dataesr/scientific-paragraphs-categorization huggingface dataset:
Relaxing rule c !RIN ab. Let us now leverage our new method to relax rule c and extract RIN ab s that are allowed to span over two or more secondary structure elements instead of exactly two for RIN abc s (rule b still prevents single SSE RIN abc s ). From Dataset 2. 92, we extract 557 RIN ab s for a total of 7709 occurrences. Comparing the collection of RIN ab s with the collection of RIN abc s is not trivial. Indeed, amongst the 557 RIN ab s , 243 are isomorphic to a RIN abc. As a consequence, 88 RIN abc s are not matched by a corresponding RIN ab. They are instead found inside larger RIN ab s (i. e. the canonical graph of the RIN abc is a subgraph of the canonical graph of at least one RIN ab ), as well as their occurrences. To put it differently, those 88 RIN abc s are still captured but are always found inside "larger contexts" that could not be perceived before because of the limitation on the number of SSEs.
e. CaRNAval). This statement is to be considered in the context of graph matching and thus NP-hard problems in general. Just like CaRNAval, our method is exponential in the worst case. However our method is able to perform the same task significantly faster than CaRNAval (0. 7h instead of 330h). Moreover, our method can extract RIN classes that are beyond the limits of the method of CaRNAval (RIN a , RIN ab , RNA 3D modules cf. section 3. 5). Table  5  displays several runtimes from our method and the method of CaRNAval on the same machine (20 CPUs) on an indicative basis. In addition to the runtimes on Dataset 2. 92, it also displays runtimes on Dataset 3. 137 that we will present in the next section 3. 4 and on RNA 3D modules that we present in section 3. 5. Please note that, for our method, producing the set of RIN abc s is equivalent to producing the set of RIN ab s and applying a filter corresponding to rule c.
RIN a s from Dataset 3. 137 Now that we have accessed our method on Dataset 2. 92, let us move to Dataset 3. 137. We will only consider RIN a in this section. As we mentioned when we introduced the two datasets, Dataset 3. 137 contains significantly more RNA chains than Dataset 2. 92 (1152 vs 843, +37%), a consequence of the four years that separate the publication of the two versions of the non-redundant RNA database maintained on RNA3DHub they are respectively based on. Our methods finds 1875 distinct RIN a s for a total of 29344 occurrences in Dataset 3. 137. Compared to the results we obtained from Dataset 2. 92 (920 RIN a s for a total of 12239 occurrences), we find 104% more RIN a s and 140% more occurrences. Those numbers might appear surprising considering that there are only 37% more structures in Dataset 3.
137 than in Dataset 2. 92. However the structures in Dataset 3. 137 are larger in average (65 nt. vs 45nt. , +44%) so there are actually nearly twice as much nucleotides in total in Dataset 3. 137 than in Dataset 2. 9. Let us mention that all our structures of interest (A-minor type I/II, Ribozipper, GNRA, A-rich Loop) are present amongst those 1875 RIN a s and that their numbers of occurrences increased similarly to the average as displayed in Table  6  Figs 11 and 12 respectively display the distribution of RIN a s by the number of long range interactions they contain and by the number of SSEs they span over. Those distributions are similar to their counterparts from Dataset 2. 92 (Figs 9 and 10). 92. Yet, some aspects ot this discussion require our latest results on Dataset 3. 137 and it was thus moved to section 3.
4 instead. We previously mentioned that the relaxation of rule c (being exactly over 2 SSEs) allowed for larger structures to be extracted. Indeed, our method does not cap the size of the structures extracted outside of the limitations fixed by the rules. As such, relaxing rule c, that directly limits the size of the structures accepted, naturally results in larger structures being found. However, the three rules x,y and z (that we amalgamated into rule a) ensure that only densely connected structures are accepted. Typically, if we apply a filter enforcing those rules to the vast majority of the RNA 2D structure graphs in the dataset (which is the role of the second filtering function f 0 RIN a cf. section 2. 5. 1), it disconnects the vast majority of them as the filter "cuts" the stems, the backbone and the danglings if those do not contain any non-canonical interactions.
Yet, relaxing rule c still drastically raised the order (i. e. number of nodes, although the same can be said for the size) of the structures found: while the largest RIN abc found in Dataset 2. 92 contained 26 nodes, 64 RIN ab s were found with more than 26 nodes on the same dataset. Amongst those 64 RIN ab s , 4 have more than 100 nodes and the largest contains 293 nodes. The numbers are the same for class RIN a on Dataset 2. 92 but not on Dataset 3. 137. On Dataset 3. 137 there are 287 RIN a s over 26 nodes, 7 over 100 and the largest RIN a now contains 376 nodes. By comparison, the number of nodes of our target structures varies between 3 (Aminor Type I) and 13 (A-rich Loop). Those large RIN a s have very limited numbers of occurrences. The 287 RIN a s with more than 26 nodes totalize 1154 occurrences for an average of 4 occurrences per large RIN a whereas the average for the whole RIN a class is 15.
Moreover, both Dataset 2. 92 and 3. 137 include multiple ribosomal chains despite being non-redundant due to those chains corresponding to different ribosomal subunits and/or organisms. Those observations on the large RIN a s suggest that part of the collection of RIN a s (typically the RIN a s found in ribosomal chains) could be used as base for a study of conserved structural elements in ribosomes. However, such study falls out of the scope of this paper. On the contrary, and as we mentioned in the previous section, 97. 5% of the network of RIN a s in Dataset 3. 137 is connected in a single component. All those 288 large RIN a s are in this giant component and thus are linked to 97. 5% of the collection. As a consequence, in the perspective we adopt in this study, those large RIN a s constitute the tail-end of our collection of RIN a s rather than a separate group.
Applications to RNA 3D module-based RNA structure prediction As described earlier, we designed our method to be versatile by separating the rule system that define what structures should be extracted from the graph matching algorithms. We illustrated this versatility in section 3. 3 with 3 RIN classes (RIN abc ,RIN ab and RIN a ). In addition to RIN  classes, we applied our method on another class of structures linked to the RNA 3D structure prediction problem: the RNA 3D modules. RNA 3D modules are small RNA substructures involved in structural organization and ligand binding processes that can be leveraged in the prediction of a full 3D structure. The fragment-based method implemented by Parisien and Major in MC-Sym  [33]  constructs a full 3D structure from an augmented secondary structure by mapping the components of this secondary structure to a database of 3D structure fragments.
The prediction of 3D modules has been shown to improve this class of methods by providing more informative fragments, namely in RNA-MoIP  [7]. Further progress has since been made in this direction with recent improvements in RNA 3D modules identification in sequences  [34, 35]. The main limitation of this type of method remains the difficulty of assembling a strong dataset of modules. RNA modules are typically identified by searching RNA 3D structures for recurrent subgraphs, a task to which CaRNAval should have been able to contribute. Unfortunately, as of now, no fragment-based method has been able to integrate long-range modules into a 3D structure prediction pipeline. Moreover, RNA modules do not need to include long range interactions, and many of the well characterized modules are entirely local, namely the kink-turn and g-bulged modules, and the published version of CaRNAval cannot be applied to the discovery of common subgraphs without long range interactions as its execution time would explode.
However, the method presented in this paper does not have such limitation as demonstrated by the extraction of the RIN a class. We adapted the set of rules of the RIN classes presented in section 3. 2 to the problem, focusing on purely local RNA 3D modules, as a first approximation. The resulting set of rules would correspond, in our notation, to the RIN xyb ̄class, with b ̄being the constraint of having no longrange interactions. We relaxed rule z as it would have invalidated structures we are interested in such as the kink-turn that contains a bulge of backbone. Using this definition, we extracted 3387 structures with a total of 39513 occurrences from Dataset 3. 137. Amongst those 3387 are structures we aimed at extracting such as the kink-turn displayed in Fig  14,  which highlight the potential of our approach.
However, those results also highlight a challenging aspect of RNA 3D modules. Indeed, several non-isomorphic structures can be labelled as a single RNA 3D modules. Please note that it is not specific to our approach: the RNA 3D Motif Atlas, the reference dataset for local modules, typically has multiple entries that match the definition of the kink-turn loop. As our method relies on exact graph matching to compare structures, an extra step of post processing needs to be added to the pipeline to process the structures found into a collection of RNA 3D modules. However, the design of this additional step is not trivial and falls out of the scope of this paper and so does the improvement of the set of rules used. Even with the shortcomings we just mentioned, those results show that the modularity and the improved complexity allow for the tackling of this problem (cf.
To demonstrate the performance of our methods, we first successfully reproduce the results presented in CaRNAval and show that we can conduct the same analysis in a matter of hours instead of months. This achievement is an important milestone towards the release of a reliable online database of motifs for structural biologists studying the architecture and evolution of RNA structures. In particular, in light of the increasing number of new RNA 3D models deposited in the structural repositories (e. g. for the first time, more than 100 RNA-only structures have been released in the RCSB Protein Data Bank in 2020), our ability to quickly update our catalog of recurrent motifs (i. e. RIN) is key to maintaining this service up-to-date. Then, we proceed to another computational experiment to highlight new opportunities offered by our technology.
Network of RIN a s in Dataset 3. 137. The network of RIN a s in Dataset 3. 137 shows the same trend as in Dataset 2. 92. One massive mesh clusters 97. 5% of RIN a s (vs. 96. 5% in Dataset 2. 92). This component still aggregates the three meshes (Pseudoknot mesh, A-minor meshe and Trans W-C/H mesh) presented in CaRNAval. The remaining 44 RIN a s are distributed in 4 small components (sizes: 7, 3, 3 and 2) and 29 singletons. Ribosomes and very large RIN a s. Note: this discussion on the very large RIN a s extracted could arguably falls into section 3. 3 as it involves the RIN abc and RIN ab s classes and thus Dataset 2.
The colored parts correspond to the 3 largest RIN a s found in Dataset 3. 137: RIN a 1984 in red, RIN a #1983 in cyan and RIN a #1982 in lime green. The overlap of two RIN a s is colored in indigo. Additional information about those RIN a s and their overlap is provided in Table 7. https://doi. org/10. 1371/journal. pcbi. 1008990. g013Thecolors correspond to ones used in Fig 13. The values for the overlaps correspond to the number of nodes shared between the RIN a s. The RNA chains are denoted using the name of the file (ex:4Y4O) plus the name of the chain (ex:2A).
3. 4. 1 3. 4. 2 341342 Network of RIN a s in Dataset 3. 137. The network of RIN a s in Dataset 3. 137 shows the same trend as in Dataset 2. 92. One massive mesh clusters 97. 5% of RIN a s (vs. 96. 5% in Dataset 2. 92). This component still aggregates the three meshes (Pseudoknot mesh, A-minor meshe and Trans W-C/H mesh) presented in CaRNAval. The remaining 44 RIN a s are distributed in 4 small components (sizes: 7, 3, 3 and 2) and 29 singletons. Ribosomes and very large RIN a s. Note: this discussion on the very large RIN a s extracted could arguably falls into section 3. 3 as it involves the RIN abc and RIN ab s classes and thus Dataset 2.
Another notable study was recently conducted by Koukouvinis et al. [45]. In that study, the authors proposed a uniform, tabulated thermodynamic approach based on the NIST-database and VLE using PC-SAFT EoS for simulation of the ECN spray A  [45]. They reported that, by using the PC-SAFT EoS and log10(P)-T tabulation approach, the results were independent of table size when bilinear interpolation was adopted, even with linear mixing rules. As a result, to sum up, most of the available studies show that the tabulation should be highly accurate, with low-storage requirements, and a fast lookup algorithm.
The effects of atmospheric icing can be anticipated by Computational Fluid Dynamics (CFD). Past studies show that the convective heat transfer influences the ice accretion and is itself a function of surface roughness. Uncertainty quantification (UQ) could help quantify the impact of surface roughness parameters on the reliability of ice accretion prediction. This paper aims to quantify ice accretion uncertainties and identify the key surface roughness correction parameters contributing the most to the uncertainties in a Reynolds-Averaged Navier-Stokes (RANS) formulation. Ice accretion simulations over a rough flat plate using two thermal correction models are used to construct a RANS database. Non-Intrusive Polynomial Chaos Expansion (NIPCE) metamodels are developed to predict the convective heat transfer and icing characteristics of the RANS database.
The technical challenge in this case is double. First, an efficient way of generating a large sample set had to be used: making a sensitivity study requires a large database. The metamodel approach is well suited to this challenge, avoiding the computational effort that computational fluid dynamics (CFD) simulations would have required. Secondly, a reliable gauge of the importance of each roughness parameter was needed. The Sobol indexes give us the opportunity of directly reading on a bar chart the classification of the parameters depending on their level of influence on ice accretion. The objective is to quantify ice accretion uncertainty and identify the key roughness parameters contributing the most to the uncertainty of the thermal correction models for RANS equations. The importance of roughness parameters uncertainty compared to the freestream temperature uncertainty on the ice accretion shape is also investigated.
The epistemic uncertainty is illustrated by two different roughness characteristics (see Figure  1b ). Given their geometrical differences, these patterns lead to different convective heat transfer coefficients (see Figure  1c ), which in turn cause differences in the characteristics of ice accretions (see Figure  1d ). The database needed to obtain the metamodels relies on RANS simulations. The geometry used in the present work is a 2-m-long horizontal flat plate. This geometry is commonly used in CFD validation  [35] , and the mesh size is selected according to a grid study presented in  [19]. The 2D mesh of the domain consists of a structured domain of 137 by 97 cells with 114 points on the flat plate surface. The growth rate in the direction perpendicular to the plate is 1. 3, and the y + value is between 0.
45 and 1. 35. In a rough configuration, with this mesh, the difference with the experimental results of Hosni et al. [13]  for the Stanton number evaluation was less than 7% for both the HAX and 2PP thermal correction models. In addition, the flat plate configuration was extensively studied both numerically  [36]  and experimentally  [14]  in the context of aircraft ice accretion studies. The complete 2. 33 m by 1 m domain is sketched in Figure  2. The leading edge of the flat plate is located 0. 33 m downstream of the domain inlet. The database needed to obtain the metamodels relies on RANS simulations. The geometry used in the present work is a 2-m-long horizontal flat plate. This geometry is commonly used in CFD validation  [35] , and the mesh size is selected according to a grid study presented in  [19].
These values correspond to the roughness elements observed in different atmospheric conditions, and which led to different surface roughness types. The parameter k s is taken from Fortin  [17] , who reviewed the validation test cases used for the LEWICE software  [42]  and concluded on a range of variations for k s between 0. 309 mm and 1. 247 mm for airfoils with a chord varying between 53. 3 cm and 91. 4 cm. Both of these ranges for k and k s are maintained in the present study and lead to a k s /k ratio ranging between 0. 288 and 0. 753. The last input parameter, S corr , is taken from Aupoix and has a value between 1 and 2. 5  [18]. For all the parameters, a uniform PDF between the margins given above is assumed. Although other values are possible in practice, Table  1  gives the distributions used in this study.
With the range of distribution of each parameter defined, the input parameters can be sampled by Latin Hypercube Sampling (LHS), which is one of the methods commonly used to obtain random, but homogeneously spread, sample points  [43]. The HAX model has 190 sample simulations and the 2PP model 120, for their respective analyses, as shown in Figure  5. These sample size ranges are in agreement with the literature, which gives a sample size between 160 and 200 for a three-input model  [25]. For instance, Solaï et al. used a sample of 115 simulations for their analysis with four input parameters  [44]. The framework used for the UQ analysis, as well as for the subsequent s analysis, is the UQLab suite  [20]. To be able to generate a large number of resul to several thousand for the sensitivity analysis) without the need to run additio simulations, three metamodels constructed from the database are used, as show 2.
A metamodel can be represented as a function M estimating the output of i from the input vector X, whose components are Xj. The method retained to create the metamodels is the NIPCE  [21]. This exist modeling tool is frequently used in UQ, and the present work specifically applie UQ for roughness study in aircraft icing. This method is suitable for models with three input parameters. The general expression of a NIPCE metamodel is given tion (10): The sampling presented gives triplets (HAX model) or doublets (2PP model) of roughness parameters that are used as inputs for the RANS simulations. The database of convective heat transfer coefficients and ice accretion characteristics allows for the construction of the metamodels. = ( ) = × ( ) The framework used for the UQ analysis, as well as for the subsequent sensitivity analysis, is the UQLab suite  [20].
To be able to generate a large number of results (i. e. , up to several thousand for the sensitivity analysis) without the need to run additional RANS simulations, three metamodels constructed from the database are used, as shown in Table  2. A metamodel can be represented as a function M estimating the output of interest Y i from the input vector X, whose components are X j. The method retained to create the metamodels is the NIPCE  [21]. This existing metamodeling tool is frequently used in UQ, and the present work specifically applies it to the UQ for roughness study in aircraft icing. This method is suitable for models with two and three input parameters. The general expression of a NIPCE metamodel is given by Equation  (10) : Y i = M(X) = ∑ α y α × ψ α (X) (10) When generating a metamodel, the leave-one-out (LOO) error value allows an estimation of the metamodel quality.
The LOO value, based on the mean square error between the observed and predicted values, quantifies the ability of the metamodel to predict an output value close to what the RANS computation would have calculated  [21]. The computation of the LOO error relies on the cross-validation technique. For an N-sample design, the main aspect of the LOO calculation is to create N metamodels M i , which are constructed on the CFD database where the ith simulation was dropped, leaving N-1 sample points. The difference between the predicted value of the dropped sample by M i and its real CFD value allows the computation of the LOO error  [45]  as in Equation  (11). LOO = ∑ N i=1 Y (i) -M i X (i) 2 ∑ N i=1 Y (i) -Y 2 (11) In Equation  (11) , X (i) and Y (i) represent the input parameters of the ith CFD sample simulation and the corresponding CFD output, respectively.
Outputs of Interest PDFs The metamodel-generated databases allow the computation of the outputs of interest PDFs. Figure  8  shows all the ice accretion shapes obtained for (a) the HAX model and (b) the 2PP model. Additionally, for each of the models, one ice accretion chosen among the set is plotted alone for clear visualization. Each of the colored lines on the top charts of Figure  8  represents one specific accretion obtained with one particular sample run. For the single plotted accretions, the specific roughness parameters for the HAX model (a) are k = 1. 14 mm, ks/k = 0. 599 and Scorr = 1. 17, and for the 2PP model (b), k = 2. 07 mm and ks/k = 0. 385. The general trend illustrated in Figure  6 , with a step observed in the accretion, is visible for all the sample accretions. Figure  8  shows that varying the roughness parameters The trend shown in Figure  7 , with all the scatter points homogeneously packed close to the identity line, suggests a metamodel making accurate predictions.
Outputs of Interest PDFs The metamodel-generated databases allow the computation of the outputs of interest PDFs. Figure  8  shows all the ice accretion shapes obtained for (a) the HAX model and (b) the 2PP model. Additionally, for each of the models, one ice accretion chosen among the set is plotted alone for clear visualization. Each of the colored lines on the top charts of Figure  8  represents one specific accretion obtained with one particular sample run. For the single plotted accretions, the specific roughness parameters for the HAX model (a) are k = 1. 14 mm, k s /k = 0. 599 and S corr = 1. 17, and for the 2PP model (b), k = 2. 07 mm and k s /k = 0. 385. The general trend illustrated in Figure  6 , with a step observed in the accretion, is visible for all the sample accretions. Figure  8  shows that varying the roughness parameters changes the position of the step, which characterizes the ice extension along the surface.
Using all 190 (HAX model) or 120 (2PP model) data sets allows for the computing of the NIPCE-predicted PDF of the outputs of interest. The NIPCE allows for an increase in sample size, but the PDF is similar to the PDF predicted, with a sample size of 190 or 120. To verify this for the 2PP model, a sample of 10,000 doublets (k, ks/k) was generated with the Latin Hypercube Sampling, and the NIPCE metamodel applied to the sample. The PDFs of the 120 and 10,000 NIPCE predictions are shown together in Figure  9 , where the bars' heights are scaled by the total number of predictions for each case (i. e. , 120 and 10,000, respectively). Using all 190 (HAX model) or 120 (2PP model) data sets allows for the computing of the NIPCE-predicted PDF of the outputs of interest. The NIPCE allows for an increase in sample size, but the PDF is similar to the PDF predicted, with a sample size of 190 or 120.