Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs). However, finding a set of neural network parameters that fulfill a PDE at the boundary and within the domain of interest can be challenging and non-unique due to the complexity of the loss landscape that needs to be traversed. Although a variety of multi-task learning and transfer learning approaches have been proposed to overcome these issues, no incremental training procedure has been proposed for PINNs. As demonstrated herein, by developing incremental PINNs (iPINNs) we can effectively mitigate such training challenges and learn multiple tasks (equations) sequentially without additional parameters for new tasks. Interestingly, we show that this also improves performance for every equation in the sequence. Our approach learns multiple PDEs starting from the simplest one by creating its own subnetwork for each PDE and allowing each subnetwork to overlap with previously learned subnetworks. We demonstrate that previous subnetworks are a good initialization for a new equation if PDEs share similarities. We also show that iPINNs achieve lower prediction error than regular PINNs for two different scenarios: (1) learning a family of equations (e.g., 1-D convection PDE); and (2) learning PDEs resulting from a combination of processes (e.g., 1-D reaction–diffusion PDE). The ability to learn all problems with a single network together with learning more complex PDEs with better generalization than regular PINNs will open new avenues in this field.

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In 1970, Hillert and Staffansson published a paper entitled “The Regular Solution Model for Stoichiometric Phases and Ionic Melts”. It was the beginning of the sublattice model that has been a key component in the development of Computational Thermodynamics. This formalism, now often called the Compound Energy Formalism (CEF), has been used to describe a great variety of phases driven by the need for accurate descriptions of thermodynamic phase stability in a wide range of materials involving many elements. The purpose of this paper is to describe the formalism, the physical meaning of its various parameters and the way they can be assessed using experimental and theoretical data. Furthermore, new developments derived from the CEF, such as the Effective Bond Energy Formalism, and other ideas for further development are presented.

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In a material under stress, grain boundaries may give rise to stress discontinuities. The stress state at grain boundaries strongly affects microscopic processes, such as diffusion and segregation, as well as failure initiation, such as fatigue, creep, and corrosion. Here the general condition of incompatibility stress at grain boundaries is studied with a bicrystal model for linear elastic materials. In materials with cubic crystal structures, it is proven that hydrostatic stress does not lead to a stress discontinuity at grain boundaries. For bicrystals with inclined grain boundaries under uniaxial stress, the extreme values of the incompatibility stress as a function of the inclination angle are obtained by a simulated annealing method. A simple criterion is proposed to classify cubic materials into three groups. For cubic crystals with at most moderate anisotropy, the highest incompatibility stress occurs when the grain boundary plane is perpendicular to the uniaxial stress. For highly anisotropic materials, such as alkali metals and polymorphic high-temperature phases, the highest incompatibility stress occurs on grain boundaries with an inclination of about 47^o.

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Current deep neural networks (DNNs) are overparameterized and use most of their neuronal connections during inference for each task. The human brain, however, developed specialized regions for different tasks and performs inference with a small fraction of its neuronal connections. We propose an iterative pruning strategy introducing a simple importance-score metric that deactivates unimportant connections, tackling overparameterization in DNNs and modulating the firing patterns. The aim is to find the smallest number of connections that is still capable of solving a given task with comparable accuracy, i.e. a simpler subnetwork. We achieve comparable performance for LeNet architectures on MNIST, and significantly higher parameter compression than state-of-the-art algorithms for VGG and ResNet architectures on CIFAR-10/100 and Tiny-ImageNet. Our approach also performs well for the two different optimizers considered—Adam and SGD. The algorithm is not designed to minimize FLOPs when considering current hardware and software implementations, although it performs reasonably when compared to the state of the art.

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Rapid developments in the field of hydrogen energy have prompted the need for safe and efficient hydrogen transportation and storage. Steels form the backbone of the current energy infrastructure and thus offer a fast and cost-effective solution. Their excellent mechanical properties are attributed to the underlying microstructure which comprises of finely dispersed nano-precipitates. However, one major factor restricting their application is their susceptibility to Hydrogen Embrittlement (HE). In the past decade, experimental and theoretical works have been carried out to understand if the nano-sized carbides can aid in reducing the susceptibility to HE along with providing strengthening. Within this ab-inito study, we investigated the effectiveness of fully coherent nano-carbides (i.e. TiC, VC and NbC) to limit the diffusible hydrogen content in bcc Fe. Our study revealed that the interplay between hydrogen and carbon vacancies, local atomic environment at interface as well as elastic strain fields at the interface can lead to significantly increased hydrogen solubilities. While in TiC, the deepest traps were found to be in the bulk of carbides, in VC and NbC, the elastic strain fields around the interface led to the strongest trapping. Further, the formation of a two-hydrogen-vacancy complex was found to be favourable in VC. Finally, the migration barriers for hydrogen trapping in bulk TiC as well as across the Fe/TiC coherent interface indicate that these deep traps in the form of carbon vacancies are fairly accessible.

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Carbide nano-precipitates are commonly used to improve mechanical properties of steel. It has been experimentally observed that TiC, NbC, and VC carbide precipitates initially form as ‘plate-like’ particles oriented in the {1 0 0} planes of the ferrite lattice. These platelets share similarities with Guinier-Preston zones in Al-Cu alloys. The clustering of group IV and V transition metal atoms (M = Ti, Zr, Hf, V, Nb, Ta) in ferrite is studied using density functional theory. It is deduced that the transition metal carbides all form in a similar way. Furthermore, the transition from an initial M–C cluster to a NaCl-structured platelet to a NaCl-structured precipitate is examined through atomistic simulations using Modified Embedded Atom Method potentials. A route is established along which transition metal carbides form and transform into precipitates that possess the Baker-Nutting orientation relation with the ferrite matrix.

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The human brain is capable of learning tasks sequentially mostly without forgetting. However, deep neural networks (DNNs) suffer from catastrophic forgetting when learning one task after another. We address this challenge considering a class-incremental learning scenario where the DNN sees test data without knowing the task from which this data originates. During training, Continual Prune-and-Select (CP&S) finds a subnetwork within the DNN that is responsible for solving a given task. Then, during inference, CP&S selects the correct subnetwork to make predictions for that task. A new task is learned by training available neuronal connections of the DNN (previously untrained) to create a new subnetwork by pruning, which can include previously trained connections belonging to other subnetwork(s) because it does not update shared connections. This enables to eliminate catastrophic forgetting by creating specialized regions in the DNN that do not conflict with each other while still allowing knowledge transfer across them. The CP&S strategy is implemented with different subnetwork selection strategies, revealing superior performance to state-of-the-art continual learning methods tested on various datasets (CIFAR-100, CUB-200-2011, ImageNet-100 and ImageNet-1000). In particular, CP&S is capable of sequentially learning 10 tasks from ImageNet-1000 keeping an accuracy around 94% with negligible forgetting, a first-of-its-kind result in class-incremental learning. To the best of the authors’ knowledge, this represents an improvement in accuracy above 10% when compared to the best alternative method.

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In this study possible routes from dissolved M and C atoms to a M-C (M = Ti, Nb) cluster are studied. Using atomistic modelling to perform relaxation simulations and molecular dynamics (MD) simulations for the Fe-M-C ternary system, the formation of clusters is studied for M. Additionally the stability of M-C clusters is assessed. The clustering of M and C atoms as observed in experiments is also found in simulations. The initial clusters found in this work have a (Fe,M)C composition with a large Fe fraction. Moreover, structurally relaxed clusters reveal that there are growth pathways with a monotone decrease in Gibbs energy, suggesting that the highest energy barrier in the formation of M-C clusters is the diffusion barrier for the atoms forming the cluster. The development of M-C clusters as found in this study suggests a formation mechanism for nano-precipitation of carbides consisting of several steps; first a C cluster forms, then M atoms attach to the C cluster forming a (Fe,M)C cluster, and in the final step the (Fe,M)C cluster transforms to a NaCl-structured carbide.

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In a material under stress, grain boundaries may give rise to stress discontinuities. Stress localization is crucial to materials' behavior such as segregation, precipitation, and void nucleation. Here, the stress state at a grain boundary perpendicular to a uniaxial external stress is studied systematically. The grain boundary with the most extreme stress discontinuity is determined for cubic materials within the elastic limit for a bicrystal model. Additionally, grain boundaries with negligible stress discontinuity are identified. The influence of the elastic tensor components, C₁₁, C₁₂, and C₄₄, and grain orientation is studied quantitatively.

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A reference-free modified embedded atom method (RF-MEAM) potential for iron has been constructed. The new potential is made to predict both bcc and fcc (α-Fe and γ-Fe) lattice properties, with a special interest in modelling in the 800-1300 K temperature range. This is the range in which transformations and key processes in steel occur. RF-MEAM potentials can be used directly in commonly used molecular dynamics simulation software (e.g. LAMMPS). The new potential is compared to several other (M)EAM potentials which are commonly used. It is demonstrated that the new potential combines good characteristics for point defect energies with free surface and stacking fault energies. Also the Nishiyama-Wassermann and Kurdjumov-Sachs orientation relation ratios and interface energies are reproduced, allowing for simulations of α-Fe and γ-Fe interphases.

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A new approximation to the equations describing Classical Nucleation and Growth Theories, is proposed providing quick, and intuitive insight. It gives a prediction of the mean precipitate radius and number density development under quasi-isothermal conditions. Current “mean-radius”, and “multi-class” approaches to modelling classical nucleation and growth theory for precipitation, require considerable computation times. An analytical approximation is proposed to solve the equations, and its results are compared to numerical simulations for quasi-isothermal precipitation. From the approximation a start and end time for the nucleation stage is predicted, as well as a time at which growth occurs and when the coarsening stage starts. Ultimately, these times, outline the numerical solution to the precipitation trajectory, providing key insight before performing numerical simulations. This insight can be used to more efficiently simulate precipitate development, as time scales at which the various stages in precipitate development occur can be predicted for individual precipitates. When these time scales are known a numerical simulation can be used for a specific goal, for instance to only simulate nucleation and growth, thus saving computational time. Moreover, for a first indication of the precipitate development in a composition under a particular heat treatment a numerical simulation is no longer necessary. This is also useful for process control as consequences of changes in treatment can be assessed on-line. Using these approximate analytical results an estimate can be made for the matrix concentration of precipitate forming elements. Additionally some dimensionless parameters are established to provide intuitive details to the precipitation trajectory.@en

Results from DFT calculations are in many cases equivalent to experimental data. They describe a set of properties of a phase at a well-defined composition and temperature, T, most often at 0 K. In order to be practically useful in materials design, such data must be fitted to a thermodynamic model for the phase to allow interpolations and extrapolations. The intention of this paper is to give a summary of the state of the art by using the Calphad technique to model thermodynamic properties and calculate phase diagrams, including some models that should be avoided. Calphad models can decribe long range ordering (LRO) using sublattices and there are model parameters that can approximate short range ordering (SRO) within the experimental uncertainty. In addition to the DFT data, there is a need for experimental data, in particular, for the phase diagram, to determine the model parameters. Very small differences in Gibbs energy of the phases, far smaller than the uncertainties in the DFT calculations, determine the set of stable phases at varying composition and T. Thus, adjustment of the DFT results is often needed in order to obtain the correct set of stable phases.@en

Single-crystal copper films on sapphire have recently been reported upon in relation to graphene growth on these films. In the present paper the kinetics of the formation of single crystal copper films is investigated. We demonstrate the importance of heating the sapphire substrate in 1000 hPa oxygen, followed by a fast cooling prior to depositing the copper film. The importance of this treatment is tentatively explained by the dissolution of oxygen in sapphire and subsequent out-diffusion during recrystallization of the copper film to form a copper-oxide interface layer. Also, the importance of avoiding oxygen incorporation in the sputter deposited film is demonstrated.

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A new generation of radiation detectors relies on the crystalline Si and amorphous B (c-Si/a-B) junctions that are prepared through chemical vapor deposition of diborane (B₂H₆) on Si at low temperature (∼400 C). The Si wafer surface is dominated by the Si{0 0 1}3 1 domains that consist of two different Si species at low temperature. Here we investigate the geometry, stability and electronic properties of the hydrogen passivated Si{0 0 1}3 1 surfaces with deposited BH_n (n = 0 to 3) radicals using parameter-free first-principles approaches. Ab initio molecular dynamics simulations using the density functional theory (DFT) including van der Waals interaction reveal that in the initial stage the BH₃ molecules/radicals deposit on the Si(-H), forming (-Si)BH₄ radicals which then decompose into (-Si)BH₂ with release of H₂ molecules. Structural optimizations provide strong local relaxation and reconstructions at the deposited Si surface. Electronic structure calculations reveal the formation of various defect states in the forbidden gap. This indicates limitations of the presently used rigid electron-counting and band-filling models. The attained information enhances our understanding of the initial stage of the PureB process and the electric properties of the products.

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γ-M₂₃C₆ carbide often forms at grain boundaries in creep-resistant steels and plays a crucial role in creep resistance by blocking microstructural changes at elevated temperatures. Given that the dislocations and stacking faults (SFs) in carbides may affect their stability, the observation of SF formation in γ-M₂₃C₆ using atomic-scale microscopy has implications for Cr-rich creep-resistant steels. Our analysis of SF energies (SFEs) derived from density functional theory calculations reveals that the SFEs are high, which suggests that the SF is not induced by external shear stress deformation but by lattice misfit between conjoined subgrains nucleated from the same austenite grain.

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It is shown that in substitutional alloys, peculiar ordered patterns can result from neighborhood-dependent diffusion activation barriers even when there are no metastable ordered phases. Lattice gases with pure phase separation character are shown to exhibit transient ordered structures that can be retained almost indefinitely, although these structures are not at thermodynamic equilibrium. It is shown that such structures can come about relatively easily by quenching from the high-temperature configurationally random solid solution.

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One of the effective potentials that has proven to be very versatile and useful for describing metals is the modified embedded atom method (MEAM) potential. The reference-free version of the MEAM (RF-MEAM) potential provides more flexibility for fitting than the 2NN-MEAM because it also describes the pair potential as an explicit function. In this work, we present a methodology to fit RF-MEAM potentials to DFT data. We then evaluate the performance of the fitted potential by comparing MD simulations with experimental and DFT data. As an example, the methodology is applied to a binary and a quaternary alloy, namely NiTi and NbMoTaW. In the case of the equi-atomic NiTi shape memory alloy, our attention focuses on designing a potential that properly captures its mechanical behavior, given that the existing potentials fail to predict elastic constants in agreement with experiments. To reach our aim, we included the stress tensors of different high temperature NiTi configurations in the fitting database. The obtained RF-MEAM potential outperforms existing EAM and MEAM potentials in predicting the lattice and elastic constants of austenitic and martensitic phases as well as the corresponding transformation temperatures. To demonstrate the suitability of this methodology also for more complex systems, a RF-MEAM potential is fitted to model the multi-component NbMoTaW high-entropy alloy. Validation is achieved through comparison between observables obtained through the MD output and ab initio data. The article also reports key improvements to the optimization code MEAMfit v2 and the freely-available LAMMPS implementation of the RF-MEAM formalism. Most notably, resorting to analytic derivatives of the objective function with respect to the potential parameters rather than derivatives through finite differences, the time necessary for fitting has decreased by an order of magnitude.

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In recently discovered self healing creep steels, open-volume creep cavities are filled by the precipitation of supersaturated solute. These creep cavities form on the grain boundaries oriented perpendicular to the applied stress. The presence of a free surface triggers a flux of solute from the matrix, over the grain boundaries towards the creep cavities. We studied the creep cavity filling by finite element modelling and found that the filling time critically depends on (i) the ratio of diffusivities in the grain boundary and the bulk, and (ii) on the ratio of the intercavity distance and the cavity size. For a relatively large intercavity spacing 3D transport is observed when the grain boundary and volume diffusivities are of a similar order of magnitude, while a 2D behaviour is observed when the grain boundary diffusivity is dominant. Instead when the intercavity distance is small, the transport behaviour tends to a 1D behaviour in all cases, as the amount of solute available in the grain boundary is insufficient. A phase diagram with the transition lines is constructed.@en

A self-consistent model is applied to predict the creep cavity growth and strain rates in metals from the perspective of self-healing. In this model, the creep cavity growth rate is intricately linked to the strain rate. The self-healing process causes precipitates to grow inside creep cavities. Due to the Kirkendall effect, a diffusional flux of vacancies is induced in the direction away from the creep cavity during this selective self-healing precipitation. This process impedes the creep cavity growth. The critical stress for self-healing can be derived, and an analysis is made of the efficiency of self-healing elements in binary Fe–Cu, Fe–Au, Fe–Mo, and Fe–W alloys. Fe–Au is found to be the most efficient self-healing alloy. Fe–Mo and Fe–W alloys provide good alternatives that have the potential to be employed at high temperatures.

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Electrolysis of molten fluorides is one of the promising methods for the recovery and recycling of rare earth metals from used magnets. Due to the dearth of phase equilibria data for molten fluoride systems, thermodynamic modelling of LiF-DyF₃-NdF₃ system using the CALPHAD approach was carried out. Gibbs energy modelling for LiF-NdF₃ and LiF-DyF₃ systems was performed using the constitutional data from literature. Ab initio calculations were used to obtain enthalpy of reaction of LiDyF₄, an intermediate phase that is found to exist in the LiF-DyF₃ system. Differential thermal analysis was carried out for selected compositions in the NdF₃-DyF₃ system, in order to determine liquidus and solidus temperatures. The Gibbs energy parameters for the limiting binaries determined in this work is used for modelling the Gibbs energy functions of equilibrium phases in the ternary system. Selected compositions of LiF-NdF₃-DyF₃ were subjected to DTA in order to validate the calculated phase temperatures involving melt.

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