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We compare a recently proposed generalized eigensystem approach and a new modified generalized eigensystem approach to more widely used truncated singular value decomposition and zero‐order Tikhonov regularization for solving multidimensional elliptic inverse problems. As a test case, we use a finite element representation of a homogeneous eccentric spheres model of the inverse problem of electrocardiography. Special attention is paid to numerical issues of accuracy, convergence, and robustness. While the new generalized eigensystem methods are substantially more demanding computationally, they exhibit improved accuracy and convergence compared with widely used methods and offer substantially better robustness.

This study aims to develop an adaptive finite element method for structural eigenproblems of cracked Euler–Bernoulli beams via the superconvergent patch recovery displacement technique. This research comprises the numerical algorithm and experimental results for free vibration problems (forward eigenproblems) and damage detection problems (inverse eigenproblems).

The weakened properties analogy is used to describe cracks in this model. The adaptive strategy proposed in this paper provides accurate, efficient and reliable eigensolutions of frequency and mode (i.e. eigenpairs as eigenvalue and eigenfunction) for Euler–Bernoulli beams with multiple cracks. Based on the frequency measurement method for damage detection, using the difference between the actual and computed frequencies of cracked beams, the inverse eigenproblems are solved iteratively for identifying the residuals of locations and sizes of the cracks by the Newton–Raphson iteration technique. In the crack detection, the estimated residuals are added to obtain reliable results, which is an iteration process that will be expedited by more accurate frequency solutions based on the proposed method for free vibration problems.

Numerical results are presented for free vibration problems and damage detection problems of representative non-uniform and geometrically stepped Euler–Bernoulli beams with multiple cracks to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.

The proposed combination of methodologies described in the paper leads to a very powerful approach for free vibration and damage detection of beams with cracks, introducing the mesh refinement, that can be extended to deal with the damage detection of frame structures.

Prediction of excess pore water pressure and estimation of soil parameters are the two key interests for consolidation problems, which can be mathematically quantified by a set of partial differential equations (PDEs). Generally, there are challenges in solving these two issues using traditional numerical algorithms, while the conventional data-driven methods require massive data sets for training and exhibit negative generalization potential. This paper aims to employ the physics-informed neural networks (PINNs) for solving both the forward and inverse problems.

A typical consolidation problem with continuous drainage boundary conditions is firstly considered. The PINNs, analytical, and finite difference method (FDM) solutions are compared for the forward problem, and the estimation of the interface parameters involved in the problem is discussed for the inverse problem. Furthermore, the authors also explore the effects of hyperparameters and noisy data on the performance of forward and inverse problems, respectively. Finally, the PINNs method is applied to the more complex consolidation problems.

The overall results indicate the excellent performance of the PINNs method in solving consolidation problems with various drainage conditions. The PINNs can provide new ideas with a broad application prospect to solve PDEs in the field of geotechnical engineering, and also exhibit a certain degree of noise resistance for estimating the soil parameters.

This study presents the potential application of PINNs for the consolidation of soils. Such a machine learning algorithm helps to obtain remarkably accurate solutions and reliable parameter estimations with fewer and average-quality data, which is beneficial in engineering practice.

The investigation of the influence of the modeling error on the solution of the inverse problem given uncertain measured data in electrical capacitance tomography (ECT).

The solution of the nonlinear inverse problem in ECT and hence, the obtainable accuracy of the reconstruction result, highly depends on the numerical modeling of the forward map and on the required regularization. The inherent discretization error propagates through the forward map, the solution of the inverse problem, the subsequent calculation of process parameters and properties and may lead to a substantial estimation error. Within this work different finite element meshes are compared in terms of obtainable reconstruction accuracy. In order to characterize the reconstruction results, two error measures are introduced, a relative integral error and the relative error in material fraction. In addition, the influence of the measurement noise given different meshes is investigated from the statistical point of view using repeated measurements.

The modeling error, the degree of regularization, and measurement uncertainties are the determining and limiting factors for the obtainable reconstruction accuracy of electrical tomography systems. The impact of these key influence factors on the calculation of process properties given both synthetic as well as measured data is quantified. Practical implications – The obtained results show that especially for measured data, the variability in calculated parameters strongly depends on the efforts put on the forward modeling, i.e. on an appropriate finite element mesh size. Hence, an investigation of the modeling error is highly recommended when real‐world tomography problems have to be solved.

The results presented in this work clearly show how the modeling error as well as inherent measurement uncertainties influence the solution of the inverse problem and the posterior calculation of certain parameters like void fraction in process tomography.

For efficient trajectory control of industrial robots, a cumbersome computation for inverse kinematics and inverse dynamics is needed, which is usually developed using spatial transformation using Denavit–Hartenberg principle and Lagrangian or Newton–Euler methods, respectively. The model is highly non-linear and needs to deal with uncertainties because of lack of accurate measurement of mechanical parameters, noise and non-inclusion of joint friction, which results in some inaccuracies in predicting accurate torque trajectories. To get a guaranteed closed form solution, the robot designers normally follow Pieper’s recommendation and compromise with the mechanical design. While this may be acceptable for the industrial robots where the aesthetic look is not that important, it is not for humanoid and social robots. To help solve this problem, this study aims to propose an alternative machine learning-based computational approach based on a multi-gated sequence model for finding appropriate mapping between Cartesian space to joint space and motion space to joint torque space.

First, the authors generate sufficient data required for the sequence model, using forward kinematics and forward dynamics by running N number of nested loops, where N is the number of joints of the robot. Subsequently, to develop a learning-based model based on sequence analysis, the authors propose to use long short-term memory (LSTM) and hence, train an LSTM model, the architecture details of which have been discussed in the paper. To make LSTM learning algorithms perform efficiently, the authors need to detect and eliminate redundant features from the data set, which the authors propose to do using an elegant statistical tool called Pearson coefficient.

To validate the proposed model, the authors have performed rigorous experiments using both hardware and simulation robots (Baxter/Anukul robot) available in their laboratory and KUKA simulation robot data set made available from Neural Learning for Robotics Laboratory. Through several characteristic plots, it has been shown that a sequence-based LSTM model of deep learning architecture with non-redundant features could help the robots to learn smooth and accurate trajectories more quickly compared to data sets having redundancy. Such data-driven modeling techniques can change the future course of direction of robotics research for solving the classical problems such as trajectory planning and motion planning for manipulating industrial as well as social humanoid robots.

The present investigation involves development of deep learning-based computation model, statistical analyses to eliminate redundant features, data creation from one hardware robot (Anukul) and one simulation robot model (KUKA), rigorously training and testing separately two computational models (specially configured two LSTM models) – one for learning inverse kinematics and one for learning inverse dynamics problem – and comparison of the inverse dynamics model with the state-of-the-art model. Hence, the authors strongly believe that the present paper is compact and complete to get published in a reputed journal so that dissemination of new ideas can benefit the researchers in the area of robotics.

The purpose of this paper is to develop a magnetic induction tomography (MIT) system as well as the conductivity reconstruction algorithms (inverse problem).

In order to define and verify the solution of the inverse problem, the forward problem is formulated using mathematical model of the system. The forward problem is solved using the finite element method. The optimization of the excitation unit is based on the numerical solutions of the direct problem. All the dimensions and shape of the excitation system are optimized in order to focus the main part of the magnetic field in the vicinity of the receiver. Finally, two formulations of the inverse problem are discussed: based on the inversion of the Biot‐Savart law; and based on the artificial neural networks.

The formulation of the forward problem of the considered MIT system is given. The construction of the exciter unit that focuses the main part of the magnetic field in the vicinity of the receiver is proposed. Two formulations of the inverse problem are discussed. First using the inversion of the Biot‐Savart law and second using the artificial neural network. The neural networks seem to be promising tools for reconstructing the MIT images.

This paper demonstrates a real‐life MIT system whose performance is satisfactorily predicted by mathematical models. The original design of the exciter is shown. The new approach to the inverse problem in MIT – the use of the artificial neural network – is presented.

A method has been developed for the solution of inverse heat diffusion problems to find the initial condition, boundary condition, and the source and sink function in the heat diffusion equation. The method has been used in the development of a source‐and‐sink method to find the boundary conditions in inverse Stefan problems. Green's functions have been used in the solution, and the problems are solved by using two approaches: a series solution approach, and a time incremental approach. Both can be used to find the boundary conditions without reliance on the flux information to be supplied at both sides of the interface. The methods are efficient in that they require less equations to be solved for the conditions. The numerical results have shown to be accurate, convergent, and stable. Most of all, the results do not degrade with time as in other time marching schemes reported in the literature. Algorithms can also be easily developed for the solution of the conditions.

Inverse problems deal with determining the causes on the basis of knowing their effects. The object of the inverse parameter estimation problem is to fix the thermal material parameters (the cause) on the strength of a given observation of the temperature history at one or more interior points (the effect). This paper demonstrates two novel approaches to the inverse problems. These approaches use two artificial intelligence mechanisms: neural network and genetic algorithm. Examples shown in this paper give a comparison of results obtained by both of these methods. The numerical technique of neural networks evolved from the effort to model the function of the human brain and the genetic algorithms model the evolutional process of nature. Both of the presented approaches can lead to a solution without having problems with the stability of the inverse task. Both methods are suitable for parallel processing and are advantageous for a multiprocessor computer architecture.

This paper aims to use different machine learning (ML) algorithms for the prediction of inverse kinematic solutions in parallel manipulators (PMs) to overcome the computational difficulties and approximations involved with the analytical methods. The results obtained from the ML algorithms and the Denavit–Hartenberg (DH) approach are compared with the experimental results to evaluate their performances. The study is performed on a novel 6-degree of freedom (DoF) PM that offers precise motions with a large workspace for the end effector.

The kinematic model for the proposed 3-PPSS PM is obtained using the modified DH approach and its inverse kinematic solutions are determined using the Levenberg–Marquardt algorithm. Various prediction algorithms such as the multiple linear regression, multi-variate polynomial regression, support vector, decision tree, random forest regression and multi-layer perceptron networks are applied to predict the inverse kinematic solutions for the manipulator. The data set required to train the network is generated experimentally by recording the poses of the end effector for different instantaneous positions of the slider using the concept of ArUco markers.

This paper fully demonstrates the possibility to use artificial intelligence for the prediction of inverse kinematic solutions especially for complex geometries.

As the analytical models derived from the geometrical method, Screw theory or numerical techniques involve approximations and needs more computational power, it is not advisable for real-time control of the manipulator. In addition, the data set obtained from the derived inverse kinematic equations to train the network may lead to inaccuracies in the predicted results. This error may generate significant deviations in the end-effector position from the desired position. The present work attempts to resolve this issue by proposing a camera-based approach that uses ArUco library and ML algorithms to create the data set experimentally and predict the inverse kinematic solutions accurately.

The application of statistical inversion theory provides a powerful approach for solving estimation problems including the ability for uncertainty quantification (UQ) by means of Markov chain Monte Carlo (MCMC) methods and Monte Carlo integration. This paper aims to analyze the application of a state reduction technique within different MCMC techniques to improve the computational efficiency and the tuning process of these algorithms.

A reduced state representation is constructed from a general prior distribution. For sampling the Metropolis Hastings (MH) Algorithm and the Gibbs sampler are used. Efficient proposal generation techniques and techniques for conditional sampling are proposed and evaluated for an exemplary inverse problem.

For the MH-algorithm, high acceptance rates can be obtained with a simple proposal kernel. For the Gibbs sampler, an efficient technique for conditional sampling was found. The state reduction scheme stabilizes the ill-posed inverse problem, allowing a solution without a dedicated prior distribution. The state reduction is suitable to represent general material distributions.

The state reduction scheme and the MCMC techniques can be applied in different imaging problems. The stabilizing nature of the state reduction improves the solution of ill-posed problems. The tuning of the MCMC methods is simplified.

The paper presents a method to improve the solution process of inverse problems within the Bayesian framework. The stabilization of the inverse problem due to the state reduction improves the solution. The approach simplifies the tuning of MCMC methods.