Kernel Approximation Research Articles

Adaptive control of reaction–diffusion PDEs via neural operator-approximated gain kernels

Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction–diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45× relative to the traditional finite difference solvers for every timestep in the simulation trajectory.

Nonlinear Meshfree Collocation Method for Solving Double-Diffusive Natural Convection in a Porous Enclosure

A meshfree collocation method formulated by using reproducing kernel shape functions with Newton–Raphson method is presented with application to analyzing a three-phase coupling system in a porous enclosure. To tackle such a highly nonlinear phenomenon caused by double-diffusive natural convection, direct collocation using local approximation in conjunction with a two-step iteration solver is employed to stably and efficiently solve this problem. It has been shown that a determined and well-conditioned system is established under the collocation framework with reproducing kernel approximation. By conducting the parametric study, the numerical results have demonstrated the robustness of the proposed method in analyzing the porous enclosure by considering various geometry, boundary conditions, and governing parameters.

TEC-CNN: Towards Efficient Compressing Convolutional Neural Nets with Low-rank Tensor Decomposition

Most state-of-the-art convolutional neural networks (CNNs) are characterised by excessive parameterisation, leading to a high computational burden. Tensor decomposition has emerged as a model reduction technique for compressing deep neural networks. Previous approaches have predominantly relied on either Tucker decomposition or Canonical Polyadic (CP) decomposition for CNNs. However, CP decomposition exhibits exceptional compression capabilities in comparison to Tucker decomposition, which results in a more pronounced accuracy loss. This paper introduces an efficient model compression method, termed TEC-CNN, designed to achieve significant compression while preserving accuracy levels comparable to those of the original models. In TEC-CNN, convolutional layers are identified to obtain convolutional kernels by analysing given models under the principles of low-rank tensor decomposition, and then, calculating the ranks of convolutional kernels. Furthermore, an efficient decomposition schema for the convolutional kernel is proposed with approximate kernel tensor for reducing parameters. Additionally, a novel format of a convolutional sequence is presented and constructed with a reduced number of parameters to replace the original convolutional layers. Finally, the effectiveness of TEC-CNN is assessed across a range of computer vision tasks. For instance, in CIFAR-100 classification, ResNet18 is compressed to 4.1 MB, while Unext, when applied to image segmentation using the International Skin Imaging Collaboration (ISIC) dataset, is reduced to 3.419 MB. When employed for fire object detection with Yolov7, TEC-CNN achieves a model size reduction of 71.6 MB. Comprehensive experimental results underscore that our approach achieves significant model compression while preserving model performance.

Structured orthogonal random features based on DCT for kernel approximation

Random Fourier features are a popular technique used to improve nonlinear kernel methods in large-scale problems. Recent studies have shown that replacing random Gaussian matrices of random feature maps with appropriately scaled random orthogonal matrices, such as SORF, can significantly improve kernel approximation performance. However, the SORF method theoretically requires zero-padding for datasets whose input feature dimension does not meet d=2p for some p∈N+, resulting in increased memory and computation time. To address this limitation, we propose a new structured orthogonal random features method based on discrete cosine transform (SORF-DCT) to approximate Gaussian kernel functions. The SORF-DCT method does not require the input feature dimension to be d=2p. The key to SORF-DCT is that we combine the DCT matrix with diagonal ”sign-flipping” matrices and scale them appropriately using a diagonal matrix whose diagonal elements follow the chi distribution, resulting in a structured orthogonal matrix whose elements follow the standard Gaussian distribution instead of a random Gaussian matrix. SORF-DCT generates D-dimensional structured orthogonal random features in O(Dlogd) time and O(D) memory by employing fast discrete cosine transform, where d and D represent the input feature dimension and expansion dimension, respectively. We prove that SORF-DCT is an unbiased estimator of the Gaussian kernel function and analyze the concentration error bounds. Experimental results on eleven benchmark datasets show that SORF-DCT achieves lower kernel approximation errors, requires less time, and has comparable nonlinear learning capabilities.

A new quasi-finite-rank approximation of compression operators on [formula omitted] with applications to sampled-data and time-delay systems: Piecewise linear kernel approximation approach

This paper provides a new quasi-finite-rank approximation (QFRA) of infinite-rank compression operators defined on the Banach space L∞[0,H), which are associated with tractable representations of infinite-dimensional systems such as time-delay and sampled-data systems. We first formulate the QFRA by an optimization problem with a matrix-valued parameter X to minimize the associated error in terms of the L∞[0,H)-induced norm. To facilitate solving the optimization problem, we next employ the piecewise linear kernel approximation (PLKA) technique, by which the optimization problem is then converted to a linear programming (LP) problem. The solution of the LP problem is shown to converge to the optimal solution of the original QFRA with the order of 1/M, where M is the PLKA parameter. The PLKA-based QFRA is shown to lead to practical methods of the stability analysis for time-delay systems and the L1 optimal controller synthesis for sampled-data systems. Finally, the overall arguments developed in this paper are demonstrated through some numerical and experimental studies.

Unified framework for open quantum dynamics with memory

The dynamics of quantum systems coupled to baths are typically studied using the Nakajima-Zwanzig memory kernel (K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\bf{{{{\\mathcal{K}}}}}}}}$$\\end{document}) or the influence functions (I), particularly when memory effects are present. Despite their significance, formal connections between the two have not been explicitly known. We establish their connections by examining the system propagator for a N-level system linearly coupled to Gaussian baths with various types of system-bath coupling. For a certain class of problems, we devised a non-perturbative, diagrammatic approach to construct K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\bf{{{{\\mathcal{K}}}}}}}}$$\\end{document} from I for (driven) systems interacting with Gaussian baths, bypassing conventional projection-free dynamics inputs. Our work provides a way to interpret approximate path integral methods in terms of approximate memory kernels. Moreover, it offers a Hamiltonian learning procedure to extract the bath spectral density from reduced system trajectories, opening new avenues in quantum sensing and engineering. The insights we provide advance our understanding of non-Markovian dynamics and will serve as a stepping stone for future theoretical and experimental developments in this area.

The value of CCTA combined with machine learning for predicting angina pectoris in the anomalous origin of the right coronary artery

BackgroundAnomalous origin of coronary artery is a common coronary artery anatomy anomaly. The anomalous origin of the coronary artery may lead to problems such as narrowing of the coronary arteries at the beginning of the coronary arteries and abnormal alignment, which may lead to myocardial ischemia due to the compression of the coronary arteries. Clinical symptoms include chest tightness and dyspnea, with angina pectoris as a common symptom that can be life-threatening. Timely and accurate diagnosis of anomalous coronary artery origin is of great importance. Coronary computed tomography angiography (CCTA) can provide detailed information on the characteristics of coronary arteries. Therefore, we combined CCTA and artificial intelligence (AI) technology to analyze the CCTA image features and clinical features of patients with anomalous origin of the right coronary artery to predict angina pectoris and the relevance of different features to angina pectoris.MethodsIn this retrospective analysis, we compiled data on 15 characteristics from 126 patients diagnosed with anomalous right coronary artery origins. The dataset encompassed both CCTA imaging attributes, such as the positioning of the right coronary artery orifices and the alignment of coronary arteries, and clinical parameters including gender and age. To identify the most salient features, we employed the Chi-square feature selection method, which filters features based on their statistical significance. We then focused on features yielding a Chi-square score exceeding a threshold of 1, thereby narrowing down the selection to seven key variables, including cardiac function and gender. Subsequently, we evaluated seven classifiers known for their efficacy in classification tasks. Through rigorous training and testing, we conducted a comparative analysis to identify the top three classifiers with the highest accuracy rates.ResultsThe top three classifiers in this study are Support Vector Machine (SVM), Ensemble Learning (EL), and Kernel Approximation Classifier. Among the SVM, EL and Kernel Approximation Classifier-based classifiers, the best performance is achieved for linear SVM, optimizable Ensembles Learning and SVM kernel, respectively. And the corresponding accuracy is 75.7%, 75.7%, and 73.0%, respectively. The AUC values are 0.77, 0.80, and 0.75, respectively.ConclusionsMachine learning (ML) models can predict angina pectoris caused by the origin anomalous of the right coronary artery, providing valuable auxiliary diagnostic information for clinicians and serving as a warning to clinicians. It is hoped that timely intervention and treatment can be realized to avoid serious consequences such as myocardial infarction.

Numerical investigation on ball-sealers transport and diversion performance in shale gas horizontal well based on semi-resolved CFD-DEM

In the staged multi-cluster fracturing of shale gas horizontal wells, ball sealers are used to ensure uniform fluid distribution among clusters, a strategy that is both cost-effective and operationally beneficial. Despite these advantages, comprehending the ball sealers' dynamics within the wellbore and their plugging behavior at perforations is still challenging. This complexity results in prediction difficulties regarding their diversion efficiency. To address this, our study utilized a semi-resolved CFD-DEM model based on kernel approximation to simulate the behavior of medium-sized ball sealers in single and multiple cluster scenarios. Our findings from a single cluster scenario reveal that the plugging probability is co-determined by velocity gradients in the fluid ingestion area near the perforation, backflow region, and inertial forces of the ball sealers. As the critical flow rate is achieved, the plugging probability negatively correlated with fluid viscosity and displacement, and positively correlated with the perforation flow ratio (PFR), the difference in particle-fluid density, ball sealer’s diameter, and the ball sealer’s offset from the pipeline center. Temporary plugging control efficiency was used to evaluate the flow balance effect among multiple clusters. The results indicate that an increased number of ball sealers enhances the fault tolerance during the temporary plugging process. Nevertheless, excessive ball sealers might undermine the temporary plugging control efficiency, as perforations with lower fluid inflow rates are unexpectedly plugging. Higher differences in fluid injection rates between clusters led to increased efficiency in temporary plugging control. Premature deployment of ball sealers cannot effectively plug perforations with marginally higher fluid inflow rates, but instead accidently plug intermediate clusters with lower fluid inflow rates. These findings offer a theoretical basis for optimizing the design of ball sealers.

Application of Improved SPH Method in Solving Time Fractional Schrödinger Equation

In this paper, a pure meshless method for solving the time fractional Schrödinger equation (TFSE) based on KDF-SPH method is presented. The method is used for the first time to numerically solve the TFSE. The method utilizes the finite difference method (FDM) to approximate the time fractional-order derivative defined in the Caputo sense. The spatial derivatives are discretized by the KDF-SPH meshless method. Expressions for the kernel approximation and the particle approximation are provided. To ensure the validity and flexibility of the numerical calculations, we conducted numerical simulations of one- and two-dimensional linear/nonlinear time Schrödinger equations (1D/2D TFLSE/TF-NLSE) in both bounded and unbounded regions. We also examined nonlinear time fractional Schrödinger equations that lack analytical solutions and compared our method with other meshless methods. Numerical results show that the proposed method can approximate to the second-order precision in space, which verifies the effectiveness and accuracy of the proposed method.

Adaptive soft sensor using stacking approximate kernel based BLS for batch processes

To deal with the highly nonlinear and time-varying characteristics of Batch Process, a model named adaptive stacking approximate kernel based broad learning system is proposed in this paper. This model innovatively introduces the approximate kernel based broad learning system (AKBLS) algorithm and the Adaptive Stacking framework, giving it strong nonlinear fitting ability, excellent generalization ability, and adaptive ability. The Broad Learning System (BLS) is known for its shorter training time for effective nonlinear processing, but the uncertainty brought by its double random mapping results in poor resistance to noisy data and unpredictable impact on performance. To address this issue, this paper proposes an AKBLS algorithm that reduces uncertainty, eliminates redundant features, and improves prediction accuracy by projecting feature nodes into the kernel space. It also significantly reduces the computation time of the kernel matrix by searching for approximate kernels to enhance its ability in industrial online applications. Extensive comparative experiments on various public datasets of different sizes validate this. The Adaptive Stacking framework utilizes the Stacking ensemble learning method, which integrates predictions from multiple AKBLS models using a meta-learner to improve generalization. Additionally, by employing the moving window method—where a fixed-length window slides through the database over time—the model gains adaptive ability, allowing it to better respond to gradual changes in industrial Batch Process. Experiments on a substantial dataset of penicillin simulations demonstrate that the proposed model significantly improves predictive accuracy compared to other common algorithms.

Semiclassical approaches to perturbative time-convolution and time-convolutionless quantum master equations for electronic transitions in multistate systems.

Understanding the dynamics of photoinduced processes in complex systems is crucial for the development of advanced energy-conversion materials. In this study, we investigate the nonadiabatic dynamics using time-convolution (TC) and time-convolutionless (TCL) quantum master equations (QMEs) based on treating electronic couplings as perturbation within the framework of multistate harmonic (MSH) models. The MSH model Hamiltonians are mapped from all-atom simulations such that all pairwise reorganization energies are consistently incorporated, leading to a heterogeneous environment that couples to the multiple electronic states differently. Our exploration encompasses the photoinduced charge transfer dynamics in organic photovoltaic carotenoid-porphyrin-C60 triad dissolved in liquid solution and the excitation energy transfer (EET) dynamics in photosynthetic Fenna-Matthews-Olson complexes. By systematically comparing the perturbative TC and TCL QME approaches with exact quantum-mechanical and various semiclassical approximate kernels, we demonstrate their efficacy and accuracy in capturing the essential features of photoinduced dynamics. Our calculations show that TC QMEs generally yield more accurate results than TCL QMEs, especially in EET, although both methods offer versatile approaches adaptable across different systems. In addition, we investigate various semiclassical approximations featuring the Wigner-transformed and classical nuclear densities as well as the governing dynamics during the quantum coherence period, highlighting the trade-off between accuracy and computational cost. This work provides valuable insights into the applicability and performance of TC and TCL QME approaches via the MSH model, offering guidance for realistic applications to condensed-phase systems on the atomistic level.

On a Spectral Method for β-particle Bound Excitation Collisions in Kilonovae

The interaction of β-particles with the weakly ionized plasma background is an important mechanism for powering the kilonova (KN) transient signal from neutron star mergers. For this purpose, we present an implementation of the approximate fast-particle collision kernel, described by Inokuti following the seminal formulation of Bethe, in a spectral solver of the Vlasov–Maxwell–Boltzmann equation. In particular, we expand the fast-particle plane-wave atomic excitation kernel into coefficients of the Hermite basis, and derive the relevant discrete spectral system. In this fast-particle limit, the approach permits the direct use of atomic data, including optical oscillator strengths, normally applied to photon–matter interaction. The resulting spectral matrix is implemented in the MASS-APP spectral solver framework, in a way that avoids full matrix storage per spatial zone. We numerically verify aspects of the matrix construction, and present a proof-of-principle 3D simulation of a 2D axisymmetric KN ejecta snapshot. Our preliminary numerical results indicate that a reasonable choice of Hermite basis parameters for β-particles in the KN is a bulk velocity parameter u = 0, a thermal velocity parameter α = 0.5c, and a 9 × 9 × 9 mode velocity basis set (Hermite orders of 0–8 in each dimension). For interior-ejecta sample zones, we estimate that the ratio of thermalization from large-angle (≳2.°5) bound excitation scattering to total thermalization is ∼0.002–0.003.

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

Overdestabilization vs Overstabilization in the Theoretical Analysis of f-Orbital Covalency.

The complex nature of the f-orbital electronic structures and their interaction with the chemical environment pose significant computational challenges. Advanced computational techniques that variationally include scalar relativities and spin-orbit coupling directly at the molecular orbital level have been developed to address this complexity. Among these, variational relativistic multiconfigurational multireference methods stand out for their high accuracy and systematic improvement in studies of f-block complexes. Additionally, these advanced methods offer the potential for calibrating low-scaling electronic structure methods such as density functional theory. However, studies on the Cl K-edge X-ray absorption spectra of the [Ce(III)Cl6]3- and [Ce(IV)Cl6]2- complexes show that time-dependent density functional theory with approximate exchange-correlation kernels can lead to inaccuracies, resulting in an overstabilization of 4f orbitals and incorrect assessments of covalency. In contrast, approaches utilizing small active space wave function methods may understate the stability of these orbitals. The results herein demonstrate the need for large active space, multireference, and variational relativistic methods in studying f-block complexes.

Machine Learning-Assisted design of boron and nitrogen doped graphene nanosheets with tailored thermomechanical properties

Doping graphene with boron and/or nitrogen enhances its potential for various applications, such as electronics and energy devices. However, these modifications impact the material's properties, influencing its response to external forces, temperature, and heating. Investigating the connection between the structure of co-doped graphene nanosheets and their thermomechanical properties is crucial for accurate predictions and tailored nanosheet engineering. In the present study, molecular dynamics (MD) simulations were performed to compute the thermal and mechanical properties of 300 randomly generated boron/nitrogen co-doped graphene nanosheets with doping concentrations in the range of 0%-15%. Next, the obtained MD results were used to train, tune, and validate a wide spectrum of supervised machine learning models, including linear regression models, regression trees, support vector machines, Gaussian process regression models, kernel approximation models, ensembles of regression trees, and regression neural networks. Our aim is to find models that better estimate, without overfitting, the structure-properties relation of boron/nitrogen co-doped nanosheets. To this end, the models with the highest coefficient of determination (R2) and the lowest root mean square error (RMSE) were utilized. Finally, an appropriate implementation of guided random search algorithms, mainly Genetic Algorithms, was used to solve the inverse problem, i.e., find the nanosheet structure which exhibits the desired set of property values. To this end, thermal conductivity, Young’s modulus, failure stress, and failure strain were specified as target values. A detailed comparison between the employed models and their validation via MD simulations is also included. The achieved results provide sufficient evidence of the accuracy and viability of the applied procedure in both prediction of thermomechanical properties as well as identification of structures with desired properties. This study provides a roadmap that could help scientists and engineers develop nanosheets-based materials with tunable thermomechanical properties, ultimately leading to the creation of innovative and high-performance technologies.

A gradient reproducing kernel based stabilized collocation method for the 5th order Korteweg–de Vries equations

This paper presents a gradient reproducing kernel based stabilized collocation method (GRK-SCM) to solve the generalized nonlinear 5th order Korteweg–de Vries (KdV) equations. By introducing gradient reproducing kernel (GRK) approximations, one can circumvent the intricacy of high-order derivatives in RK approximations, while still satisfying high-order consistency requirements. This leads to the high accuracy and efficiency. GRKs are very good approximation candidates for addressing high order partial differential equations that require higher order derivatives formulated as strong form. The stabilized collocation method (SCM) effectively meets high-order integration constraints which can achieve accurate integration in the subdomains. This ensures both high precision and optimal convergence. Von Neumann analysis is utilized to establish the stability criteria for GRK-SCM when combined with forward difference temporal discretization. Numerical solutions for Sawada–Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are studied where the solitary wave migration and collision, and periodic waves are represented. A fifth order forced KdV equation is also considered. The presented method's high accuracy and convergence are demonstrated through comparative studies with analytical solutions, and investigations of invariants and corresponding errors determine the good conservation properties of this algorithm.

Uniform approximation of common Gaussian process kernels using equispaced Fourier grids

The high efficiency of a recently proposed method for computing with Gaussian processes relies on expanding a (translationally invariant) covariance kernel into complex exponentials, with frequencies lying on a Cartesian equispaced grid. Here we provide rigorous error bounds for this approximation for two popular kernels—Matérn and squared exponential—in terms of the grid spacing and size. The kernel error bounds are uniform over a hypercube centered at the origin. Our tools include a split into aliasing and truncation errors, and bounds on sums of Gaussians or modified Bessel functions over various lattices. For the Matérn case, motivated by numerical study, we conjecture a stronger Frobenius-norm bound on the covariance matrix error for randomly-distributed data points. Lastly, we prove bounds on, and study numerically, the ill-conditioning of the linear systems arising in such regression problems.

Enhancement in the safety and reliability of Pressurized Water reactors using Machine Learning approach

The increasing interest in artificial intelligence and automation within the nuclear industry stems from the hope of elevating the safety and reliability of nuclear power plants by minimizing the impact of human errors. This paper explores the capabilities of machine-learning based fault detection and diagnosis (FDD) models in accurately classifying transient events in a nuclear power plant using the Classification Learners Application from MATLAB. The scope of the research is narrowed to the identification of transient events of the same nature occurring at various locations within the plant. To accomplish this, the Generic Pressurized Water Reactor (GPWR) from the Western Services Cooperation (WSC) was used to simulate 14 transients in the primary system and 10 transients in the secondary system, resulting in two distinct datasets of 108,000 and 79,200 observations. Nine types of classifiers, including Decision Trees, Discriminant Analysis, Support Vector Machines, Logistic Regression, Nearest Neighbors, Naive Bayes, Kernel Approximation, Ensembles, and Neural Networks, with a total of 33 predefined models, were trained and validated. The K-Nearest Neighbors model achieved the highest accuracy of 97% in the primary system, while the Efficient Linear Discriminant and Logistic Regression models achieved the highest average accuracy of 99% in the secondary system. The assessment of the top-performing models was conducted through a comprehensive analysis employing key classification model evaluation metrics, such as the confusion matrix, accuracy, precision, recall, and the F1 score. In an effort to optimize the models, the study integrated feature selection using the minimum Redundancy Maximum Relevance (mRMR) algorithm. Additionally, the investigation explored different validation schemes, including both holdout and k-fold cross-validation schemes, resulting in a substantial reduction in training time without significantly compromising the overall accuracy of the models. The optimized models in this study demonstrated remarkable prediction accuracies, consistently exceeding 94%. Despite the complexity of the models and the intricacies involved in the training process, the training times ranged from 10 to 1800 s, reflecting the efficiency and robustness of the implemented optimization techniques. The results of this study underscore the potential of machine learning models in identifying transient events in nuclear reactors, showcasing their promising capabilities while maintaining low computational and execution costs. This suggests their efficacy in optimizing safe and efficient operational practices within nuclear facilities.

A coupled hydro-thermo-mechanical model based on TLF-SPH for simulating crack propagation in fractured rock mass

In this paper, a hydro-thermo-mechanical coupling model based on the smoothed particle hydrodynamics with total Lagrangian formula (HTM-TLF-SPH) is proposed to simulate the crack propagation and instability process of fractured rock mass. TLF-SPH uses the Lagrangian kernel approximation, that is, the kernel function and its gradient need only be calculated once in the initial configuration, which is much more efficient than the smoothed particle hydrodynamics (SPH) based on the Euler kernel approximation. In TLF-SPH, particles interact with each other through virtual link, and the crack propagation path of rock mass is tracked dynamically by capturing the fracture of virtual link. Firstly, the accuracy and robustness of the HTM-TLF-SPH coupling model are verified by a reference example of drilling cold shock, and the simulation results agree well with the analytical solutions. Then, the crack propagation law of surrounding rock and the evolution characteristics of physical fields (displacement, seepage and temperature fields) after excavation and unloading of deep roadway under the coupling condition of hydro-thermo-mechanical are investigated. In addition, the seepage and heat transfer processes of the surrounding rock of Daqiang coal mine under different coupling conditions are successfully simulated. Meanwhile, the effect of the boundary water pressure difference on the temperature and seepage fields under the hydro-thermal coupling condition is quantitatively analyzed.

A new method to design energy-conserving surrogate models for the coupled, nonlinear responses of intervertebral discs

The aim of this study was to design physics-preserving and precise surrogate models of the nonlinear elastic behaviour of an intervertebral disc (IVD). Based on artificial force-displacement data sets from detailed finite element (FE) disc models, we used greedy kernel and polynomial approximations of second, third and fourth order to train surrogate models for the scalar force-torque -potential. Doing so, the resulting models of the elastic IVD responses ensured the conservation of mechanical energy through their structure. At the same time, they were capable of predicting disc forces in a physiological range of motion and for the coupling of all six degrees of freedom of an intervertebral joint. The performance of all surrogate models for a subject-specific L4|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert$$\\end{document}5 disc geometry was evaluated both on training and test data obtained from uncoupled (one-dimensional), weakly coupled (two-dimensional), and random movement trajectories in the entire six-dimensional (6d) physiological displacement range, as well as on synthetic kinematic data. We observed highest precisions for the kernel surrogate followed by the fourth-order polynomial model. Both clearly outperformed the second-order polynomial model which is equivalent to the commonly used stiffness matrix in neuro-musculoskeletal simulations. Hence, the proposed model architectures have the potential to improve the accuracy and, therewith, validity of load predictions in neuro-musculoskeletal spine models.