dataesr/scientific-paragraphs-categorization huggingface dataset:
The "total" (1α)-reliable ℓ 2 -loss of the "simplified" adaptive recovery is then O σ s 2 log[1/α] + s log[L/α]. The latter bound is a simple corollary of the oracle inequalities of Proposition 3. 1 and the following statement. Lemma 4. 1. Let m ∈ Z + , ν > 1, and let H s [ω] be the set of harmonic oscillations x with the minimal frequency separation satisfying δ min ≥ 2πν 2m + 1. ( 24 ) Then there exists a filter φ o ∈ C m (Z) satisfying x = φ o * ∆ m x for all x ∈ H s [ω] and such that φ o 2 ≤ Qs 2m + 1 , where Q = ν + 1 ν -1. In particular, whenever δ min ≥ 4π 2m+1 , one has φ o 2 ≤ 3s 2m + 1. Dmitrii Ostrovskii was supported by ERCIM Alain Bensoussan Scholarship while finalizing this project. Zaid Harchaoui received support from NSF CCF 1740551. Research of Anatoli Juditsky and Arkadi Nemirovski was supported by MIAI Grenoble Alpes (ANR-19-P3IA-0003).
1. 3). The section ends with an useful overview of the method is Section 3. 2. In Section 4 we propose a novel physically inspired DNN architecture tailored to exploit particular physical symmetries in the reduced-basis generation (Section 4. 2). Finally, Section 5 and Section 6 are dedicated to the numerical implementation of the proposed methodology, concerning dataset construction, DNN model training, and relevant numerical examples in coupled micro-macro simulations, respectively.
Easiness of implementation: The very mathematical structure of the local problem is unchanged, only the BCs are touched. Our method leads to non-homogeneous Dirichlet BCs, not leading to any additional difficulties in terms of implementation. As usual in RB-ROM strategies, DeepBND can be structured into offline and online phases. Having chosen some meaningful microstructural parametrisation for the HF problem at hand, the main goal of the offline phase is to find a mapping that translates parameters from the high-fidelity problem into an accurate BC of the problem solved in the online phase. It consists in the following pipeline: i) the dataset acquisition by simulating those models for some representative sampling of the parameter space, ii) the RB construction using the previous dataset, iii) extraction of relevant features of the dataset through the orthogonal projections onto this RB and, iv) the training of a DNN model aiming to predict these projections.
Homogenisation of the obtained solution Algorithm 2: General workflow of the DeepBND method detailing Offline and Online phases. It is worth mentioning that the RB-ROM and DNN concentrate most of the computational efforts in the offline phase rather than in the online phase. To generate the dataset, some parametrisation of the microstructure should be chosen, together with the range of these parameters, their probability distributions, and sample strategy. This is a problem-dependent choice and we discuss in Section 5 the examples treated in the manuscript. In turn, the final cost added in comparison to a standard coupled multiscale simulation is the evaluation of a DNN model to determine the BC that feeds the corrector problem solver. The DNN evaluation step (prediction) is many orders of magnitude less computationally demanding than the corrector problem numerical solution, by using FEM for example.
Designing the Neural Network In this section, we introduce the DNN model to predict the BCs. As mentioned before, we aim at characterising the nonlinear mapping from the parameter space into RB coefficients N : R Np → R N rb. In terms of DNN architecture, our model is based on two instances of the standard fully connected Multilayer Perceptron (MLP) model, which is revisited in Section 4. 1, combined seamlessly to exploit: i) the splitting between macro-strain and microscale constitutive parameters and ii) the problem symmetries. We also explore such principles for dataset and RB generation (Section 4. 2), which, in fact, naturally leads to a specific format for combining MLP submodels, as explained in Section 4. 3. Regarding the loss function, we also adopt a non-standard setting motivated from the RB-ROM (see Section 4.
Basic notation for Deep Neural Networks Here we review some basic concepts and notation for supervised learning using DNN. For the interested reader, we refer to  [18, 13]  for a deeper presentation. Let D = {(α (i) , β (i) ) ∈ R m × R n ; i = 1,. , N s } be a dataset with N s samples (or snapshots). We aim at finding the mapping N (θ; •) : R m → R n such that it minimises the regularised empirical risk min θ 1 N s Ns i=1 L(N (θ; α (i) ), β (i) ) + λR(θ), (30) where L : R m × R m → R + is the so-called cost or loss function, and R : R p × R m → R + is a regularising function controlled by a intensity factor λ ∈ R +. The vector θ should be understood as a collection of parameters that defines the mapping N (θ; •), which is taken accordingly with the usual definition of the MLP architecture for DNN  [13].
Dataset and RB construction strategies As seen in Section 3. 1. 2, the trace w of the fluctuations on ∂Ω R µ , is the linear combination of auxiliary solutions obtained to each load direction (w (I) ), for I = 1, 2, 3. Obtaining them involve fewer parameters, since the loads are fixed in unitary directions, which lead to a reduced parameter space to sample. In other words, exploiting this property in the dataset generation means that for a fixed number of high-fidelity snapshots, we can sample better a smaller parameter set P r ⊂ R N b instead of P ⊂ R Np , where N p = N b + 3 (in 2D). In this regard, it is also important to maximise the covering of P r by using a smart sampling strategy, for example one of the Quasi-Monte Carlo Family such as the Latin Hypercube Sampling (LHS) used in this work.
DeepBND-NN model: definition and combination of the axial and shear DNN submodels As already commented, before defining the DNN model, we need to obtain a DNN submodel for the axial unitary macro-strain (without loss of generality, chosen here as the horizontal one) and another DNN submodel for shear unitary macro-strain. Following the previous conventions, we denote them as N (1) , N (3) : R N b → R N rb , for the axially horizontal and shear loading, respectively. In turn, for the axially vertical loading, we invoke  (31)  to define N (2)  (p) = N (1) π -1 π 2 (p). Finally, the BC prediction for some p ∈ R Np reads as ũN µ (p)| ∂Ω R µ = 3 I=1 p ε,I   N rb j=1 [N (I) (p c )] j ξ (I) j   ∈ L 2 (∂Ω R µ ). ( 32 ) It is important to mention that by taking into account the physical structure of the problem into the DNN architecture (reviewed in Figure  6 ), we are able to: i) minimise the size of datasets needed since the size of parameter space (input) is smaller for each submodel; ii) use fewer number of neurons and layers (smaller complexity of the DNN architecture) since the prediction of directional load-wise solutions are also less complex; iii) profit from the full advantage of the RB representation due to their specialisation according to the load; iv) use fewer RB modes for the same accuracy; v) naturally take into account symmetries concerning horizontal and vertical loads and vi) training of 2 instead of 3 submodels.
DNN model construction: datasets, RB, and training In this section, we specify the choice of numerical parameters adopted to assess our method. We choose to constrain the more general setting of Sections 3. 1 and 4. As already discussed, the DNN model construction encompasses the RB extraction and the training of a DNN for a given dataset, denoted here as training/RB dataset, with size N trb s = 51200. Moreover, from the point of view of DNN training, we also need to simulate auxiliary datasets for validation and testing reasons, in which the size is assumed to be respectively 20% (N val s = 10240) and 10% (N tes s = 5120) of the training/RB dataset size. Importantly, each dataset is considered for the axial and shear loads separately, as summarised in Table  1. The fixed coefficients used in all simulations are presented in Table  2 , while the random sampling of the inclusion sizes, which are the random parameters of the problem, are shown next.
Indeed, each realisation is not independent from each other. We use the default LHS implementation provided scikit-opt library  [4] , with three different seeds for random generation and sample sizes according to Table  1 , yielding the samples P (1) , P (2) , and P (3). Examples of some realisations of microstructures are found in Figure  7. It is widely known in the multiscale solid mechanics literature that the volume fraction φ = N b i=1 πr 2 i (L HF µ ) 2 is a random variable that plays a crucial role in the final solution. Consequently it is worth looking at how this variable is indirectly sampled by the assumed random generation algorithm, which focuses only on individual radii. In Figure  8 , checking the histograms of φ for P (1) , P (2) , and P (3) , we can observe that, although not implicitly enforced, the resulting underlying probability distribution seems to be consistent along the several datasets.
Furthermore,       Finally, by applying POD procedure of Algorithm 1 to the datasets 1 and 2, we can evaluate the committed error in the sense of  (10) , as shown in Figure  9. This error represents the best expected approximation of the BC in the sense of L 2 (∂Ω R µ ), i. e. , with no error due to the DNN approximation. We can verify these two errors are comparable, with a slight advantage to the shear over the axial model, for a given number of RB functions. As result, we have adopted the same DNN architecture and number of RB in both models, since they have comparable complexity. To define the models N (1) and N (3) , it is worth studying the sensibility of the total error E T with the number of RB (see  (33) ). It is reasonable to expect that at some point the decreasing of E P OD may not compensate the increasing of E DN N as N rb gets larger.
0 × 10 -5 in epoch 5000 (last). It is worth mentioning that we reached these values by systematically testing combinations of smaller architectures, regularisation values, and activation functions. Therefore, such values should be understood as guidance for problems with similar complexity and dataset sizes, although there might be still room for small tunings. Finally, we used Tensorflow  [1]  for the implementation, and the final model is assumed to be the one yielding to the least loss function value in the validation dataset. In Figures  10(a  The loss function evaluated on training and validation differ considerably as result of regularisation, which is not used in validation, the latter turning out to be smaller. For the last epochs, the validation reaches a plateau while the training error continues to decrease slightly.
This indicates that the model reached its full capacity while avoiding overfitting. The very same comments apply to Figures  11(a     from the validation and test datasets from Section 5, already at hand. They are sorted among the elements without repetition, which is possible since there are fewer integration points in the finer mesh than the pre-simulated structures microstructures available. To obtain more reliable results, we always report results considering the average over 10 realisations, where each realisation should be understood as a different sorting of MDs along the integration points. Importantly, also notice that due the scale separation, the MDs choices keeps no neighbouring relation. As first study, we consider the convergence of the vertical displacement of the tip A (Figure  13(a) ) with respect to the mesh divisions in the vertical direction.
This number refers only to the embarrassingly parallel part of the problem, which in this case is the assembly of FEM matrices for the macroscale problem, specifically the solution of the local problems. The remaining fixed time for solving the macroscale problem is the same, hence no reason of including it here. Also, the mesh generation is not taken into account, but if taken, the speed-up would be even higher. The DeepBND was not compared directly with periodic BCs times, since due to the more efficient native Fenics/Multiphenics implementation of enforcing Dirichlet BCs with DeepBND, instead of the periodicity which is imposed by Lagrange Multipliers with additional pure Python routines. (g) σ HF 12. (h) σ dnn 12. (i) σ per 12. Figure  17 : Comparison Cauchy stress components in the Cook membrane.
Clamped bar using DNS The last test of the new multiscale methodology aims at assessing its performance in terms of accuracy and computational cost with respect to other multiscale methods in comparison to direct numerical simulations (DNS). In this second numerical example, for easiness of microstructures constructions, we adopt a rectangular bar Ω = [0, L x = 4] × [0, L y = 1] as macroscopic domain such that it is clamped on the left and loaded with a uniform shear on the right t = (0, -0. 2), as in Figure  18. The material is organised by a N DN S x × N DN S y grid of H DN S = L x /N DN S x = L y /N DN S y squared blocks in which each block contains a random sized circular inclusion. The mesh for the DNS simulations is generated with the characteristic length h ≈ H DN S /15, which is similar to the one used in the dataset generation.
Results of scientific work in chemistry can usually be obtained in the form of materials and data. A big step towards transparency and reproducibility of the scientific work can be gained if scientists publish their data in a FAIR (Findable, Accessible, Interoperable, Reusable) manner in research data repositories. Nevertheless, in order to make chemistry as a discipline sustainable, obtaining FAIR data is insufficient and a comprehensive concept including the preservation of materials is needed. We describe in this article how we combined two infrastructures, a repository for research data (Chemotion repository) and an archive for chemical compounds (Molecule Archive), in order to offer a comprehensive infrastructure to find and access data and materials that were generated in chemistry projects.
INTRODUCTION Sustainable work and the sustainable provision of research results for others is an essential criterion for efficient work in science. Only if results are accessible to the entire scientific community and thereby reusable, scientific progress can be accelerated in a targeted manner. Since the publication of the FAIR data principles  1  , more and more scientists and stakeholders such as https://doi. org/10. 26434/chemrxiv-2023-2dd4c ORCID: https://orcid. org/0000-0001-9513-2468 Content not peer-reviewed by ChemRxiv. License: CC BY 4. 0 funding agencies and journals/publishers support the generation and provision of FAIR data, regardless of the discipline. Accordingly, data should be Findable, Accessible, Interoperable and Reusable (FAIR), which is particularly relevant for providing research data that form the basis of publications.
In chemistry and materials science, a variety of initiatives promote the provision of FAIR data or provide assistance for the implementation of FAIR data measures. Established stakeholders that supported the concepts of FAIR data even before their explicit publication are e. g. IUPAC  2  , RDA  3  or CODATA  4. These have been joined by other important groups such as EOSC  5  , and in Germany by the National Research Data Infrastructure (NFDI)  6  , in particular the consortium for Chemistry (NFDI4Chem)  7. In the future, NFDI4Chem will strengthen the chemical community by providing infrastructure for the generation and provision of FAIR data. Adhering to the FAIR data principles and aligning research processes to obtain FAIR data can be the basis for substantial improvement in data availability and quality.
FAIR data can strengthen trust in research results through transparency and promote their systemic and barrier-free subsequent use. In synthetic chemistry, the FAIR principles must not be reduced to data and descriptive metadata alone. Chemists can provide more than data for documentation and subsequent use. Very often, chemists can substantiate results by synthesis products and thus provide physical evidence of the research work and its quality. Where the reaction products obtained are stable, they can be collected, stored and registered so that, if suitable, the work result can be used directly for further studies. Examples of such direct re-use could be the independent reproduction of experiments, the use of the chemical samples for reactions or the analysis of the samples by characterization or screening techniques.
Principles and concept design: Scientific outcome in chemistry consists of data and materials and new studies also depend on both. Therefore, we suggest complementing the FAIR (meta)data principles with a concept for materials that address sustainable access to and re-use of physical objects such as chemical samples wherever possible. The concept should make it possible to secure https://doi. org/10. 26434/chemrxiv-2023-2dd4c ORCID: https://orcid. org/0000-0001-9513-2468 Content not peer-reviewed by ChemRxiv. License: CC BY 4. 0 physical research results, verify the gained results and increase the re-usability of the materials in addition to the re-use of already well-established data. To this aim, chemical compounds or more precisely samples of chemical compoundswhich are the starting point of analytical studies or the outcome of synthetic studies in chemistryshould be preserved and made available.
As a first step, the metadata of samples would need to be Findable, Accessible, Interoperable, and Reusable (= FAIR metadata for samples). In the second step, the samples would need to be registered and stored in a materials archive to be Accessible, and made available under suitable access policies and rules to be physically Reusable (= AR material for samples). This concept is further referred to as the FAIR-AR samples concept (Figure  1 ). The FAIR-AR sample concept in detail. Metadata for a sample can be seen as a virtual representation of the sample or in other words a digital object to which a globally unique and persistent identifier can be assigned. Metadata that are part of such a virtual sample representation https://doi. org/10. 26434/chemrxiv-2023-2dd4c ORCID: https://orcid. org/0000-0001-9513-2468 Content not peer-reviewed by ChemRxiv.
License: CC BY 4. 0 include information on the sample's provenance, components/content, and properties. Most of the metadata assigned to such a virtual sample representation are relevant for a FAIR data approach if it comes to a comprehensive description of analytical measurements of chemical samples. This is because chemistry samples can usually be described with standardized metadata, to which both, the physical sample and the gained data refer. Efficient concepts that intend to consider the FAIR deposition of research data and the deposit of FAIR-AR samples could, therefore, closely link research data deposition to sample deposition. A design to this is depicted in Figure  2 , showing a virtual sample representation described by rich standardized metadata, which is complemented by (1) information on further relevant data assigned to the sample (Figure  2 , left), and (2) information on available samples' location and its unique registry or reference number (Figure  2 , right).
License: CC BY 4. 0 The implementation in the form of infrastructure: The combined approach for research data and materials/samples makes sense due to the obvious methodical connection of both via the virtual sample representation (Figure  2 ) and the often observed issue that chemical samples are only efficiently reusable by others if they are described sufficiently -which also includes the full set of available research data or analytical data. Therefore, methodical aspects and scientific reasons can speak in favor of a combination of a sample archival system and a research data repository. Such a combination was realized with the Chemotion repository and the Molecule Archive at KIT (Karlsruhe Institute of Technology). The Chemotion repository is a repository for research data related to chemistry, particularly experimental chemistry.
Scientists can either upload data directly or interoperably transfer it from https://doi. org/10. 26434/chemrxiv-2023-2dd4c ORCID: https://orcid. org/0000-0001-9513-2468 Content not peer-reviewed by ChemRxiv. License: CC BY 4. 0 an electronic lab notebook to the repository without losing information. The data are peer-reviewed in combination with automatic checks and can then be disclosed with persistent identifiers. 14  Especially for organic chemistry, the repository offers additional functions for direct viewing and analysis of the stored data by discipline-specific research software  15  and thus enables an easy way of re-using the data. The repository follows an open-access policy and is part of the strategy of NFDI4Chem in Germany. The Molecule Archive is a facility of the Karlsruhe Institute of Technology (KIT) which enables the registration, validation, and collection of chemical substances.
The protocol supports multiple metadata formats, such as Dublin Core and DataCite which allow easy interoperability of the Chemotion repository with other systems such as the NFDI4Chem search service  30. Once the user finds the required sample information, all available metadata are directly accessible by their DOI identifier without further authentication and authorization processes. The metadata are accessible, even when the materials are no longer available, and allow the constant link to the analytical data. Interoperable sample metadata: Interoperable metadata are needed in particular to compare the materials with other available materials and to query additional databases. The sample metadata in the Chemotion repository include standardized molecule descriptors following domain-specific standards such as InChI, InChI keys and canonical SMILES strings wherever applicable.
org/0000-0001-9513-2468 Content not peer-reviewed by ChemRxiv. License: CC BY 4. 0 the respective organizational units of the partners involved, if a legal basis for material re-use is to be created comparable to the licensing of research data. Despite the obvious challenges of providing materials, the combination of FAIR data and FAIR-AR materials reveals enormous potential for more transparent and sustainable research. In particular, experimental disciplines in natural sciences may benefit from a concept introducing models for a systematic provision of samples. If samples are submitted before the publication of results, the benefits could be increased as the sample deposit can be directly linked to the publication. This strengthens the trust in the work and allows a direct link of a publication to all gained resultsthe research data and the available material.
Knowing that the implementation of the FAIR data principles has not been achieved among all scientists, the additional claim for FAIR-AR samples might seem to be challenging. Nevertheless, the establishment of a standardized process for FAIR-AR samples can be very easy once the concept of FAIR data is adopted: the infrastructure supporting the access and re-use of chemical samples already exists, and the effort for the single scientist as the material producer is low if the initial agreements between the partner sites and the Molecule Archive are done. While the deposition of FAIR data currently lacks incentives for the providing scientists, the provision of materials offers special scientific advantages such as publications with materials' re-users -this makes materials storage and provision an attractive aim that could foster the broad application of the FAIR-AR concept.