Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems

We present a method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy.In this procedure,a coarse grid of training points is used at the initial training stages,while more points are added at later stages based on the value of the residual at a larger set of evaluation points.This method increases the robustness of the neural network approximation and can result in significant computational savings,particularly when the solution is non-smooth.Numerical results are presented for benchmark problems for scalar-valued PDEs,namely Poisson and Helmholtz equations,as well as for an inverse acoustics problem.

Computational Machine Learning Representation for the Flexoelectricity Effect in Truncated Pyramid Structures

In this study,machine learning representation is introduced to evaluate the flexoelectricity effect in truncated pyramid nanostructure under compression.A Non-Uniform Rational B-spline(NURBS)based IGA formulation is employed to model the flexoelectricity.We investigate 2D system with an isotropic linear elastic material under plane strain conditions discretized by 45×30 grid of B-spline elements.Six input parameters are selected to construct a deep neural network(DNN)model.They are the Young's modulus,two dielectric permittivity constants,the longitudinal and transversal flexoelectric coefficients and the order of the shape function.The outputs of interest are the strain in the stress direction and the electric potential due flexoelectricity.The dataset are generated from the forward analysis of the flexoelectric model.80%of the dataset is used for training purpose while the remaining is used for validation by checking the mean squared error.In addition to the input and output layers,the developed DNN model is composed of four hidden layers.The results showed high predictions capabilities of the proposed method with much lower computational time in comparison to the numerical model.

Mechanical Properties of All MoS_(2) Monolayer Heterostructures:Crack Propagation and Existing Notch Study

The outstanding thermal,optical,electrical and mechanical properties of molybdenum disolphide(MoS_(2))heterostructures make them exceptional candidates for an extensive area of applications.Nevertheless,despite considerable technological and academic interest,there is presently a fewinformation regarding the mechanical properties of these novel two-dimensional(2D)materials in the presence of the defects.In thismanuscript,we performed extensive molecular dynamics simulations on pre-cracked and pre-notched all-molybdenum disolphide(MoS_(2))heterostructure systems using ReaxFF force field.Therefore,we study the influence of several central-crack lengths and notch diameters on the mechanical response of 2H phase,1T phase and composite 2H/1T MoS_(2) monolayers with different concentrations of 1T phase in 2H phase,under uniaxial tensile loading at room temperature.Our ReaxFF models reveal that larger cracks and notches decrease the strength of all 2D MoS_(2) single-layer heterostructures.Additionally,for all studied crack and notch sizes,2H phase of MoS_(2) films exhibits the largest strength.Maximum tensile stress of composite 2H/1T MoS_(2) nanosheet with different concentrations are higher than those for the equivalent 1T phase,which implies that the pre-cracked composite structure is remarkably stronger than the equivalent 1T phase.The comparison of the results for cracked and notched all-MoS_(2) nanosheet heterostructures reveal that the load bearing capacity of the notched samples of monolayerMoS_(2) are higher than the cracked ones.