Electrical impedance tomography (EIT) is a non-invasive imaging modality used for estimating the conductivity of an object Ω from boundary electrode measurements. In recent years, researchers ...achieved substantial progress in analytical and numerical methods for the EIT inverse problem. Despite the success, numerical instability is still a major hurdle due to many factors, including the discretization error of the problem. Furthermore, most algorithms with good performance are relatively time consuming and do not allow real-time applications. In our approach, the goal is to separate the unknown conductivity into two regions, namely the region of homogeneous background conductivity and the region of non-homogeneous conductivity. Therefore, we pose and solve the problem of shape reconstruction using machine learning. We propose a novel and simple jet intriguing neural network architecture capable of solving the EIT inverse problem. It addresses previous difficulties, including instability, and is easily adaptable to other ill-posed coefficient inverse problems. That is, the proposed model estimates the probability for a point of whether the conductivity belongs to the background region or to the non-homogeneous region on the continuous space Rd∩Ω with d∈{2,3}. The proposed model does not make assumptions about the forward model and allows for solving the inverse problem in real time. The proposed machine learning approach for shape reconstruction is also used to improve gradient-based methods for estimating the unknown conductivity. In this paper, we propose a piece-wise constant reconstruction method that is novel in the inverse problem setting but inspired by recent approaches from the 3D vision community. We also extend this method into a novel constrained reconstruction method. We present extensive numerical experiments to show the performance of the architecture and compare the proposed method with previous analytic algorithms, mainly the monotonicity-based shape reconstruction algorithm and iteratively regularized Gauss-Newton method.
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In engineering design, oftentimes a system’s dynamic response is known or can be measured, but the source generating these responses is not known. The mathematical problem where the focus is on ...inferring the source terms of the governing equations from the set of observations is known as an inverse source problem (ISP). ISPs are traditionally solved by optimization techniques with regularization, but in the past few years, there has been a lot of interest in approaching these problems from a deep-learning viewpoint. In this paper, we propose a deep learning approach—infused with physics information—to recover the forcing function (source term) of systems with one degree of freedom from the response data. We test our architecture first to recover smooth forcing functions, and later functions involving abruptly changing gradient and jump discontinuities in the case of a linear system. Finally, we recover the harmonic, the sum of two harmonics, and the gaussian function, in the case of a non-linear system. The results obtained are promising and demonstrate the efficacy of this approach in recovering the forcing functions from the data.
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The application of deep neural networks towards solving problems in science and engineering has demonstrated encouraging results with the recent formulation of physics-informed neural networks ...(PINNs). Through the development of refined machine learning techniques, the high computational cost of obtaining numerical solutions for partial differential equations governing complicated physical systems can be mitigated. However, solutions are not guaranteed to be unique, and are subject to uncertainty caused by the choice of network model parameters. For critical systems with significant consequences for errors, assessing and quantifying this model uncertainty is essential. In this paper, an application of PINN for laser bio-effects with limited training data is provided for uncertainty quantification analysis. Additionally, an efficacy study is performed to investigate the impact of the relative weights of the loss components of the PINN and how the uncertainty in the predictions depends on these weights. Network ensembles are constructed to empirically investigate the diversity of solutions across an extensive sweep of hyper-parameters to determine the model that consistently reproduces a high-fidelity numerical simulation.
•A physics informed neural network is designed for solving the heat diffusion equation.•An ensemble method increases accuracy of predictions and quantifies uncertainty.•A weighting heuristic automatically normalizes individual components of loss function.•Equitable convergence amongst competing minimization objectives is enforced.•Network design parameters are optimized for both accuracy and reliability.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Modeling of physical processes as partial differential equations (PDEs) is often carried out with computationally expensive numerical solvers. A common, and important, process to model is that of ...laser interaction with biological tissues. Physics-informed neural networks (PINNs) have been used to model many physical processes, though none have demonstrated an approximation involving a source term in a PDE, which modeling laser-tissue interactions requires. In this work, a numerical solver for simulating tissue interactions with lasers was surrogated using PINNs while testing various boundary conditions, one with a radiative source term involved. Models were tested using differing activation function combinations in their architectures for comparison. The best combinations of activation functions were different for cases with and without a source term, and R2 scores and average relative errors for the predictions of the best PINN models indicate that it is an accurate surrogate model for corresponding solvers. PINNs appear to be valid replacements for numerical solvers for one-dimensional tissue interactions with electromagnetic radiation.
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In this study, we develop a physics-based autonomous vehicle longitudinal driver model (PAVL-DM) to move an automated vehicle (AV) forward with a minimum following gap while considering safety and ...passenger comfort. Unlike existing driver models for longitudinal control, PAVL-DM parameters do not need to be calibrated for different traffic states, such as congested and non-congested traffic conditions. First, we present the concept, theoretical considerations and mathematical formulations of the PAVL-DM longitudinal control model for AVs. Second, we theoretically analyze the PAVL-DM model equilibria and robustness for safety assurance and string stability. Third, we present numerical analysis related to riding comfort and a dynamic minimum following gap function for each automated vehicle in a platoon while considering the safety for the automated platoon. Our analyses suggest that the PAVL-DM (i) is able to maintain safety between a subject AV and an immediate front vehicle using a newly defined safe gap function depending on the speed and reaction time of an AV; (ii) shows local stability and string stability; and (iii) provides riding comfort for a range of autonomous driving aggressiveness depending on passengers' corresponding preferences on driving pattern. In addition, we conducted a case study using a real-world dataset, which proves that an AV with the PAVL-DM model maintains a minimum following gap between a subject AV and an immediate front vehicle without compromising safety and passenger comfort.
This paper aims to determine whether regularization improves image reconstruction in electrical impedance tomography (EIT) using a radial basis network. The primary purpose is to investigate the ...effect of regularization to estimate the network parameters of the radial basis function network to solve the inverse problem in EIT. Our approach to studying the efficacy of the radial basis network with regularization is to compare the performance among several different regularizations, mainly Tikhonov, Lasso, and Elastic Net regularization. We vary the network parameters, including the fixed and variable widths for the Gaussian used for the network. We also perform a robustness study for comparison of the different regularizations used. Our results include (1) determining the optimal number of radial basis functions in the network to avoid overfitting; (2) comparison of fixed versus variable Gaussian width with or without regularization; (3) comparison of image reconstruction with or without regularization, in particular, no regularization, Tikhonov, Lasso, and Elastic Net; (4) comparison of both mean square and mean absolute error and the corresponding variance; and (5) comparison of robustness, in particular, the performance of the different methods concerning noise level. We conclude that by looking at the R2 score, one can determine the optimal number of radial basis functions. The fixed-width radial basis function network with regularization results in improved performance. The fixed-width Gaussian with Tikhonov regularization performs very well. The regularization helps reconstruct the images outside of the training data set. The regularization may cause the quality of the reconstruction to deteriorate; however, the stability is much improved. In terms of robustness, the RBF with Lasso and Elastic Net seem very robust compared to Tikhonov.
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Malaria is the world’s most fatal and challenging parasitic disease, caused by the Plasmodium parasite and transmitted to humans by the bites of infected female mosquitos. This study presents a ...stochastic process representing the evolution in time of an unexpected phenomenon. We use different probability techniques to assume the possible outcome in a Malaria model since state variables or parameters are random in a stochastic model. This study considers a deterministic vector-host malaria model and converts this model into the corresponding stochastic model with a Continuous-time Markov chain (CTMC) and Stochastic differential equation (SDE). We prove the positivity and boundedness of solutions and calculate the equilibrium points. The threshold values are evaluated using different approaches. Moreover, the Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) procedures are used to analyze the sensitivity of each model parameter. Finally, a transient numerical simulation is discussed, and their outcomes strongly correlate with the actual scenario.
•Develop a continuous-time Markov chain and stochastic differential equations approach for modeling malaria propagation.•Present positivity and boundedness of solutions and discuss disease-free equilibrium.•Conduct sensitivity analysis using Latin hypercube sampling and partial rank correlation coefficients procedure.•Suggest the best strategy for successfully controlling malaria transmission.•Show such analysis can help the decision-makers understand the impact of various parameters in the malaria model.
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Population diffusion in river-ocean ecologies and for wild animals, including birds, mainly depends on the availability of resources and habitats. This study explores the dynamics of the ...resource-based competition model for two interacting species in order to investigate the spatiotemporal effects in a spatially distributed heterogeneous environment with no-flux boundary conditions. The main focus of this study is on the diffusion strategy, under conditions where the carrying capacity for two competing species is considered to be unequal. The same growth function is associated with both species, but they have different migration coefficients. The stability of global coexistence and quasi-trivial equilibria are also studied under different conditions with respect to resource function and carrying capacity. Furthermore, we investigate the case of competitive exclusion for various linear combinations of resource function and carrying capacity. Additionally, we extend the study to the instance where a higher migration rate negatively impacts population growth in competition. The efficacy of the model in the cases of one- and two-dimensional space is also demonstrated through a numerical study.
AMS subject classification 2010
92D25, 35K57, 35K50, 37N25, 53C35.
Population movements are necessary to survive the individuals in many cases and depend on available resources, good habitat, global warming, climate change, supporting the environment, and many other ...issues. This study explores the spatiotemporal effect on the dynamics of the reaction-diffusion model for two interacting populations in a heterogeneous habitat. Both species are assumed to compete for different fundamental resources, and the diffusion strategies of both organisms follow the resource-based diffusion toward a positive distribution function for a large variety of growth functions. Depending on the values of spatially distributed interspecific competition coefficients, the study is conducted for two cases: weak competition and strong competition, which do not perform earlier in the existing literature. The stability of global attractors is studied for different conditions of resource function and carrying capacity. We investigated that in the case of weak competition, coexistence is attainable, while strong competition leads to competitive exclusion. This is an emphasis on how resource-based diffusion in the niche impacts selection. When natural resources are in sharing, either competition or predator-prey interaction leads to competitive exclusion or coexistence of competing species. However, we concentrate on the situation in which the ideal free pair is achieved without imposing any other additional conditions on the model's parameters. The effectiveness of the model is accomplished by numerical computation for both one and two space dimension cases, which is very important for biological consideration.
