Kernel-based Methods Research Articles

Metabolomic age (MileAge) predicts health and life span: A comparison of multiple machine learning algorithms.

Biological aging clocks produce age estimates that can track with age-related health outcomes. This study aimed to benchmark machine learning algorithms, including regularized regression, kernel-based methods, and ensembles, for developing metabolomic aging clocks from nuclear magnetic resonance spectroscopy data. The UK Biobank data, including 168 plasma metabolites from up to N = 225,212 middle-aged and older adults (mean age, 56.97 years), were used to train and internally validate 17 algorithms. Metabolomic age (MileAge) delta, the difference between metabolite-predicted and chronological age, from a Cubist rule-based regression model showed the strongest associations with health and aging markers. Individuals with an older MileAge were frailer, had shorter telomeres, were more likely to suffer from chronic illness, rated their health worse, and had a higher all-cause mortality hazard (HR = 1.51; 95% CI, 1.43 to 1.59; P < 0.001). This metabolomic aging clock (MileAge) can be applied in research and may find use in health assessments, risk stratification, and proactive health tracking.

A Family of Hyperparameter Estimators Linking EB and SURE for Kernel-Based Regularization Methods

A Family of Hyperparameter Estimators Linking EB and SURE for Kernel-Based Regularization Methods

Online Attentive Kernel-Based Off-Policy Temporal Difference Learning

Temporal difference (TD) learning is a powerful framework for value function approximation in reinforcement learning. However, standard TD methods often struggle with feature representation and off-policy learning challenges. In this paper, we propose a novel framework, online attentive kernel-based off-policy TD learning, and in combination with well-known algorithms, introduce OAKGTD2, OAKTDC, and OAKETD. This framework uses two-timescale optimization. In the slow-timescale, a sparse representation of state features is learned using an online attentive kernel-based method. In the fast-timescale, auxiliary variables are used to update the value function parameters under the off-policy setting. We theoretically prove the convergence of all three algorithms. Through experiments conducted in several standard reinforcement learning environments, we demonstrate the effectiveness of the improved algorithms and compare their performance with existing algorithms. Specifically, from the perspective of cumulative rewards, the proposed algorithm achieves an average improvement of 15% compared to on-policy algorithms and an average improvement of 25% compared to common off-policy algorithms.

Operator learning with Gaussian processes

Operator learning focuses on approximating mappings G†:U→V between infinite-dimensional spaces of functions, such as u:Ωu→R and v:Ωv→R. This makes it particularly suitable for solving parametric nonlinear partial differential equations (PDEs). Recent advancements in machine learning (ML) have brought operator learning to the forefront of research. While most progress in this area has been driven by variants of deep neural networks (NNs), recent studies have demonstrated that Gaussian process (GP)/kernel-based methods can also be competitive. These methods offer advantages in terms of interpretability and provide theoretical and computational guarantees. In this article, we introduce a hybrid GP/NN-based framework for operator learning, leveraging the strengths of both deep neural networks and kernel methods. Instead of directly approximating the function-valued operator G†, we use a GP to approximate its associated real-valued bilinear form G˜†:U×V∗→R. This bilinear form is defined by the dual pairing G˜†(u,φ)≔[φ,G†(u)], which allows us to recover the operator G† through G†(u)(y)=G˜†(u,δy). The mean function of the GP can be set to zero or parameterized by a neural operator and for each setting we develop a robust and scalable training mechanism based on maximum likelihood estimation (MLE) that can optionally leverage the physics involved. Numerical benchmarks demonstrate our method’s scope, scalability, efficiency, and robustness; showing that (1) it enhances the performance of a base neural operator by using it as the mean function of a GP, and (2) it enables the construction of zero-shot data-driven models that can make accurate predictions without any prior training. Additionally, our framework (a) naturally extends to cases where G†:U→∏s=1SVs maps into a vector of functions, and (b) benefits from computational speed-ups achieved through product kernel structures and Kronecker product matrix representations of the underlying kernel matrices.11GitHub repository: https://github.com/Bostanabad-Research-Group/GP-for-Operator-Learning.

Tooth point cloud resampling method based on divergence index and improved Euclidean clustering rule.

In endodontic therapy, 3D Cone Beam Computerized Tomography (CBCT) and oral scan fusion models allow exact root canal channels and guidance. However, the point cloud model from CBCT has few data points and poor model features, limiting 3D fusion with oral scan data. Our aim to build a sub-regional point cloud resampling method and evaluate the precision of merging it with three-dimensional oral scan data.&#xD;Approach: Two molars and four incisors were resampled for this investigation. Based on point cloud density and curvature, the rebuilt model was separated into the crown and cervical cavities. Using crown surface morphology, Divergence Index (DI) was employed to determine resampling points based on point dispersion. Improved Euclidean Clustering Rule (IECR) downsamples each point using its weight and joins the two halves using Iterative Nearest Neighbour (ICP) to create a complete resampled point cloud. After aligning with the oral scanning model, the maximum error, maximum distance, average distance, and other characteristics are calculated to assess resampling. Additionally, a cross-entropy kernel-based point cloud reconstruction depth selection method is given to determine the appropriate reconstruction depth.&#xD;Main results: Applying the DI-IECR technique reduces the average distance between the resampled tooth point cloud and the point cloud generated by the dental scanner by around 20%. The maximum error remains same to that of the widely used method. This study also demonstrates that the use of the DI-IECR approach guarantees the complete representation of the coronal characteristics of the resampled reconstructed 3D model, rather than excessively focusing processing resources on pertinent but insignificant areas.&#xD;Significance: Point cloud data and crown features are balanced using DI-IECR. When registered with the oral scan model, CBCT-generated point clouds are more accurate and timely, making them a better intraoperative navigation model.

Kernel-Based Versus Tree-Based Data-Driven Models: On Applying Suspended Sediment Load Estimation

River sediment load estimation poses a critical challenge for water engineers due to its complex and nonlinear hydrological processes. This study assessed the amount of suspended sediment at the Bagh-e-Kalayeh hydrometric station on the Alamut River in the Qazvin province of Iran using two hydrological and meteorological variables, including discharge and rainfall, by considering three scenarios (discharge, discharge + monthly rainfall, and discharge + monthly rainfall + daily rainfall). For modeling, kernel-based data-driven methods, including Gaussian process regression (GPR) and support vector regression (SVR), and tree models, including the M5 tree, random forest (RF), random tree (RT), extra trees, reduced error pruning tree (REPT), and multi-search methods, were used. The results showed that the best performance was achieved by the SVR, with r = 0.948, Wilmot index = 0.965, and RMSE = 0.011 in the first scenario (only discharge). Discharge had the most significant impact on sediment estimation compared to rainfall. It was determined that the suspended sediment load in the Alamut River can be successfully estimated by the SVR method, where only the discharge was used as the input parameter. Additionally, the results indicated that given its characteristics and inherent features, the multi-search method can be used as a complementary approach in sediment modeling, especially in situations where the data volume is not extensive.

Greedy trial subspace selection in meshfree time-stepping scheme with applications in coupled bulk-surface pattern formations

Combining kernel-based collocation methods with time-stepping methods to solve parabolic partial differential equations can potentially introduce challenges in balancing temporal and spatial discretization errors. Typically, using kernels with high orders of smoothness on some sufficiently dense set of trial centers provides high spatial approximation accuracy that can exceed the accuracy of finite difference methods in time. The paper proposes a greedy approach for selecting trial subspaces in the kernel-based collocation method applied to time-stepping to balance errors in both well-conditioned and ill-conditioned scenarios. The approach involves selecting trial centers using a fast block-greedy algorithm with new stopping criteria that aim to balance temporal and spatial errors. Numerical simulations of coupled bulk-surface pattern formations, a system involving two functions in the domain and two on the boundary, illustrate the effectiveness of the proposed method in reducing trial space dimensions while maintaining accuracy.

A Comparative Study of Pavement Roughness Prediction Models under Different Climatic Conditions

Predicting the International Roughness Index (IRI) is crucial for maintaining road quality and ensuring the safety and comfort of road users. Accurate IRI predictions help in the timely identification of road sections that require maintenance, thus preventing further deterioration and reducing overall maintenance costs. This study aims to develop robust predictive models for the IRI using advanced machine learning techniques across different climatic conditions. Data were sourced from the Ministry of Energy and Infrastructure in the UAE for localized conditions coupled with the Long-Term Pavement Performance (LTPP) database for comparison and validation purposes. This study evaluates several machine learning models, including regression trees, support vector machines (SVMs), ensemble trees, Gaussian process regression (GPR), artificial neural networks (ANNs), and kernel-based methods. Among the models tested, GPR, particularly with rational quadratic specifications, consistently demonstrated superior performance with the lowest Root Mean Square Error (RMSE) and highest R-squared values across all datasets. Sensitivity analysis identified age, total pavement thickness, precipitation, temperature, and Annual Average Daily Truck Traffic (AADTT) as key factors influencing the IRI. The results indicate that pavement age and higher traffic loads significantly increase roughness, while thicker pavements contribute to smoother surfaces. Climatic factors such as temperature and precipitation showed varying impacts depending on the regional conditions. The developed models provide a powerful tool for predicting pavement roughness, enabling more accurate maintenance planning and resource allocation. The findings highlight the necessity of tailoring pavement management practices to specific environmental and traffic conditions to enhance road quality and longevity. This research offers a comprehensive framework for understanding and predicting pavement performance, with implications for infrastructure management both locally and worldwide.

Kernel-Based Minimal Distributed Charges: A Conformationally Dependent ESP-Model for Molecular Simulations.

A kernel-based method (kernelized minimal distributed charge model (kMDCM)) to represent the molecular electrostatic potential (ESP) in terms of off-center point charges is introduced. The positions of the charges adapt to the molecular geometry and allow the description of intramolecular charge flow. Using Gaussian kernels and atom-atom distances as the features, the ESPs for water and methanol are shown to improve by at least a factor of 2 compared with point charge models fit to an ensemble of structures. The conformationally fluctuating molecular dipole moment of water is reproduced almost twice as accurately using kMDCM compared with static PCs, despite not fitting to the dipole directly. The role of hyperparameters in the kernelization is investigated and their implication on model performance and simulation stability is discussed. Combining kMDCM for the electrostatics and reproducing kernels for the bonded terms allows energy-conserving simulations of 2000 water molecules with periodic boundary conditions on the nanosecond time scale. These MD simulations sample geometries outside the training set but remain stable, which demonstrates the robustness of the model and its implementation.

Double-kernel based Bayesian approximation broad learning system with dropout

Broad learning system (BLS) is an efficient incremental learning machine algorithm. However, there are some disadvantages in such an algorithm. For example, the number of hidden layer nodes needs to be manually adjusted during the training process, meanwhile the large uncertainty will be caused by two random mappings. To solve these problems, based on the optimization ability of the kernel function, a double-kernel broad learning system (DKBLS) is proposed to eliminate the uncertainty of random mapping by using additive kernel strategy. Meanwhile, to reduce the computing costs and training time of DKBLS, a double-kernel based bayesian approximation broad learning system with dropout (Dropout-DKBLS) is further proposed. Ablation experiments show that the output accuracy of Dropout-DKBLS does not decrease even if the node is dropped. In addition, function approximation experiments show that DKBLS and Dropout-DKBLS have good robustness and can accurately predict noise data. The regression and classification experiments on multiple datasets are compared with the latest kernel-based learning methods. The comparison results show that both DKBLS and Dropout-DKBLS have good regression and classification performance. By further comparing the training time of these kernel-based learning methods, we prove that the Dropout-DKBLS can reduce the computational cost while ensuring the output accuracy.

An unsupervised [formula omitted]-means machine learning algorithm via overlapping to improve the nodes selection for solving elliptic problems

We propose an overlapping algorithm utilizing the K-means clustering technique to group scattered data nodes for discretizing elliptic partial differential equations. Unlike conventional kernel-based approximation methods, which select the closest points from the entire region for each center, our algorithm selects only the nearest points within each overlapping cluster. We present computational results to demonstrate the efficiency of our algorithm for both two-dimensional and three-dimensional problems. For evaluation and validation, these results are compared with results obtained using the RBF-FD+polynomial method with different kernels.

Kernel-based framework for improved prediction of discharge coefficient in vertically supported cylindrical weirs

ABSTRACT The present study represents the first use of kernel-based models to predict discharge coefficient (Cd) for two distinct types of cylindrical weirs, featuring vertical support and a 30-degree upstream ramp. For this purpose, kernel-based methods, including support vector machine, Gaussian process regression (GPR), Kernel extreme learning machine, and Kernel ridge regression, were used, as they offer notable advantages compared to other machine learning models, such as flexibility in handling various data patterns, robustness against overfitting, and effectiveness in high-dimensional data scenarios. The results indicated that the GPR model, with statistical metrics of R = 0.967, Nash–Sutcliffe efficiency (NSE) = 0.935, and root-mean-square error (RMSE) = 0.027, demonstrates superior accuracy in modeling the overall dataset collected from two distinct types of weirs. Through a conducted sensitivity analysis, it was identified that the upstream Froude number is pivotal in accurately predicting the Cd of a cylindrical weir. The modeling conducted for two distinct weir types revealed that a cylindrical weir with vertical support exhibits enhanced predictive capabilities (R = 0.997, NSE = 0.994, and RMSE = 0.007) for Cd. The findings indicate that the introduction of the upstream ramp alters hydraulic conditions, resulting in reduced modeling accuracy (R = 0.760, NSE = 0.529, and RMSE = 0.060).

Consensus local graph for multiple kernel clustering

Due to the ability to represent the relationship among data, graph has received extensive attention in clustering field. As illustrated by existing studies, learning graph in kernel space is an effective way to capture the structure of data. Among kernel-based methods, multiple kernel learning (MKL) is increasingly popular since it can automatically utilize the complementary information contained in base kernels. Although many existing MKL-based graph learning algorithms have promising learning abilities, we observe that they (1) put too much attention on learning the consensus kernel which probably contains lots of redundant information and loses the diversity of base kernels; (2) ignore the local structure of the representations in multiple kernel spaces. To eliminate the bad effect of these issues, we propose a novel graph learning method, termed consensus local graph based on MKL (CLGMKL), for clustering. In CLGMKL, a low-rank kernel matrix is used to extract the discriminative information of each base kernel and a local graph is constructed to capture the local structure contained in each new kernel. Then CLGMKL combines these local graphs in a self-weighed way. Since the above processes are associated with each other, we jointly optimize them to obtain the overall optimality. An effective learning scheme with proved convergence is developed to optimize the combined objective function. Finally, extensive experiments on some popular datasets are conducted to test the effectiveness of the presented method. As illustrated, CLGMKL is more competitive than the state-of-the-art algorithms.

A time-to-event modelling of sputum conversion within two months after antituberculosis initiation among drug-susceptible smear positive pulmonary tuberculosis patients: Implementation of internal and external validation

Delayed sputum conversion has been associated with a higher risk of treatment failure or relapse among drug susceptible smear-positive pulmonary tuberculosis patients. Several contributing factors have been identified in many studies, but the results varied across regions and countries. Therefore, the current study aimed to develop a predictive model that explained the factors affecting time to sputum conversion within two months after initiating antituberculosis agents among Malaysian with drug-susceptible smear-positive pulmonary tuberculosis patients. Retrospective data of pulmonary tuberculosis patients followed up at a tertiary hospital in the Northern region of Malaysia from 2013 until 2018 were collected and analysed. Nonlinear mixed-effect modelling software (NONMEM 7.3.0) was used to develop parametric survival models. The final model was further validated using Kaplan-Meier-visual predictive check (KM-VPC) approach, kernel-based hazard rate estimation method and sampling-importance resampling (SIR) method. A total of 224 patients were included in the study, with 34.4 % (77/224) of the patients remained positive at the end of 2 months of the intensive phase. Gompertz hazard function best described the data. The hazard of sputum conversion decreased by 39 % and 33 % for moderate and advanced lesions as compared to minimal baseline of chest X-ray severity, respectively (adjusted hazard ratio (aHR), 0.61; 95 % confidence intervals (95 % CI), (0.44–0.84) and 0.67, 95 % CI (0.53–0.84)). Meanwhile, the hazard also decreased by 59 % (aHR, 0.41; 95 % CI, (0.23–0.73)) and 48 % (aHR, 0.52; 95 % CI, (0.35–0.79)) between active and former drug abusers as compared to non-drug abuser, respectively. The successful development of the internally and externally validated final model allows a better estimation of the time to sputum conversion and provides a better understanding of the relationship with its predictors.

Life-cycle production optimization with nonlinear constraints using a least-squares support-vector regression proxy

When nonlinear constraints such as field liquid or water production rate, injection pressures, etc., as functions of time need to be honored in addition to linear ones, the life-cycle production optimization problem, a component of a closed loop reservoir management, becomes challenging and computationally expensive to perform using a high-fidelity reservoir simulator with the existing gradient-based methods using the adjoint or stochastic approximate gradient methods. Therefore, the objective of this study is to present computationally efficient methods for deterministic production optimization under nonlinear constraints using a kernel-based machine learning method where the cost function is the net present value (NPV). We use the least-squares support-vector regression (LSSVR) to approximate the NPV function. To achieve computational efficiency, we generate a set of output values of the NPV and nonlinear constraint functions, which are field liquid production rate (FLPR) and water production rate (FWPR) in this study, by running the high-fidelity simulator for a broad set of input design variables (well controls) and then using the collection of input/output data to train LSSVR proxy models to replace the high-fidelity simulator to compute NPV and nonlinear state constraint functions during iterations of sequential quadratic programming (SQP). To obtain improved (higher) estimated optimal NPV values, we use the existing so-called iterative sampling refinement (ISR) method to update the LSSVR proxy so that the updated proxy remains predictive toward promising regions of search space during the optimization. Direct and indirect ways of constructing LSSVR-based NPVs as well as different combinations of input data, including nonlinear state constraints and/or the bottomhole pressures (BHPs) and water injection rates, are tested as alternative feature spaces. The results obtained from our proposed LSSVR-based optimization methods are compared with those obtained from our in-house stochastic simplex approximate gradient (StoSAG)-based line-search SQP programming (LS-SQP-StoSAG) algorithm that uses directly a high-fidelity simulator to compute the gradients of the objective function and the nonlinear state functions with StoSAG for the Brugge reservoir model. The results show that nonlinear constrained optimization with the LSSVR ISR with SQP is computationally 3.25 fold more efficient than LS-SQP-StoSAG. In addition, the results show that constructing NPV indirectly using the field liquid and water rates for a waterflooding problem where inputs come from LSSVR proxies of the nonlinear state constraints requires significantly fewer training samples than the method constructing NPV directly from the NPVs computed from a high-fidelity simulator. LSSVR has advantages for its computational efficiency which is the main goal in our research, and robustness against overfitting especially for the cases where we have limited data. However, DNNs and random forest require large training size and require much more computational resources and longer training times. Also, Random Forests and Gradient Boosting Machines can be prone to overfitting and become computationally intensive.

Impact of emotional states on tinnitus sound therapy efficacy based on ECG signals and emotion recognition model

BackgroundDiagnosis and severity assessment of tinnitus are mostly based on the patient's descriptions and subjective questionnaires, which lacks objective means of diagnosis and assessment bases, the accuracy of which fluctuates with the clarity of the patient's description. This complicates the timely modification of treatment strategies or therapeutic music to improve treatment efficacy. New methodWe employed a novel random convolutional kernel-based method for electrocardiogram (ECG) signal analysis to identify patients' emotional states during Music Tinnitus Sound Therapy (Music-TST) sessions. Then analyzed correlations between emotional changes in different treatment phase and Tinnitus Handicap Inventory (THI) score differences to determine the impact of emotions on tinnitus treatment efficacy. ResultsThis study revealed a significant correlation between patients' emotion changes during Music-TST and the therapy's effectiveness. Changes in arousal and dominance dimension, were strongly linked to THI variations. These findings highlight the substantial impact of emotional responses on sound therapy's efficacy, offering a new perspective for understanding and optimizing tinnitus treatment. Comparison with existing methodsCompared to existing methods, we proposed an objective indicator to assess the progress of sound therapy, the indicator could also be used to provide feedback to optimize sound therapy music. ConclusionsThis study revealed the critical role of emotion changes in tinnitus sound therapy. By integrating objective ECG-based emotion analysis with traditional subjective scale like THI, we present an innovative approach to assess and potentially optimize therapy effectiveness. This finding could lead to more personalized and effective treatment strategies for tinnitus sound therapy.

EMD-based time-frequency analysis methods of audio signals

Using appropriate signal processing tools to analyze time series data accurately is essential for correctly interpreting the underlying processes. Commonly employed methods include kernel-based transforms that utilize base functions and modifications to depict time series data. This paper refers to the analysis of audio data using two such transforms: the Fourier transform and the wavelet transform, both based on assumptions regarding the signal's linearity and stationarity. However, in audio engineering, these assumptions often do not hold as the statistical characteristics of most audio signals vary over time, making them unsuitable for treatment as outputs from a Linear Time-Invariant (LTI) system. Consequently, more recent methods have shifted towards breaking down signals into various modes in an adaptive, data-specific manner, potentially offering benefits over traditional kernel-based methods. Techniques like empirical mode decomposition and Holo-Hilbert Spectral Analysis are examples of this. The effectiveness of these methods was tested through simulations using speech signals for both kernel-based and adaptive decomposition methods, demonstrating that these adaptive methods are effective for analyzing audio data that is both nonstationary and an output of the nonlinear system.

Comparison of Glomerular Filtration Rate Equations in a Rural New Mexico Cohort: Results from the COMPASS Study.

The NKF-ASN Task Force recommends accurate kidney function estimation avoiding biases through racial adjustments. We explored the use of multiple kidney function biomarkers and hence estimated glomerular filtration rate (eGFR) equations to improve kidney function calculations in an ethnically diverse patient population. Prospective community cohort study. rural New Mexico clinic with patients > 18 yo. Markers of kidney function, IDMS-Creatinine (SCr), chemiluminescence Beta-2 Microglobulin (B2M), Nephelometry-calibrated ELISA Cystatin C (CysC), inflammation, glucose tolerance, demographics, BUN/UACR from the baseline visit of the COMPASS cohort, were analyzed by Kernel-based Virtual Machine learning methods. Among 205 participants, the mean age was 50.1, 62% were female, 54.1% Hispanic American and 30.2% Native American. Average kidney function biomarkers were: SCr 0.9 mg/dl, B2M 1.8 mg/L, and CysC 0.7 mg/dl. The highest agreement was observed between SCr and B2M-based eGFR equations [mean difference in eGFRs: (4.48 ml/min/1.73m2], and the lowest agreement between B2M and CysC-based eGFR equations (-24.75 ml/min/1.73m2). There was no pattern of association between the differences in eGFR measures and gender. In the continuous analyses, the absolute eGFR value (p<2 x 10-16) and serum albumin (p =6.4 x 10-5) predicted the difference between B2M- and SCr-based e-GFR. The absolute eGFR value (p<2 x 10-16) and age (p =7.6 x 10-5) predicted the difference between CysC- and SCr-based e-GFR. Relatively small sample size, elevated inflammatory state in majority of study participants and no inulin excretion rate measurements. B2M should be strongly considered as a kidney function biomarker fulfilling the criteria for the NKF-ASN. B2M's eGFR equation does not need adjustment for gender or race and showed the highest agreement with SCr-based eGFR equations.

Estimating Position-Dependent and Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories: Existing Methods and Future Outlook.

Confinement can substantially alter the physicochemical properties of materials by breaking translational isotropy and rendering all physical properties position-dependent. Molecular dynamics (MD) simulations have proven instrumental in characterizing such spatial heterogeneities and probing the impact of confinement on materials' properties. For static properties, this is a straightforward task and can be achieved via simple spatial binning. Such an approach, however, cannot be readily applied to transport coefficients due to lack of natural extensions of autocorrelations used for their calculation in the bulk. The prime example of this challenge is diffusivity, which, in the bulk, can be readily estimated from the particles' mobility statistics, which satisfy the Fokker-Planck equation. Under confinement, however, such statistics will follow the Smoluchowski equation, which lacks a closed-form analytical solution. This brief review explores the rich history of estimating profiles of the diffusivity tensor from MD simulations and discusses various approximate methods and algorithms developed for this purpose. Besides discussing heuristic extensions of bulk methods, we overview more rigorous algorithms, including kernel-based methods, Bayesian approaches, and operator discretization techniques. Additionally, we outline methods based on applying biasing potentials or imposing constraints on tracer particles. Finally, we discuss approaches that estimate diffusivity from mean first passage time or committor probability profiles, a conceptual framework originally developed in the context of collective variable spaces describing rare events in computational chemistry and biology. In summary, this paper offers a concise survey of diverse approaches for estimating diffusivity from MD trajectories, highlighting challenges and opportunities in this area.

Complexity of many-body interactions in transition metals via machine-learned force fields from the TM23 data set

This work examines challenges associated with the accuracy of machine-learned force fields (MLFFs) for bulk solid and liquid phases of d-block elements. In exhaustive detail, we contrast the performance of force, energy, and stress predictions across the transition metals for two leading MLFF models: a kernel-based atomic cluster expansion method implemented using sparse Gaussian processes (FLARE), and an equivariant message-passing neural network (NequIP). Early transition metals present higher relative errors and are more difficult to learn relative to late platinum- and coinage-group elements, and this trend persists across model architectures. Trends in complexity of interatomic interactions for different metals are revealed via comparison of the performance of representations with different many-body order and angular resolution. Using arguments based on perturbation theory on the occupied and unoccupied d states near the Fermi level, we determine that the large, sharp d density of states both above and below the Fermi level in early transition metals leads to a more complex, harder-to-learn potential energy surface for these metals. Increasing the fictitious electronic temperature (smearing) modifies the angular sensitivity of forces and makes the early transition metal forces easier to learn. This work illustrates challenges in capturing intricate properties of metallic bonding with current leading MLFFs and provides a reference data set for transition metals, aimed at benchmarking the accuracy and improving the development of emerging machine-learned approximations.