Forward Problem Research Articles

Identification Source Term of the Distributed Order Time-Space Fractional Diffusion Equation

Abstract The inverse source problem for time-distributed order spatial fractional diffusion equations is examined in this study. The forward problem is solved by combining finite difference methods with matrix transformation techniques. The Tikhonov regularization approach is then applied to convert the inverse problem into a variational problem. The optimal perturbation approach is used to find the minimizer of the variational problem, leading to an approximation solution that depends solely on the time-dependent source term. The algorithm’s effectiveness and stability are demonstrated using numerical examples in one-dimensional domains.

AUTOLYCUS: Exploiting Explainable Artificial Intelligence (XAI) for Model Extraction Attacks against Interpretable Models

Explainable Artificial Intelligence (XAI) aims to uncover the decision-making processes of AI models. However, the data used for such explanations can pose security and privacy risks. Existing literature identifies attacks on machine learning models, including membership inference, model inversion, and model extraction attacks. These attacks target either the model or the training data, depending on the settings and parties involved. XAI tools can increase the vulnerability of model extraction attacks, which is a concern when model owners prefer black-box access, thereby keeping model parameters and architecture private. To exploit this risk, we propose AUTOLYCUS, a novel retraining (learning) based model extraction attack framework against interpretable models under black-box settings. As XAI tools, we exploit Local Interpretable Model-Agnostic Explanations (LIME) and Shapley values (SHAP) to infer decision boundaries and create surrogate models that replicate the functionality of the target model. LIME and SHAP are mainly chosen for their realistic yet information-rich explanations, coupled with their extensive adoption, simplicity, and usability. We evaluate AUTOLYCUS on six machine learning datasets, measuring the accuracy and similarity of the surrogate model to the target model. The results show that AUTOLYCUS is highly effective, requiring significantly fewer queries compared to state-of-the-art attacks, while maintaining comparable accuracy and similarity. We validate its performance and transferability on multiple interpretable ML models, including decision trees, logistic regression, naive bayes, and k-nearest neighbor. Additionally, we show the resilience of AUTOLYCUS against proposed countermeasures.

Backward Stability of the Schur Decomposition under Small Perturbations

In the present paper, we show the backward stability of the Schur decomposition for a given matrix under small perturbations. We also present the forward stability problem for matrices with multiple eigenvalues and show conditions under which the forward stability fails.

Time Series Analysis of Sugarcane Plant Height and Planting Area Identification Based on DPRVI and RF Algorithm

With the modernisation of agriculture, the time series analysis of sugarcane growth and the identification of planting areas play an important role in the investigation and detection of agriculture. The study conducted a time series analysis of sugarcane plant height and identification of planting areas based on dual polarisation radar vegetation index and random forest algorithm. Firstly, dual-polarisation radar vegetation index data of sugarcane were obtained using Sentienl-1A radar remote sensing data for time-series analysis of sugarcane plant height. Then the random forest model was applied to the regression and classification of Sentinel-2A remote sensing images for sugarcane plantation area identification. According to the findings, sugarcane field 4 had an R2 value of 0.835 in the quadratic regression model inversion, and the average absolute difference between the model’s predicted plant height and the actual value was only 7.3%. It shows that the dual-polarization radar vegetation index has great reliability and is capable of accurately predicting sugarcane plant height during the pre-growth stage. The user and producer accuracies are as high as 0.911 and 0.989, respectively, and the overall accuracies and Kappa coefficients are as high as 0.976 and 0.932, respectively, indicating that the Random Forest with multi-temporal phase and multi-feature is able to effectively identify sugarcane planting areas. The accuracy under multi-temporal phase and multi-feature is better than the single-temporal phase. It offers a trustworthy way for developing contemporary agricultural policy and development planning.

Active Maneuver Load Alleviation for a Pitching Wing via Spanwise-Distributed Camber Morphing

This paper presents the design and verification of a nonlinear model inversion (NMI) controller for the maneuver load alleviation of a pitching oscillating wing based on spanwise-distributed active camber morphing. Recurrent neural networks (RNNs) are used to predict nonlinear and unsteady aerodynamic forces due to wing's large amplitude pitching maneuver, and a fully connected neural network is introduced to build the dynamic inversion of the aeroelastic system for control law design. The inversed system is concatenated with a PI controller to assemble a nonlinear active controller. The controller is first utilized in an offline environment for a 1DoF pitching finite-span wing with spanwise-distributed active camber morphing and then verified in CFD-based fluid-structure-control coupling simulation. The results show that the offline controller could eliminate the maneuver load. In the online CFD-based fluid-structure-control simulation, the bending moment can be alleviated by 38%.

Determination of biogeochemical properties in sea waters using the inversion of the three-stream irradiance model

Inversion models, in the context of oceanography, relate the observed ocean color to the concentrations of the different biogeochemical components present in the water of the ocean. However, building accurate inversion models can be quite complex due to the many factors that can influence the observed ocean color, such as variations in the composition or the optical properties of biogeochemical products. Here we assess the feasibility of the inversion approach, by implementing the three-stream light inversion model in a one-dimensional water column configuration, represented at the BOUSSOLE site in the northwestern Mediterranean Sea. Moreover, we provide a comprehensive sensitivity analysis of the model’s skill by perturbing the parameters of the bio-optical properties and phytoplankton physiology. Analysis of the inversion indicates that the model is able to reconstruct the variability of the optical constituents. Results indicate that chlorophyll-a and coloured dissolved organic matter play a major role in light modulation. The sensitivity analysis shows that the parameterization of the ratio of chlorophyll-a to carbon is important for the performance of the inversion model. A coherent inversion model, as presented, can be used as an observational operator to assimilate remote sensing reflectance.

AsPINN: Adaptive symmetry-recomposition physics-informed neural networks

Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs). However, PINNs’ loss, the regularization terms, can only guarantee that the prediction results conform to the physical constraints in the average sense, which results in PINNs’ inability to strictly adhere to implied physical laws such as conservation laws and symmetries. This limits the optimization speed and accuracy of PINNs. Although some feature-enhanced PINNs attempt to address this issue by adding explicit constraints, their generality is limited due to specific question settings. To overcome this limitation, our study proposes the adaptive symmetry-recomposition PINN (AsPINN). By analyzing the parameter-sharing patterns of fully connected PINNs, specific network structures are developed to provide predictions with strict symmetry constraints. These structures are incorporated into diverse subnetworks to provide constrained intermediate outputs, then a specialized multi-head attention mechanism is attached to evaluate and composite them into final predictions adaptively. Thus, AsPINN maintains precise constraints while addressing the inability of individual structural subnetworks’ generality. This method is then applied to address several physically significant PDEs, including both forward and inverse problems. The numerical results demonstrates AsPINN’s mathematical consistency and generality, offering advantages in terms of optimization speed and accuracy with a reduced number of trainable parameters. The results also manifest that AsPINN mitigates the impact of ill-conditioned data.

A Review of Factors Influencing Syntactic Priming in Chinese English Learners Over the Past Decade

Syntactic priming plays a crucial role in language teaching and learning. This paper reviews studies on syntactic priming among Chinese English learners over the past decade, highlighting the various research paradigms employed, such as picture-description paradigm, sentence-fragment completion paradigm and continuation paradigm, as well as their respective advantages and disadvantages. It also delves into three key factors that influence syntactic priming: learner characteristics, material features, and test task, examining these factors from multiple perspectives, including working memory, repetitive words, and task intensity. By comparing different studies, the paper points out how these factors function and discusses whether they have a positive relationship with syntactic priming. Also, possible reasons for the contradictory findings in some studies have been explored. The paper concludes by summarizing current research finding, putting forward problems existing in present studies, and identifying gaps for future exploration, aiming to offer insights and directions for ongoing research in this area.

Forward and inverse problems for the Benney–Luke type fractional equations

Forward and inverse problems for the Benney–Luke type fractional equations

Greedy selection of optimal location of sensors for uncertainty reduction in seismic moment tensor inversion

We address an optimal sensor placement problem through Bayesian experimental design for seismic full waveform inversion for the recovery of the associated moment tensor. The objective is that of optimally choosing the location of the sensors (stations) from which to collect the observed data. The Shannon expected information gain is used as the objective function to search for the optimal network of sensors. A closed form for such objective is available due to the linear structure of the forward problem, as well as the Gaussian modeling of the observational errors and prior distribution. The resulting problem being inherently combinatorial, a greedy algorithm is deployed to sequentially select the sensor locations that form the best network for learning the moment tensor. Numerical results are presented to display the optimal network of sensors and how the uncertainty in the inferred seismic moment tensor contracts. The scenario of full three-dimensional velocity models or unknown earthquake source locations is treated as nuisance uncertainty, contributing to the overall uncertainty without being the focus of the inversion. This is addressed using a consensus approach over a set of realizations of the nuisance parameter. We analyzed the resulting network of stations for the moment tensor inversion under model misspecification, which reflects realistic data-generating processes.

Sentinel-2 Multispectral Satellite Remote Sensing Retrieval of Soil Cu Content Changes at Different pH Levels

With the development of multispectral imaging technology, retrieving soil heavy metal content using multispectral remote sensing images has become possible. However, factors such as soil pH and spectral resolution affect the accuracy of model inversion, leading to low precision. In this study, 242 soil samples were collected from a typical area of the Pearl River Delta, and the Cu content in the soil was detected in the laboratory. Simultaneously, Sentinel-2 remote sensing image data were collected, and two-dimensional and three-dimensional spectral indices were established. Constructing independent decision trees based on pH values, using the Successive Projections Algorithm (SPA) combined with the Boruta algorithm to select the characteristic bands for soil Cu content, and this was combined with Optuna automatic hyperparameter optimization for ensemble learning models to establish a model for estimating Cu content in soil. The research results indicated that in the SPA combined with the Boruta feature selection algorithm, the characteristic spectral indices were mainly concentrated in the spectral transformation forms of TBI2 and TBI4. Full-sample modeling lacked predictive ability, but after classifying the samples based on soil pH value, the R2 of the RF and XGBoost models constructed with the samples with pH values between 5.85 and 7.75 was 0.54 and 0.76, respectively, with corresponding RMSE values of 22.48 and 16.12 and RPD values of 1.51 and 2.11. This study shows that the inversion of soil Cu content under different pH conditions exhibits significant differences, and determining the optimal pH range can effectively improve inversion accuracy. This research provides a reference for further achieving the efficient and accurate remote sensing of heavy metal pollution in agricultural soil.

Product Design Incorporating Competition Relations: A Network-Based Design Framework Considering Local Dependencies

Abstract System design has been facing the challenges of incorporating complex dependencies between individual entities into design formulations. For example, while the decision-based design framework successfully integrated customer preference modeling into optimal design, the problem was formulated from a single entity’s perspective, and the competition between multiple enterprises was not considered in the formulation. Network science has offered several solutions for studying interdependencies in various system contexts. However, efforts have primarily focused on analysis (i.e., the forward problem). The inverse problem still remains: How can we achieve the desired system-level performance by promoting the formation of targeted relations among local entities? In this study, we answer this question by developing a network-based design framework. This framework uses network representations to characterize and capture dependencies and relations between individual entities in complex systems and integrate these representations into design formulations to find optimal decisions for the desired performance of a system. To demonstrate its utility, we applied this framework to the design for market systems with a case study on vacuum cleaners. The objective is to increase the sales of a vacuum cleaner or its market share by optimizing its design attributes, such as suction power and weight, with the consideration of market competition relations, such as inter-brand triadic competition involving three products from different brands. We solve this problem by integrating an exponential random graph model (ERGM) with a genetic algorithm. The results indicate that the new designs, which consider market competition, can effectively increase the purchase frequency of specific vacuum cleaner models and the proposed network-based design method outperforms traditional design optimization.

Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.

Adjoint-based inversion for stress and frictional parameters in earthquake modeling

We present an adjoint-based optimization method to invert for stress and frictional parameters used in earthquake modeling. The forward problem is linear elastodynamics with nonlinear rate-and-state frictional faults. The misfit functional quantifies the difference between simulated and measured particle displacements or velocities at receiver locations. The misfit may include windowing or filtering operators. We derive the corresponding adjoint problem, which is linear elasticity with linearized rate-and-state friction and, for forward problems involving fault normal stress changes, nonzero fault opening, with time-dependent coefficients derived from the forward solution. The gradient of the misfit is efficiently computed by convolving forward and adjoint variables on the fault. The method thus extends the framework of full-waveform inversion to include frictional faults with rate-and-state friction. In addition, we present a space-time dual-consistent discretization of a dynamic rupture problem with a rough fault in antiplane shear, using high-order accurate summation-by-parts finite differences in combination with explicit Runge–Kutta time integration. The dual consistency of the discretization ensures that the discrete adjoint-based gradient is the exact gradient of the discrete misfit functional as well as a consistent approximation of the continuous gradient. Our theoretical results are corroborated by inversions with synthetic data. We anticipate that adjoint-based inversion of seismic and/or geodetic data will be a powerful tool for studying earthquake source processes; it can also be used to interpret laboratory friction experiments.

Exact recovery of community detection in k-community Gaussian mixture models

Abstract We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $k\geq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the necessary and sufficient conditions for the exact recovery of the maximum likelihood estimation, which can give a sharp phase transition for the exact recovery even though the Gaussian perturbations are not identically distributed; see Section 7. Applications include the community detection on hypergraphs.

Semi-analytic three-shell forward calculation for magnetoencephalography

In previous studies, the magnetic lead field theorem in the quasi-static approximation was derived and used for the development of a method for the forward problem of MEG. It was applied and tested on a single-shell model of the human head and the question whether one shell is adequate enough for the calculation of the magnetic field is the main reason for this study. This forward method is based on the fundamental concept that one can calculate the lead field for MEG by decomposing it into two parts: the lead field of an arbitrary volume conductor that is already known and the gradient of basis functions that have to be harmonic, here derived from spherical harmonics. The problem then is reduced to evaluating the coefficients found in the basis functions. In this research we aim to improve the accuracy of the forward model, hence improving the localization accuracy in inverse methods by introducing a more detailed realistic head model. We here generalize the algorithm developed for a single-shell volume conductor to a three-shell volume conductor representing the brain, the skull and the skin with homogenous and isotropic conductivities in realistic ratios. The expansion to three shells could be tested as the three-shell algorithm is approaching the single-shell with high accuracy in special cases where three-shell solutions can also be calculated using a single-shell solution, especially for higher levels of expansion. The deviation in the calculation of the lead field is also evaluated when using three shells with realistic conductivities. The magnetic field turned out to differ to an important measurable extend in particular for deeper sources, making the three-shell algorithm substantially more accurate for these dipole locations.

Numerical methods for shape optimal design of fluid–structure interaction problems

We consider the method of mappings for performing shape optimization for unsteady fluid–structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on problems derived from an FSI benchmark to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.

Developing a Framework for Safe AI Model Development on Sensitive Healthcare Data

The Dementias Platform UK (DPUK) Data Portal holds over 60 cohort datasets from over 3 million participants with a range of multi-modal data including neuroimaging and genomics. This has meant we have seen an increasing interest in the development of AI models with the potential for clinical implementation. However, this presents a unique challenge to disclosure control when it comes to assessing these AI models for release due to the risk of attacks such as membership inference, model inversion or even just vulnerabilities in the models like overfitting. This is why we hosted a series of workshops bringing together members of the public, expert researchers, and data owners across the UK to build a framework for allowing the safe development and release of AI models trained on sensitive healthcare data. From this, we have put together recommendations and guidelines for the use of privacy-preserving techniques in AI models to protect patient privacy, and to allow the safe deployment of these models outside of trusted research environments. We also identified the unique challenges to privacy and implementation of privacy-preserving techniques in AI models regarding the use of complex data such as neuroimaging and genomics. This framework has created a path forward for supporting safe AI model development which takes into consideration rapidly evolving ethical, legal and privacy considerations.

Application of a surrogate model for condition monitoring of a digital twin gas turbine

Condition monitoring technology plays a crucial role in ensuring the reliable operation of gas turbines. Digital twin has propelled condition monitoring research into a new phase. This paper established a surrogate model of gas turbines for condition monitoring based on Markov-projection approximation subspace tracking. Furthermore, it explores the application of surrogate model in developing digital twin for gas turbines. The study initially establishes a Markov matrix and acquires an observation vector, utilizing the framework of the linear model. Utilizing real-time measurement data of gas turbine, the signal subspace of the observation vector autocorrelation matrix is updated through the projection approximation subspace tracking. By aligning this signal subspace with the generalized observability matrix, the identification results of the surrogate model parameters are obtained online. Furthermore, a variable weight projection approximation subspace tracking method has been proposed to enhance the algorithm robustness. Simulation and real experiment demonstrate that the surrogate model output effectively tracks the real-time changes in gas turbine measurement data. When faults and degradation arise, condition monitoring can be achieved by analyzing the evolution of model parameters to obtain feedback information from the gas turbine. The proposed method maintains its robustness in the presence of impulsive noise. These features offer a novel approach for the development of gas turbine digital twin.

PINN for solving forward and inverse problems involving the integrable two dimensional nonlocal equations

Abstract In this paper, the physics informed neural network (PINN) deep learning method is applied to solve the two dimensional nonlocal equations including partial reverse space y-nonlocal Mel'nikov equation, partial reverse space-time nonlocal Mel'nikov equation and nonlocal two dimensional NLS equation. By PINN method, we successfully derive data-driven two soliton solution, lump solution and rogue wave solution. Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small, which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equation.
Moreover, the parameter discovery of the partial reverse space-time nonlocal Mel'nikov equation is analysed in terms of its soliton solution for the first time.