Intelligent visualization-driven materials design via two-dimensional symbolic feature generation

Abstract

Machine learning (ML) for complex materials problems suffers from high-dimensional data, while traditional “black-box” dimensionality reduction techniques generally fail to balance predictive accuracy with visualization and interpretability. This work presents a novel method, named two-dimensional symbolic feature generation (2D-SFG), based on symbolic regression and genetic algorithms. This approach facilitates ML by providing quantifiable interpretability and enabling visualization-driven materials design. Evaluated across diverse classification and regression tasks in materials science, the proposed method demonstrates notable success in three critical aspects. First, it significantly improves predictive accuracy. Specifically, the classification accuracies for ferroelectric perovskites and high-entropy alloy (HEA) phases improved from 85.6% and 84.5% to 94.2% and 88.4%, respectively. Correspondingly, the prediction errors for shape memory alloys and copper alloys were reduced from 2.9 K, 5.9%, and 9.7% to 1.1 K, 3.7%, and 6.2%, respectively. Second, the method ensures interpretability by constructing explicit mathematical expressions that transform the original high-dimensional features into two new symbolic ones, avoiding opaque spatial transformations. Third, it enables model visualization through two-dimensional (2D) contour maps that relate the constructed features to the target properties, thereby offering intuitive insights into feature–property relationships. Leveraging these “design roadmaps”, a refractory HEA with a single-phase solid solution and a precipitation-strengthened copper alloy with an optimized property trade-off were successfully designed. The 2D symbolic feature generation framework thus addresses key limitations in accuracy, interpretability, and visualization within materials informatics, establishing a new paradigm for transparent and visual materials design.

Keywords

Machine learning / feature construction / interpretability / visualization-driven materials design

References

[1] Rahman A,Siddique A.Review: machine learning approaches for diverse alloy systems.J Mater Sci2025;60:12189-221

[2] Hu M,Knibbe R.Recent applications of machine learning in alloy design: a review.Mater Sci Eng R Rep2023;155:100746

[3] Kumar A,Yadav TP.Recent progresses on high entropy alloy development using machine learning: a review.Comput Mater Today2025;8:100038

[4] Cheng M,Okabe R.Artificial intelligence-driven approaches for materials design and discovery.Nat Mater2026;25:174-90

[5] Hart GLW,Toher C.Machine learning for alloys.Nat Rev Mater2021;6:730-55

[6] Merchant A,Schoenholz SS,Cheon G.Scaling deep learning for materials discovery.Nature2023;624:80-5 PMCID:PMC10700131

[7] Feng L,Lu Q.Accelerated development of high-strength and high-conductivity Cu-Cr-Ti alloys based on data-driven design and experimental validation.Mater Design2025;253:113948

[8] Yin J,Li X.A novel neural network-based alloy design strategy: gated recurrent unit machine learning modeling integrated with orthogonal experiment design and data augmentation.Acta Mater2023;243:118420

[9] Zhang H,Zhu S,Xie J.Machine learning assisted composition effective design for precipitation strengthened copper alloys.Acta Mater2021;215:117118

[10] Zhang H,He X.Dramatically enhanced combination of ultimate tensile strength and electric conductivity of alloys via machine learning screening.Acta Mater2020;200:803-10

[11] Li L,Yu H.A machine learning strategy to achieve strength-conductivity-ductility synergy of high‐performance Cu-Ni-Co-Si alloys via rolling and aging process.J Mater Sci Technol2026;267:184-97

[12] Sohail Y,Xue D.Machine-learning design of ductile FeNiCoAlTa alloys with high strength.Nature2025;643:119-24 PMCID:PMC12221976

[13] Vazquez G,Gurrola R.A deep neural network regressor for phase constitution estimation in the high entropy alloy system Al-Co-Cr-Fe-Mn-Nb-Ni.npj Comput Mater2023;9:1021

[14] Li H,Xu Q.High-strength medium-entropy alloy designed by precipitation-strengthening mechanism via machine learning.Mater Sci Eng A2023;882:145443

[15] Vela B,Acemi C,Arróyave R.Data-augmented modeling for yield strength of refractory high entropy alloys: a Bayesian approach.Acta Mater2023;261:119351

[16] Wang J,Kim HS.A neural network model for high entropy alloy design.npj Comput Mater2023;9:1010

[17] Yin J,Wu D.Interpretable predicting creep rupture life of superalloys: enhanced by domain-specific knowledge.Adv Sci2024;11:e2307982 PMCID:PMC10953578

[18] Lian L,Xiong Q.Intelligent design of crack-resistant nickel-based superalloys for additive manufacturing by machine learning and multilayer filtering strategy.Mater Today Commun2025;46:112387

[19] Zhuang X,Li W,Li L.Alloying effects and effective alloy design of high-Cr CoNi-based superalloys via a high-throughput experiments and machine learning framework.Acta Mater2023;243:118525

[20] Ma Q,Xin R.Thermodynamic calculation and machine learning aided composition design of new nickel-based superalloys.J Mater Res Technol2023;26:4168-78

[21] Yang F,Ru Y.Deep learning accelerates the development of Ni-based single crystal superalloys: a physical-constrained neural network for creep rupture life prediction.Mater Design2023;232:112174

[22] Xin Y,Xie A,Qiu G.Building an effective deep learning model for mechanical properties prediction of steel.Mater Lett2026;402:139262

[23] Kannan R.Accelerated alloy discovery using synthetic data generation and data mining.Scr Mater2023;228:115335

[24] Wei X,Jia Z,Xu W.On the use of transfer modeling to design new steels with excellent rotating bending fatigue resistance even in the case of very small calibration datasets.Acta Mater2022;235:118103

[25] Ren D,Wei X,Xu W.Building a quantitative composition-microstructure-property relationship of dual-phase steels via multimodal data mining.Acta Mater2023;252:118954

[26] Wu J,Hu M.Inverse design workflow discovers hole-transport materials tailored for perovskite solar cells.Science2024;386:1256-64

[27] Xu P,Lu T,Li M.Search for ABO₃ type ferroelectric perovskites with targeted multi-properties by machine learning strategies.J Chem Inf Model2022;62:5038-49

[28] He J,Liu C.Machine learning identified materials descriptors for ferroelectricity.Acta Mater2021;209:116815

[29] Min K.Accelerated discovery of potential ferroelectric perovskite via active learning.J Mater Chem C2020;8:7866-72

[30] Balachandran PV,Sehirlioglu A.Experimental search for high-temperature ferroelectric perovskites guided by two-step machine learning.Nat Commun2018;9:1668 PMCID:PMC5920103

[31] Liu Q,Kim SY.Beyond designer’s knowledge: generating materials design hypotheses via a large language model.Acta Mater2025;297:121307

[32] Tian S,Wang W.Steel design based on a large language model.Acta Mater2025;285:120663

[33] Wang P,Liao W.Generalizable descriptors for automatic titanium alloys design by learning from texts via large language model.Acta Mater2025;296:121275

[34] Polak MP.Extracting accurate materials data from research papers with conversational language models and prompt engineering.Nat Commun2024;15:1569 PMCID:PMC10882009

[35] Senior AW,Jumper J.Improved protein structure prediction using potentials from deep learning.Nature2020;577:706-10

[36] Tep P.High-fidelity grain growth modeling: leveraging deep learning for fast computations.Acta Mater2025;301:121486

[37] Yang H,Li C.Deep learning-based X-ray computed tomography image reconstruction and prediction of compression behavior of 3D printed lattice structures.Addit Manuf2022;54:102774

[38] Lundberg S. A unified approach to interpreting model predictions. arXiv 2017, arXiv:1705.07874. https://doi.org/10.48550/arXiv.1705.07874. (accessed 2026-05-21)

[39] Ribeiro MT,Guestrin C. “Why should I trust you?”: Explaining the predictions of any classifier. arXiv 2016, arXiv:1602.04938. https://doi.org/10.48550/arXiv.1602.04938. (accessed 2026-05-21)

[40] Alibagheri E,Khazaei M,Vaez Allaei SM.Remarkable optoelectronic characteristics of synthesizable square‐octagon haeckelite structures: machine learning materials discovery.Adv Funct Mater2024;34:2402390

[41] Xu H,Peng RL.Machine learning enabled the prediction of γ′-depleted depth during interdiffusion of bond-coated IN792 superalloy.Surf Coat Technol2025;513:132448

[42] Chen W,Bokas G,Jacques PJ.A map of single-phase high-entropy alloys.Nat Commun2023;14:2856 PMCID:PMC10199023

[43] Pearson K.LIII. On lines and planes of closest fit to systems of points in space.Lond Edinb Dublin Philos Mag J Sci1901;2:559-72

[44] Srinivasan S,Zhang R.Mapping chemical selection pathways for designing multicomponent alloys: an informatics framework for materials design.Sci Rep2015;5:17960 PMCID:PMC4683530

[45] Tenenbaum JB,Langford JC.A global geometric framework for nonlinear dimensionality reduction.Science2000;290:2319-23

[46] McInnes L,Melville J. UMAP: uniform manifold approximation and projection for dimension reduction. arXiv 2018, arXiv:1802.03426. https://doi.org/10.48550/arXiv.1802.03426. (accessed 2026-05-21)

[47] Zhao S,Wang J,Yuan R.Closed-loop inverse design of high entropy alloys using symbolic regression-oriented optimization.Mater Today2025;88:263-71

[48] Xue D,Yuan R.An informatics approach to transformation temperatures of NiTi-based shape memory alloys.Acta Mater2017;125:532-41

[49] Pokorny VJ,Rawls E.Impact of reduced-dimensionality independent components analysis on event-related potential measurements.Psychophysiology2023;60:e14223 PMCID:PMC10085851

[50] Miracle D.A critical review of high entropy alloys and related concepts.Acta Mater2017;122:448-511

[51] Zhang Y,Wang C.Phase prediction in high entropy alloys with a rational selection of materials descriptors and machine learning models.Acta Mater2020;185:528-39

[52] Machaka R,Raganya LM,Chikosha S.Machine learning-based prediction of phases in high-entropy alloys: a data article.Data Brief2021;38:107346 PMCID:PMC8477136

[53] Xue D,Hogden J,Xue D.Accelerated search for materials with targeted properties by adaptive design.Nat Commun2016;7:11241 PMCID:PMC4835535

[54] Schölkopf B,Müller K.Nonlinear component analysis as a kernel eigenvalue problem.Neural Comput1998;10:1299-319

[55] Roweis ST.Nonlinear dimensionality reduction by locally linear embedding.Science2000;290:2323-6

[56] Hefner R.Warren S. Torgerson, Theory and methods of scaling. New York: John Wiley and Sons, Inc., 1958. Pp. 460.Syst Res1959;4:245-7

[57] Tao Q,Li M.Machine learning for perovskite materials design and discovery.npj Comput Mater2021;7:495

[58] Zhang Y,Lin J,Liaw P.Solid‐solution phase formation rules for multi‐component alloys.Adv Eng Mater2008;10:534-8

[59] Zhang Y,Tang Z.Microstructures and properties of high-entropy alloys.Prog Mater Sci2014;61:1-93

[60] Zarinejad M.Dependence of transformation temperatures of NiTi‐based shape‐memory alloys on the number and concentration of valence electrons.Adv Funct Mater2008;18:2789-94

[61] Liu Y,Yu Q,Liu J.Significant reduction of phase-transition hysteresis for magnetocaloric (La_1-xCe_x)₂Fe₁₁Si₂H_y alloys by microstructural manipulation.Acta Mater2021;207:116687

[62] Zhou Y,Hong M.Orchestrating phase transition in GeTe thermoelectrics: an investigation into the role of electronegativity.Nano Energy2024;127:109723

[63] Zurcher R,Sachslehner F,Zehetbauer M.Dislocation resistivity in Cu: dependence of the deviations from Matthiessen’s rule on temperature, dislocation density and impurity content.J Phys Condens Matter1995;7:3515-28

[64] Žnidarič M.Modified Matthiessen’s rule: more scattering leads to less resistance.Phys Rev B2022;105:045140

[65] Ho CY,Wu KY.Electrical resistivity of ten selected binary alloy systems.J Phys Chem Ref Data1983;12:183-322