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DOE OSTI2022

Thermodynamically Driven Synthetic Optimization for Cation-Disordered Rock Salt Cathodes

Relating the synthesis conditions of materials to their functional performance has long been an experience-based trial-and-error process. However, this methodology is not always efficient in identifying an appropriate protocol and can lead to overlooked opportunities for the performance optimization of materials through simple modifications of the synthesis process. In this work, we systematically track the structural evolution in the synthesis of a representative disordered rocksalt (a promising next-generation Li-ion cathode material) at the scale of both the long-range crystal structure and the short-range atomic structure using various in situ and ex situ techniques, including transmission electron microscopy, X-ray diffraction, and pair distribution function analysis. Further, an optimization strategy is proposed for the synthesis protocol, leading to a remarkably enhanced capacity (specific energy) of 313 mAh g –1 (987 Wh kg –1 ) at a low rate (20 mA g –1 ), with a capacity of more than 140 mAh g –1 retained even at a very high cycling rate of 2,000 mA g –1 . This strategy is further rationalized using ab initio calculations, and important opportunities for synthetic optimization demonstrated in this study are highlighted.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Spatiotemporal Studies of Soluble Inorganic Nanostructures with X‐rays and Neutrons

This Review addresses the use of X-ray and neutron scattering as well as X-ray absorption to describe how inorganic nanostructured materials assemble, evolve, and function in solution. We first provide an overview of techniques and instrumentation (both large user facilities and benchtop). We review recent studies of soluble inorganic nanostructure assembly, covering the disciplines of materials synthesis, processes in nature, nuclear materials, and the widely applicable fundamental processes of hydrophobic interactions and ion pairing. Reviewed studies cover size regimes and length scales ranging from sub-Ångström (coordination chemistry and ion pairing) to several nanometers (molecular clusters, i.e. polyoxometalates, polyoxocations, and metal-organic polyhedra), to the mesoscale (supramolecular assembly processes). Reviewed studies predominantly exploit 1) SAXS/WAXS/SANS (small- and wide-angle X-ray or neutron scattering), 2) PDF (pair-distribution function analysis of X-ray total scattering), and 3) XANES and EXAFS (X-ray absorption near-edge structure and extended X-ray absorption fine structure, respectively). While the scattering techniques provide structural information, X-ray absorption yields the oxidation state in addition to the local coordination. Our goal for this Review is to provide information and inspiration for the inorganic/materials science communities that may benefit from elucidating the role of solution speciation in natural and synthetic processes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Generalized parton distributions through universal moment parameterization: zero skewness case

We present a global analysis program for the generalized parton distributions (GPDs) based on conformal moment expansion. We apply the strategy of universal moment parameterization to fit both the collinear parton distribution functions (PDFs) from phenomenology and generalized form factors from lattice calculations, and show that the parameterization is flexible enough to accommodate these constraints. In addition, we can also fit direct lattice calculations of GPDs from large-momentum effective theory. In this work we focus on the analysis of t-dependent PDFs which correspond to GPDs in the ξ → 0 limit. The strategy also applies to the ξ ≠ 0 region with extra parameters, and therefore can be fitted to experimental observables in the future. With a demonstrative example of fitted GPDs, we exhibit the quark transverse angular momentum densities of the proton as well as the impact parameter space distributions of quarks in both unpolarized and transversely polarized protons.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Resilience of the Aurivillius structure upon La and Cr doping in a Bi 5 Ti 3 FeO 15 multiferroic

Here, combining the experimental techniques of high-resolution X-ray diffraction, magnetometry, specific heat measurement, and X-ray photoelectron, Raman and dielectric spectroscopy techniques, we have studied the influence of La and Cr doping on the crystal structure and magnetism of the room temperature Aurivillius multiferroic Bi 5 Ti 3 FeO 15 by investigating the physical properties of (Bi 4 La)Ti 3 FeO 15 and Bi 5 Ti 3 (Fe 0.5 Cr 0.5 )O 15 . The parent (Bi 5 Ti 3 FeO 15 ) and the doped ((Bi 4 La)Ti 3 FeO 15 and Bi 5 Ti 3 (Fe 0.5 Cr 0.5 )O 15 ) compounds crystallize in the A2 1 am space group, which is confirmed through our analysis of high-resolution synchrotron X-ray diffraction data obtained on phase-pure polycrystalline powders. We determined the oxidation states of the metal atoms in the studied compounds as Fe 3+ , Ti 4+ , Cr 3+ , and La 3+ through the analysis of X-ray photoelectron spectroscopy data. The magnetic susceptibilities of the three compounds are marked by the absence of a long-range ordered ground state, but dominated by superparamagnetic clusters with dominant antiferromagnetic interactions. This signature of short-range magnetism is also seen in specific heat as a low temperature enhancement which is suppressed upon the application of external magnetic fields up to 8 T. Our dielectric spectroscopy experiments showed that the three studied compounds display similar features in the dielectric constant measured as a function of frequency. However, upon doping La at the Bi site, the width of the ferroelectric hysteresis loop increases for (Bi 4 La)Ti 3 FeO 15 compared to that of the parent compound Bi 5 Ti 3 FeO 15 , and with Cr doping, Bi 5 Ti 3 (Fe 0.5 Cr 0.5 )O 15 becomes a leaky dielectric. The resilience of the Aurivillius crystal structure towards doping of La at the Bi site and Cr at the Fe site is clearly seen in the bulk properties of magnetic susceptibility, specific heat and the average crystal structure. The relevance of changes in the local structure is evident from our Raman spectroscopy and X-ray pair distribution function studies.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

Orthorhombic Structures as Inorganic Halide Perovskite Models for High-Throughput DFT Investigations

Although high-symmetry Pm-3m space group cubic models are computationally efficient for high-throughput density functional theory (DFT) calculations of inorganic ternary (ABX 3 ) halide perovskites (HPs), they frequently predict band gaps (E g ) that disagree with experiment. Conversely, while low-symmetry cubic polymorphous networks (PN) comprised of 160 to 320 atoms incur significantly greater computational cost, they predict E g ’s that are more closely aligned with experiment. Here, in this study, we compare the DFT total energies and E g ’s predicted by four high-symmetry structure models (Pnma orthorhombic, R3m trigonal, P4/mbm tetragonal, and Pm-3m cubic) to cubic PNs for 5 experimentally characterized ternary HPs and find that the orthorhombic model computes Eg’s with the smallest MAD of 0.23 eV relative to the PNs. Pair distribution functions and DFT-computed total energies show that octahedral tilting, which is present in the 20-atom orthorhombic and 160-atom cubic PN models but not in the 5-atom cubic models, stabilizes all 5 compositions in our benchmarking set. We also find that imposing PN constraints when generating and optimizing these orthorhombic structures by fixing the unit cell lattice vectors and displacing the atoms prior to ionic relaxation with DFT, which we call the orthorhombic surrogate model (OSM), lowers the MAD of E g predictions to 0.09 eV. Our OSM predicts the PN band gaps of an additional 95 theoretical inorganic ternary HPs with MAD of 0.08 eV, supporting its usage in high-throughput DFT investigations to closely estimate PN band gaps with much less computational expense.

14 SOLAR ENERGY↗

DOE OSTI2024

Robust Machine Learning Inference from X-ray Absorption Near Edge Spectra through Featurization

X-ray absorption spectroscopy (XAS) is a commonly employed technique for characterizing functional materials. In particular, X-ray absorption near edge spectra (XANES) encode local coordination and electronic information, and machine learning approaches to extract this information are of significant interest. To date, most ML approaches for XANES have primarily focused on using the raw spectral intensities as input, overlooking the potential benefits of incorporating spectral transformations and dimensionality reduction techniques into ML predictions. Here, in this work, we focused on systematically comparing the impact of different featurization methods on the performance of ML models for XAS analysis. We evaluated the classification and regression capabilities of these models on computed data sets and validated their performance on previously unseen experimental data sets. Our analysis revealed an intriguing discovery: the cumulative distribution function feature achieves both high prediction accuracy and exceptional transferability. This remarkably robust performance can be attributed to its tolerance to horizontal shifts in the spectra, which is crucial when validating models using experimental data. While this work exclusively focuses on XANES analysis, we anticipate that the methodology presented here will hold promise as a versatile asset to the broader spectroscopy community.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Extraction of the neutron F 2 structure function from inclusive proton and deuteron deep-inelastic scattering data

The available world deep-inelastic scattering (DIS) data on proton and deuteron structure functions F 2 p , F 2 d , and their ratios are leveraged to extract the free neutron F 2 n structure function, the F 2 n / F 2 p ratio, and associated uncertainties using the latest nuclear effect calculations in the deuteron. Special attention is devoted to the normalization of the proton and deuteron experimental datasets and to the treatment of correlated systematic errors, as well as the quantification of procedural and theoretical uncertainties. The extracted F 2 n dataset is utilized to evaluate the Q 2 dependence of the Gottfried sum rule and the nonsinglet F 2 p − F 2 n moments. To facilitate replication of our study, as well as for general applications, we provide a comprehensive DIS database including all recent Jefferson Lab 6 GeV measurements, the extracted F n 2 , a modified CTEQ-JLab global parton distribution function fit named CJ15nlo_mod, and grids with calculated proton, neutron, and deuteron DIS structure functions. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

Measurement of beauty production via non-prompt charm hadrons in p–Pb collisions at $\sqrt{s_{\textrm{NN}}}$ = 5.02 TeV

The production cross sections of D 0 , D + , and Λ$^{+}_{c}$ hadrons originating from beauty-hadron decays (i.e. non-prompt) were measured for the first time at midrapidity in proton–lead (p–Pb) collisions at the center-of-mass energy per nucleon pair of $\sqrt{s_{\textrm{NN}}}$ = 5.02 TeV. Nuclear modification factors (R pPb ) of non-prompt D 0 , D + , and Λ$^{+}_{c}$ are calculated as a function of the transverse momentum (p T ) to investigate the modification of the momentum spectra measured in p–Pb collisions with respect to those measured in proton–proton (pp) collisions at the same energy. The R pPb measurements are compatible with unity and with the measurements in the prompt charm sector, and do not show a significant p T dependence. The p T -integrated cross sections and p T -integrated R pPb of non-prompt D 0 and D + mesons are also computed by extrapolating the visible cross sections down to p T = 0. The non-prompt D-meson R pPb integrated over p T is compatible with unity and with model calculations implementing modification of the parton distribution functions of nucleons bound in nuclei with respect to free nucleons. The non-prompt Λ$^{+}_{c}$/D 0 and D + /D 0 production ratios are computed to investigate hadronisation mechanisms of beauty quarks into mesons and baryons. The measured ratios as a function of p T display a similar trend to that measured for charm hadrons in the same collision system.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

The Information Length Concept Applied to Plasma Turbulence

A methodology to study statistical properties of anomalous transport in fusion plasma is investigated. Three time traces generated by the full-f gyrokinetic code GKNET are analyzed for this purpose. The time traces consist of heat flux as a function of the radial position, which is studied in a novel manner using statistical methods. The simulation data exhibit transport processes with both medium and long correlation length along the radius. A typical example of a phenomenon with long correlation length is avalanches. In order to investigate the evolution of the turbulent state, two basic configurations are studied, one flux-driven and one gradient-driven with decaying turbulence. The information length concept in tandem with Boltzmann–Gibbs and Tsallis entropy is used in the investigation. It is found that the dynamical states in both flux-driven and gradient-driven cases are surprisingly similar, but the Tsallis entropy reveals differences between them. This indicates that the types of probability distribution function are nevertheless quite different since the higher moments are significantly different.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Optimizing iodine adsorption in functionalized metal-organic frameworks via an unprecedented positional isomerism strategy

Porous metal–organic frameworks (MOFs) have emerged as highly promising adsorbents for capturing radioiodine, a predominant fission product released during nuclear fuel reprocessing. However, systematic investigations into the correlation between MOF structure and iodine uptake capacity remain scarce. Here, we present a novel approach to enhance the iodine adsorption capacity of MOFs by optimizing linker functionalization. Using ligand-functionalized thorium-based MOFs as a structural platform, we demonstrate that ortho-amino-substitution near the node of the dicarboxylate linker significantly increases iodine adsorption capacity compared to meta-amino-substitution, where the amino groups are directed away from the node. Specifically, ortho-substituted Th-UiO-68-3,3”-(NH 2 ) 2 exhibits higher iodine uptake capacities than the meta-substituted Th-UiO-68-2,2”-(NH 2 ) 2 via both vapor diffusion-based (2.042 vs. 1.087 g/g) and solution-based (0.841 vs. 0.784 g/g) processes. Notably, the I 2 vapor adsorption capacity (2.042 g/g) of Th-UiO-68-3,3”-(NH 2 ) 2 represents the second highest among all reported Th-MOFs. Pair distribution function (PDF) studies reveal that the superior iodine uptake performance of ortho-functionalized MOFs can be attributed to the reduced steric hindrance of the amino groups compared with the meta-substituted variants. Finally, this research highlights how positional isomerism and its subtle alterations can significantly influence host–guest interactions, extending beyond simple structural considerations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

A two-dimensional numerical study of ion-acoustic turbulence

We investigate the linear and nonlinear evolution of the current-driven ion-acoustic instability in a collisionless plasma via two-dimensional (2-D) Vlasov–Poisson numerical simulations. We initialise the system in a stable state and gradually drive it towards instability with an imposed, weak external electric field, thus avoiding physically unrealisable super-critical initial conditions. A comprehensive analysis of the nonlinear evolution of ion-acoustic turbulence (IAT) is presented, including the detailed characteristics of the evolution of the particles’ distribution functions, (2-D) wave spectrum and the resulting anomalous resistivity. Our findings reveal the dominance of 2-D quasi-linear effects around saturation, with nonlinear effects, such as particle trapping and nonlinear frequency shifts, becoming pronounced during the later stages of the system's nonlinear evolution. Remarkably, the Kadomtsev–Petviashvili (KP) spectrum is observed immediately after the saturation of the instability. Another crucial and noteworthy result is that no steady saturated nonlinear state is ever reached: strong ion heating suppresses the instability, which implies that the anomalous resistivity associated with IAT is transient and short-lived, challenging earlier theoretical results. Towards the conclusion of the simulation, electron-acoustic waves are triggered by the formation of a double layer and strong modifications to the particle distribution induced by IAT.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

Characterization of thermal transport and evolution of Au plasma in ICF experiments by Thomson scattering

Here, this paper demonstrates the capability of optical Thomson scattering (OTS) to measure thermal transport, and provides support to radiation hydrodynamic and kinetic simulations of electron thermal transport and plasma evolution. OTS theory and plasma simulations are applied to the interpretation of experimental measurements of laser-produced plasma from spherical gold targets on the OMEGA laser facility. The dynamical form factor, S($\vec{k}$, ω), of electron density fluctuations that is used in the fitting of Thomson scattering spectra includes ion–ion collisions and effects of non-Maxwellian distribution functions. OTS measurements and their interpretation are consistent with the nonlocal transport model in radiation hydrodynamic simulations as well as with kinetic simulations in the second half of the probe pulse duration. In particular, the reversal of heat transport during cooling is observed to be consistent with simulations, while some discrepancies are noted during the initial heating of the Au targets.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Measurement of the double-differential inclusive jet cross section in proton-proton collisions at $\sqrt{s}$ = 5.02 TeV

The inclusive jet cross section is measured as a function of jet transverse momentum pT and rapidity y. The measurement is performed using proton-proton collision data at $\sqrt{s}$ = 5.02 TeV, recorded by the CMS experiment at the LHC, corresponding to an integrated luminosity of 27.4 pb −1 . The jets are reconstructed with the anti-k T algorithm using a distance parameter of R = 0.4, within the rapidity interval |y| < 2, and across the kinematic range 0.06 < p T < 1 TeV. The jet cross section is unfolded from detector to particle level using the determined jet response and resolution. The results are compared to predictions of perturbative quantum chromodynamics, calculated at both next-to-leading order and next-to-next-to-leading order. The predictions are corrected for nonperturbative effects, and presented for a variety of parton distribution functions and choices of the renormalization/factorization scales and the strong coupling α S .

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

Synthesis of Free-Standing Pd-Ni-P Metallic Glass Nanoparticles with Durable Medium-Range Ordered Structure for Enhanced Electrocatalytic Properties

Topologically disordered metallic glass nanoparticles (MGNPs) with highly active and tailorable surface chemistries have immense potential for functional uses. The synthesis of free-standing MGNPs is crucial and intensively pursued because their activity strongly depends on their exposed surfaces. Herein, a novel laser-evaporated inert-gas condensation method is designed and successfully developed for synthesizing free-standing MGNPs without substrates or capping agents, which is implemented via pulse laser-induced atomic vapor deposition under an inert helium atmosphere. In this way, the metallic atoms vaporized from the targets collide with helium atoms and then condense into short-range-order (SRO) clusters, which mutually assemble to form the MGNPs. Using this method, free-standing Pd 40 Ni 40 P 20 MGNPs with a spherical morphology are synthesized, which demonstrates satisfactory electrocatalytic activity and durability in oxygen reduction reactions. Moreover, local structure investigations using synchrotron pair distribution function techniques reveal the transformation of SRO cluster connection motifs of the MGNPs from face-sharing to edge-sharing modes during cyclic voltammetry cycles, which enhances the electrochemical stability by blocking crystallization. Further, this approach provides a general strategy for preparing free-standing MGNPs with high surface activities, which may have widespread functional applications

36 MATERIALS SCIENCE↗

DOE OSTI2025

Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF

We present the first lattice quantum chromodynamics (QCD) calculation of the pion valence-quark transverse-momentum-dependent parton distribution function (TMDPDF) within the framework of large-momentum effective theory (LaMET). Using correlators fixed in the Coulomb gauge (CG), we computed the quasi-TMD beam function for a pion with a mass of 300 MeV, a fine lattice spacing of 𝑎 =0.06 fm, and multiple large momenta up to 3 GeV. The intrinsic soft functions in the CG approach are extracted from form factors with large momentum transfer, and as a byproduct, we also obtain the corresponding Collins-Soper (CS) kernel. Our determinations of both the soft function and the CS kernel agree with perturbation theory at small transverse separations (𝑏 ⊥ ) between the quarks. At larger 𝑏 ⊥ , the CS kernel remains consistent with recent results obtained using both CG and gauge-invariant TMD correlators in the literature. By combining next-to-leading logarithmic factorization of the quasi-TMD beam function and the soft function, we obtain an 𝑥-dependent pion valence-quark TMDPDF for transverse separations 𝑏 ⊥ ≳1 fm. Interestingly, we find that the 𝑏 ⊥ dependence of the phenomenological parametrizations of TMDPDF for moderate values of 𝑥 are in reasonable agreement with our QCD determinations. In addition, we present results for the transverse-momentum-dependent wave function for a heavier pion with 670 MeV mass.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

Quantifying a partial polyamorphic transition in a cerium-based metallic glass during cooling

Cerium-based metallic glasses are prototype polyamorphous systems with pressure-induced polyamorphic transitions extensively reported. Cooling typically has a similar effect on materials as compression with regard to reducing volume. However, previous studies show dramatically different behavior of Ce-based metallic glasses between cooling and compression, whose origin remains unclear. Here, using in situ low-temperature synchrotron high-energy x-ray diffraction, the structural evolution of a Ce68Al10Cu20Co2 metallic glass is accurately determined and analyzed by a structure factor and a reduced pair distribution function (PDF) during cooling from 298 to 83 K. An unusually large linear thermal expansion coefficient is revealed, which is associated with both continuous but inconsistent structural changes between the two subpeaks of the first atomic shell in terms of average bond lengths and coordination numbers. These phenomena are suggested to be attributed to a gradual 4f electron delocalization of only a minimal amount (∼2.6% at 83 K) of Ce atoms by quantitative analysis of the PDF data. However, a previously expected global polymorphic transition from a low-density amorphous state to a high-density amorphous state with an abrupt volume collapse is not observed. Moreover, electrical resistivity also shows a continuous increase during cooling without any sharp change. It is clarified that cryogenic temperatures could facilitate but are not powerful enough alone to trigger a global polymorphic transition in the Ce68Al10Cu20Co2 metallic glass, suggesting a wide distribution of its local atomic environment.

Chen, Zhi↗

DOE OSTI2020

As a Matter of State: The Role of Thermodynamics in Magnetohydrodynamic Turbulence

Turbulence simulations play a key role in advancing the general understanding of the physical properties of turbulence and in interpreting astrophysical observations of turbulent plasmas. For the sake of simplicity, however, turbulence simulations are often conducted in the isothermal limit. Given that the majority of astrophysical systems are not governed by isothermal dynamics, we aim to quantify the impact of thermodynamics on the physics of turbulence, through varying adiabatic index, γ, combined with a range of optically thin cooling functions. Here, we present a suite of ideal magnetohydrodynamics simulations of thermally balanced stationary turbulence in the subsonic, super-Alfvénic, high ${\beta }_{{\rm{p}}}$ (ratio of thermal to magnetic pressure) regime, where turbulent dissipation is balanced by two idealized cooling functions (approximating linear cooling and free–free emission) and examine the impact of the equation of state by considering cases that correspond to isothermal, monatomic, and diatomic gases. We find a strong anticorrelation between thermal and magnetic pressure independent of thermodynamics, whereas the strong anticorrelation between density and magnetic field found in the isothermal case weakens with increasing γ. Similarly, the linear relation between variations in density and thermal pressure with sonic Mach number becomes steeper with increasing γ. This suggests that there exists a degeneracy in these relations with respect to thermodynamics and Mach number in this regime, which is dominated by slow magnetosonic modes. These results have implications for attempts to infer (e.g.,) Mach numbers from (e.g.,) Faraday rotation measurements, without additional information regarding the thermodynamics of the plasma. However, our results suggest that this degeneracy can be broken by utilizing higher-order moments of observable distribution functions.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

Robust Online Sequential RVFLNs for Data Modeling of Dynamic Time-Varying Systems with Application of an Ironmaking Blast Furnace

In a world where the increasing complexity of modern industrial processes brings difficulties for accurate mathematical modeling, taking advantage of data has become an efficient solution to complex dynamic process modeling issue. In this paper, we develop a novel robust online sequential version of random vector functional-link networks (RVFLNs) for data-driven modeling of dynamic time-varying system and applied it in a blast furnace (BF) ironmaking process. First, to overcome the time-varying dynamics of process and to enable the RVFLNs to learn online with avoiding data saturation, an improved online sequential version of RVFLNs (OS-RFVLNs) is first presented by online sequential learning with forgetting factor. This improved OS-RVFLNs algorithm is not only suitable for the real-time and large data transfer situation, but also can adjust the sensitivity of the algorithm to different samples with the help of the introduced forgetting factor. Second, since the output weights of the improved OS-RVFLNs as well as other RVFLNs algorithms are obtained by the least squares approach, a robustness problem may occur when the training dataset is contaminated with various outliers. To solve this problem, a Cauchy distribution weighted M-estimator is introduced to improve the robustness of the improved OS- RVFLNs. For this proposed robust OS-RVFLNs (R-OS- RVFLNs), since the weights of different outlier data are properly determined by the Cauchy distribution function, their corresponding contribution on modeling can be properly distinguished. Thus robust and better modeling results can be achieved. Experiments using actual industrial data of BF ironmaking process and comparative studies have demonstrated that the proposed method produces a better estimation accuracy and stronger robustness than other methods.

Blast furnace (BF), Modelling, Dynamic systems↗