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The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametric PDE. The central component in our approach is an efficient neural-network-weighted Minimal-Residual formulation, which, after training, provides Galerkin-based approximations in standard discrete spaces that have accurate quantities of interest, regardless of the coarseness of the discrete space. [ABSTRACT FROM AUTHOR]

In this continuum theory, we propose a mathematical framework to study the mechanical interplay of bulk-surface materials undergoing deformation and phase segregation. To this end, we devise a principle of virtual powers with a bulk-surface dynamics, which is postulated on a material body P where the boundary ∂ P may lose smoothness, that is, the normal field may be discontinuous on an edge ∂ 2 P. The final set of equations somewhat resemble the Navier–Stokes–Cahn–Hilliard equation for the bulk and the surface. Aside from the systematical treatment based on a specialized version of the virtual power principle and free-energy imbalances for bulk-surface theories, we consider two additional ingredients: an explicit dependency of the apparent surface density on the surface thickness and mixed boundary conditions for the velocity, chemical potential, and microstructure. • A mathematical framework for the mechanical interplay of bulk-surface materials. • Explicit dependency of the apparent surface density on the surface thickness. • A principle of virtual powers with bulk-surface dynamics. • Bulk-surface Lyapunov decay relations. [ABSTRACT FROM AUTHOR]

One of the main approaches to increase the surface area of a substrate is through depositing a film of a porous materials such as Zeolite imidazole framework (ZIF). ZIF films have shown surpassing capabilities because of their zeolite-like features, including porosity, homogeneous pore size, structural tunability and remarkable thermal and chemical stability. Many methods have been proposed and tested to form such films. One of the techniques that have been documented is electrosynthesis which is considered to be the most practical, quick, and mildly conditioned. In this study, ZIF-67 films were cathodically electrodeposited on an electrically conductive Indium Tin Oxide (ITO) coated Polyethylene Terephthalate (PET) substrates. Unlike previous reports, in this study, water was used as a solvent. In addition to this, effect of crucial operating parameters such as applied potential, molar ratio of reactants and solution pH, on the formation of ZIF-67, was investigated for the first time. It was found that increasing the applied potential, increased the surface coverage and decreased the formed ZIF-67 crystal size. Changing the molar ratio between the organic ligand and the metal salt had a profound influence on the formed phase and crystal shape. It was also found that at neutral and mildly basic solution pH, ZIF-67 could not be formed. Also, statistical analyses were carried out showing low p-values (≪0.05), expressing strong relation between variables and robustness and reliability of the data collected in this study. Finally, mathematical expressions were fitted to experimental data to reveal the relation between applied potential, molar ratios of reactants, pH and conductivity of solutions, surface coverage and crystal size. This study revealed that surface coverage, crystal shape and size can be controlled by manipulating operating conditions. Cathodic electrodeposition of ZIF-67 films in three-electrodes cell. [Display omitted] • Cathodic electrodeposition of ZIF-67 using water. • Parametric study effect on crystal size and surface coverage of ZIF-67 films. • Increasing applied potential increased surface coverage and decreased crystal size. • Linker to metal molar ratio has a profound effect on crystal shape and size. [ABSTRACT FROM AUTHOR]