroscale shell. The bending response can be chosen in dependence of the mesh size to penalize the fine out‐of‐plane wrinkling of the membrane. After verifying and validating the different components of the proposed approach, we introduce several examples involving membranes subjected to tension and shear to demonstrate its versatility and good performance.

Carbon use efficiency (CUE) is a key indicator in coupled biological-abiotic systems that accounts for their capacity of effectively retain carbon, offering insights of ecosystem functioning and the dynamics of carbon cycle. Generally, CUE assessments have been limited to an autotrophic perspective, quantifying plant efficiency while neglecting carbon losses from heterotrophic respiration. This provides an incomplete view of ecosystem carbon retention. To address this critical gap, we offer a global quantification of a more holistic ecosystem-level CUE, that incorporates all respiratory fluxes. This paper proposes a methodology for mapping ecosystem CUE at global scale from in situ data, remote sensing observations and machine learning. This data-driven approach exploits a Gaussian Processes Regression (GPR) model trained with CUE from eddy-covariance towers and concomitant observations from the Moderate resolution Imaging Spectroradiometer (MODIS). The performance of the model shows high correspondence (R2 = 0.84) and low error and bias (RMSE = 0.1, ME = 0.01) regarding in situ data. The execution of the GPR model upscaled CUE and associated uncertainty from tower level to global scale and provided multitemporal global CUE estimates from 2001 to 2023. The GPR model reports a mean global CUE of 0.43 ± 0.08 for this period. A preliminary analysis carried out for different climatic zones and biomes illustrates the increase of mean CUE from tropical (0.36 ± 0.08) to cold (0.55 ± 0.08) zones. The lowest mean CUE is found over evergreen broadleaved forests (0.37 ± 0.04), whereas the largest mean CUE is found over open shrublands (0.53 ± 0.12). Finally, a global trend of (-1.2 ± 0.3) × 10-3 yr- 1 is reported for mean global CUE from 2001 to 2023. The results of this work highlight the dependence of CUE on both climate and biome type, as well as the decreasing carbon sequestration power of vegetation at global scale, which is key to better understand the effects of climate change.