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    + + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +6731 +10.21105/joss.06731 + +nimCSO: A Nim package for Compositional Space +Optimization + + + +https://orcid.org/0000-0002-2266-0099 + +Krajewski +Adam M. + + +* + + +https://orcid.org/0000-0001-9427-4499 + +Debnath +Arindam + + + + +https://orcid.org/0000-0001-7256-2123 + +Reinhart +Wesley F. + + + + + +https://orcid.org/0000-0002-7022-3387 + +Beese +Allison M. + + + + +https://orcid.org/0000-0003-3346-3696 + +Liu +Zi-Kui + + + + + +Department of Materials Science and Engineering, The +Pennsylvania State University, United States of America +04p491231 + + + + +Institute for Computational and Data Sciences, The +Pennsylvania State University, United States of America +04p491231 + + + + +* E-mail: + + +14 +9 +2023 + +9 +103 +6731 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2024 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +Nim +materials science +data science +compositionally complex materials +high entropy alloys +high performance computing +metaprogramming + + + + + + Summary +

    nimCSO is a high-performance tool + implementing several methods for selecting components (data + dimensions) in compositional datasets, which optimize the data + availability and density for applications such as machine learning. + Making said choice is a combinatorically hard problem for complex + compositions existing in high-dimensional spaces due to the + interdependency of components being present. Such spaces are + encountered across many scientific disciplines (see + Statement of + Need), including materials science, where datasets on + Compositionally Complex Materials (CCMs) often span 20-45 chemical + elements, 5-10 processing types, and several temperature regimes, for + up to 60 total data dimensions. This challenge also exists in everyday + contexts, such as study of cooking ingredients + (Ahn et + al., 2011), which interact in various recipes, giving rise to + questions like “Given 100 spices at the supermarket, which 20, + 30, or 40 should I stock in my pantry to maximize the number of unique + dishes I can spice according to recipe?”. Critically, this is + not as simple as frequency-based selection because, e.g., removing + less common nutmeg and cinnamon from your shopping list will prevent + many recipes with the frequent vanilla, but won’t affect those using + black pepper + (Edmisten, + 2022).

    +

    At its core, nimCSO leverages the + metaprogramming ability of the Nim language + (Rumpf, + 2023) to optimize itself at compile time, both in terms of + speed and memory handling, to the specific problem statement and + dataset at hand based on a human-readable configuration file. As + demonstrated in the + Methods + and Performance section, nimCSO reaches + the physical limits of the hardware (L1 cache latency) and can + outperform an efficient native Python implementation over 100 times in + terms of speed and 50 times in terms of memory usage + (not counting the interpreter), while also + outperforming the NumPy + (Harris + et al., 2020) based implementation 37 and 17 times, + respectively, when checking a candidate solution.

    +

    nimCSO is designed to be both (1) a + user-ready tool, implementing two efficient brute-force approaches + (for handling up to 25 dimensions), a custom search algorithm (for up + to 40 dimensions), and a genetic algorithm (for any dimensionality), + and (2) a scaffold for building even more elaborate methods in the + future, including heuristics going beyond data availability. All + configuration is done with a simple human-readable + YAML file and plain text data files, making it + easy to modify the search method and its parameters with no knowledge + of programming and only basic command line skills.

    +
    + + Statement of Need +

    nimCSO is an interdisciplinary tool + applicable to any field where data is composed of a large number of + independent components and their interaction is of interest in a + modeling effort, ranging from economics where factor selection affects + performance of analytical + (Fan et + al., 2013) and ML + (Peng + et al., 2021) models, through medicine where drug interactions + can have a significant impact on the treatment + (Maher + et al., 2014) (an escalating problem + (Guthrie + et al., 2015)) and understanding of microbial interactions can + help fight gastrointenstant problems + (Berg + et al., 2022; + Leeuwen + et al., 2023), to materials science, where the composition and + processing history are critical to resulting properties. The latter + has been the root motivation for the development of + nimCSO within the + ULTERA + Project (ultera.org) carried under the + US + DOE ARPA-E ULTIMATE program, which aims to develop a new + generation of ultra-high temperature materials for aerospace + applications, through generative machine learning models + (Debnath + et al., 2021) driving thermodynamic modeling, alloy design, and + manufacturing + (Li et + al., 2024).

    +

    One of the most promising materials for such applications are the + aforementioned CCMs and their metal-focused subset of Refractory High + Entropy Alloys (RHEAs) + (Senkov + et al., 2018), which have rapidly grown since first proposed by + Cantor et al. + (2004) + and Yeh et al. + (2004). + Contrary to most of the traditional alloys, they contain many chemical + elements (typically 4-9) in similar proportions in the hope of + thermodynamically stabilizing the material by increasing its + configurational entropy ( + + ΔSconf=ΣiNxilnxi + for ideal mixing of + + N + elements with fractions + + xi), + which encourages sampling from a large palette of chemical elements. + At the time of writing, the ULTERA Database is the largest collection + of HEA data, containing over 7,000 points manually extracted from 560 + publications. It covers 37 chemical elements resulting in extremely + large compositional spaces + (Krajewski + et al., 2024); thus, it becomes critical to answer questions + like “Which combination of how many elements will unlock the + most expansive and simultaneously dense dataset?” which has + + + 2371 + or 137 billion possible solutions.

    +

    Another significant example of intended use is to perform similar + optimizations over large (many millions) datasets of quantum mechanics + calculations spanning 93 chemical elements and accessible through the + OPTIMADE API + (Evans + et al., 2024).

    +
    + + Methods and Performance + + Overview +

    As shown in Figure + [fig:main], + nimCSO can be used as a user-tool based on + human-readable configuration and a data file containing data + “elements” which can be any strings representing problem-specific + names of, e.g., spices, market stocks, drug names, or chemical + systems. A single command is used to recompile + (nim c -f) and run + (-r) problem + (-d:configPath=config.yaml) with + nimCSO (src/nimcso) + using one of several methods. Advanced users can also quickly + customize the provided methods with brief scripts using the + nimCSO as a data-centric library.

    + +

    Schematic of core nimCSO data flow with a description + of key methods. Metaprogramming is used to compile the software + optimized to the human-readable data and configuration files at + hand.

    + +
    +

    Internally, nimCSO is built around storing + the data and solutions in one of two ways. The first is as bits + inside an integer (uint64), which allows for + the highest speed and lowest memory consumption possible but is + limited to 64 dimensions and does not allow for easy extension to + other use cases; thus, as of publication, it is used only in a + particular bruteForceInt routine. The second + one, used in bruteForce, + algorithmSearch, and + geneticSearch, implements a custom easily + extensible ElSolution type containing + heuristic value and BitArray payload, which + is defined at compile time based on the configuration file to + minimize necessary overheads. As shown in Table 1, both encodings + outperform typical native Python and NumPy implementations, + typically used in this application by the scientific community, + which can be found in benchmarks + directory.

    + + +

    Benchmarks of (1) the average time to evaluate how many + datapoints would be lost if 5 selected components were removed + from a dataset with 2,150 data points spanning 37 components, + averaged over 10,000 runs, and (2) the size of the data + structure representing the dataset. Values were obtained by + running scripts in benchmarks directory + on Apple M2 Max CPU. Pre-processing time is excluded, as it has + negligible impact on larger, realistic problems.

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    ToolObjectTime per DatasetTime per Entry + (Relative)Database Size + (Relative)
    Python3.11set107.5 µs50.0 ns (x1)871.5 kB (x1)
    NumPy1.26array36.4 µs16.9 ns (x3.0)79.7 kB (x10.9)
    nimCSO0.6BitArray6.9 µs3.2 ns (x15.6)50.4 kB (x17.3)
    nimCSO0.6uint640.98 µs0.456 ns (x110)16.8 kB (x52)
    +
    +
    + + Brute-Force Search +

    The brute-force search is a naïve method of evaluating all + possibilities; however, its near-zero overhead can make it the most + efficient for small problems. In this implementation, all entries in + the power set of + + N + considered elements are represented as a range of integers from + + + 0 + to + + 2N1, + and used to initialize + uint64/BitArrays on + the fly. To minimize the memory footprint of solutions, the + algorithm only keeps track of the best solution for a given number + of elements present in the solution. Current implementations are + limited to 64 elements, as it is not feasible beyond approximately + 30 elements; however, the one based on + BitArray could be easily extended if + needed.

    +
    + + Algorithm-Based Search +

    The algorithm implemented in the + algorithmSearch routine, targeting high + dimensional problems (20-50), iteratively expands and evaluates + candidates from a priority queue (implemented through an efficient + binary heap + (Williams, + 1964)) while leveraging the fact that the number of + data points lost when removing elements A and + B from the dataset has to be at least as + large as when removing either A or + B alone to delay exploration of + candidates until they can contribute to the + solution. Furthermore, to (1) avoid revisiting the same candidate + without keeping track of visited states and to (2) avoid exhaustive + search by inhibiting the exploration of unlikely candidates, the + algorithm assumes that while searching for a given + order of solution, elements present in already expanded solutions + will not improve those not yet expanded. This effectively prunes + candidate branches requiring two or more levels of backtracking, + making the algorithm computationally feasible in higher-dimensional + problems at the cost of potentially suboptimal results. In the + authors’ tests over HEAs, this algorithm has generated correct + results (agreeing with an exhaustive + bruteForce exploration), except for + occasional differences in the last explored solution, where almost + all data was discarded, and a few remaining points were highly + independent.

    +
    + + Genetic Search +

    Beyond 50 components, the + algorithm-based + method will likely run out of memory on most personal systems. The + geneticSearch routine resolves this issue + through an evolution strategy to iteratively improve solutions based + on custom mutate and + crossover procedures. Both are of uniform + type + (Goldberg, + 1989) with additional constraint of Hamming weight + (Knuth, + 2009) preservation in order to preserve number of considered + elements in parents and offspring. In mutate + this is achieved by using purely random bit swapping, rather than + more common flipping, as demonstrated in the Figure + [fig:mutate].

    + +

    Schematic of mutate procedure + where bits are swapping randomly, so that (1) bit can swap itself, + (2) bits can swap causing a flip, or (3) bits can swap with no + effect.

    + +
    +

    Meanwhile, in crossover, this constraint + is satisfied by passing overlapping bits directly, while + non-overlapping bits are shuffled and distributed at positions + present in one of the parents, as shown in Figure + [fig:crossover].

    + +

    Schematic of uniform crossover + procedure preserving Hamming weight implemented in + nimCSO. +

    + +
    +

    The above are applied iteratively, with best solutions carried to + next generation, until the solution converges or the maximum number + of iterations is reached. Unlike the other methods, the present + method is not limited by the number of components and lets user + control both time and memory requirements, either to make big + problems feasible or to get a good-enough solution quickly in small + problems. However, it comes with no optimality guarantees.

    +
    +
    + + Use Examples +

    The tool comes with two pre-defined example problems to demonstrate + its use. The first one is defined in the default + config.yaml file and goes through the complete + dataset of 2,150 data points spanning 37 components in + dataList.txt based on the ULTERA Dataset + (Debnath + et al., 2021). It is intended to showcase + algorithmSearch/-as and + geneticSearch/-gs + methods, as brute-forcing would take around one day. The second one is + defined in config_rhea.yaml and uses the same + dataset but a limited scope of components critical to RHEAs + (Senkov + et al., 2018) and is intended to showcase + bruteForce/-bf and + bruteForceInt/-bfi + methods. With four simple commands (see Table 2), the user can compare + the methods’ performance and the solutions’ quality.

    + + +

    Four example tasks alongside typical CPU time and memory usage + on Apple M2 Max.

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    Task Definition + (nim c -r -f -d:release ...)Time (s)Memory (MB)
    -d:configPath=config.yaml src/nimcso -as308s488 MB
    -d:configPath=config.yaml src/nimcso -gs5.10s3.2 MB
    -d:configPath=config_rhea.yaml src/nimcso -as0.073s2.2 MB
    -d:configPath=config_rhea.yaml src/nimcso -gs0.426s2.1 MB
    -d:configPath=config_rhea.yaml src/nimcso -bf3.726s2.0 MB
    -d:configPath=config_rhea.yaml src/nimcso -bfi0.495s2.0 MB
    +
    +

    In case of issues, the help message can be accessed by running the + tool with -h flag or by refering to + documentation at + amkrajewski.github.io/nimCSO.

    +
    + + Contributions +

    A.M.K. was reponsible for conceptualization, methodology, software, + testing and validation, writing of manuscript, and visualization; A.D. + was responsible for testing software and results in training machine + learning models; A.M.B., W.F.R., Z-K.L. were responsible for funding + acquisition, review, and editing. Z-K.L. was also supervising the + work.

    +
    + + Acknowledgements +

    This work has been funded through grants: DOE-ARPA-E + DE-AR0001435, NSF-POSE FAIN-2229690, and ONR + N00014-23-2721. We would also like to acknowledge Dr. Jonathan + Siegel at Texas A&M University for several valuable discussions + and feedback on the project, and Dr. Bradley Dice for his code + contributions to the project during the review process, such as + optimization of the native Python implementation.

    +
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