Deep learning-enhanced electrochemical impedance spectroscopy analysis and classification

Abstract

Systems and methods for machine learning models to classify electrochemical impedance spectroscopy data. Systems and methods for preprocessing electrochemical impedance spectroscopy data before analysis is by a machine learning model. Systems and methods for preprocessing electrochemical impedance spectroscopy data utilizing different representation and normalization combinations.

Description

DEEP LEARNING-ENHANCED ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY ANALYSIS AND CLASSIFICATIONFIELD OF THE INVENTION

[0001] This disclosure generally refers to systems and methods of machine learning analysis of electrochemical impedance spectroscopy data.CROSS-REFERENCES TO RELATED APPLICATIONS

[0002] The current application claims priority to Provisional Patent Application No.63 / 741,359, filed January 2, 2025, the disclosure of which is incorporated herein by reference.BACKGROUND

[0003] Electrochemical impedance spectroscopy (EIS) is pivotal in probing material properties and unraveling intricate reaction dynamics in electrochemical domains such as batteries, sensors, and catalysts. The integration of machine learning (ML) in EIS necessitates identifying the optimal data representation for enhanced accuracy and analytical capabilities. ML presents a promising solution for advancing mechanistic analysis in the electrochemical system, leveraging its proficiency in uncovering hidden patterns and providing data-driven insights with minimal human interference. Recent advancements showcase ML’s capability in automatically analyzing EIS data, specifically for classification of equivalent circuit models and parameter regression. Notably, a successful deep-learning-based Variational Autoencoders (VAE) model trained on the real and imaginary part of the EIS data, yielding an 82.0% accuracy in classifying five commonly encountered equivalent circuit models, outperforming other ML algorithms including Deep Neural Network, Random Forest, Support Vector Machine, and AdaBoost with corresponding accuracy of 78.9%, 58.2%, 48%, and 45.3%, respectively. Despite the extensive exploration of various ML algorithms for automated EIS analysis, there is a current lack of comprehensive evaluation for the data preprocessing step, including data representation question and data normalization strategies. There are two common modesof EIS data representation, Cartesian coordinate and polar coordinate, both affecting how the data set is read by the ML model. The EIS data is normalized utilizing different strategies based on min-max scaling due to the large variation in the applied potentials. This evaluation is crucial, at least in the context of ML model establishment, as an optimal preprocessing step involving data representation and normalization can significantly enhance the overall performance of ML algorithms.SUMMARY OF THE INVENTION

[0004] Systems and methods in accordance with some embodiments of the invention are directed to preprocessing methods of electrochemical impedance spectroscopy data.

[0005] In some embodiments, the techniques described herein relate to a method for analyzing electrochemical impedance spectroscopy (EIS) data including: providing an EIS data set; representing the EIS data set; transforming the EIS data set into a tensor array; inputting the tensor array into a residual neural network; and receiving an output from the residual neural network.

[0006] In some embodiments, the techniques described herein relate to a method, wherein the step of representing the EIS data set further includes representing the EIS dataset as a Cartesian plot.

[0007] In some embodiments, the techniques described herein relate to a method, wherein the Cartesian plot includes a Nyquist plot.

[0008] In some embodiments, the techniques described herein relate to a method, wherein the step of representing the EIS data set further includes representing the EIS data set as a polar plot.

[0009] In some embodiments, the techniques described herein relate to a method, wherein the polar plot is derived into a magnitude-vs-phase plot.

[0010] In some embodiments, the techniques described herein relate to a method, further includes normalizing the EIS data set.

[0011] In some embodiments, the techniques described herein relate to a method, wherein the step of normalizing the EIS data set includes layer normalization.

[0012] In some embodiments, the techniques described herein relate to a method, wherein the step of normalizing the EIS data set includes instance normalization.

[0013] In some embodiments, the techniques described herein relate to a method, wherein the tensor array includes 3 elements; wherein a first tensor array element is defined by a number of applied potentials of the EIS data set; wherein a second tensor array element is defined by the representation of the EIS dataset and a value of applied potentials of the EIS data set; and wherein a third tensor array element is defined by a number of frequencies of the EIS data set.

[0014] In some embodiments, the techniques described herein relate to a method, wherein the residual neural network is trained to classify EIS data based on a mechanism category.

[0015] In some embodiments, the techniques described herein relate to a method, wherein the mechanism categories include of E, EC, CE, ECE, and DISP1.

[0016] In some embodiments, the techniques described herein relate to a method, wherein the output received yields an output vector based on the mechanism categories.

[0017] In some embodiments, the techniques described herein relate to a system for classifying electrochemical impedance spectroscopy (EIS) data, including: an EIS data set; a processor programed to condition the EIS data set, wherein conditioning includes: representing the EIS data set under at least one of a Nyquist Plot and a Bode Plot; normalizing the EIS data set; and transforming the EIS data set into a tensor array; and a processor programmed to: receive the tensor array; receive a training data set including simulated EIS data; and input the tensor array and the training data set into a machine learning algorithm, wherein the machine learning algorithm is configured to output a classified EIS data set.

[0018] In some embodiments, the techniques described herein relate to a system, wherein normalizing the EIS data set includes layer normalizing.

[0019] In some embodiments, the techniques described herein relate to a system, wherein normalizing the EIS data set includes instance normalization.

[0020] In some embodiments, the techniques described herein relate to a system, wherein the tensor array includes three elements; wherein a first tensor array element isdefined by a number of applied potentials of the EIS data set; wherein a second tensor array element is defined by the representation of the EIS data set and a value of applied potentials of the EIS data set; and wherein a third tensor array element is defined by a number of frequencies of the EIS data set.

[0021] In some embodiments, the techniques described herein relate to a system, wherein the training data set is classified simulated EIS data.

[0022] In some embodiments, the techniques described herein relate to a system, wherein the classified simulated EIS data is classified into mechanism categories including at least one of: E, EC, CE, ECE, and DISP1.

[0023] In some embodiments, the techniques described herein relate to a system, wherein the classified EIS data set is classified into mechanism categories including at least one of: E, EC, CE, ECE, and DISP1.

[0024] In some embodiments, the techniques described herein relate to a system, wherein the machine learning algorithm is a residual neural network.

[0025] Additional embodiments and features are set forth in part in the description that follows, and in part will become apparent to those skilled in the art upon examination of the specification or may be learned by the practice of the disclosure. A further understanding of the nature and advantages of the present disclosure may be realized by reference to the remaining portions of the specification and the drawings, which forms a part of this disclosure.BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The description will be more fully understood with reference to the following figures, which are presented as embodiments of the invention and should not be construed as a complete recitation of the scope of the invention, wherein:

[0027] FIG. 1 illustrates preprocessing of EIS data for ML analysis.

[0028] FIG. 2 illustrates the experimental and mathematical foundation of EIS technique.

[0029] FIG. 3A and FIG. 3B illustrate different representation schemes of EIS data.

[0030] FIG. 4A to FIG. 4C illustrate different normalization of EIS data.

[0031] FIG. 5A to FIG. 5E illustrate five targeted homogenous electrochemical mechanisms.

[0032] FIG. 6A and FIG. 6B illustrate training of ML algorithms for EIS data analysis.

[0033] FIG. 7 illustrates the process of analyzing EIS data utilizing a ML model and the output vector.

[0034] FIG. 8 illustrates the EIS data input into a ML model for analysis.

[0035] FIG. 9 illustrates a box and whisker plot of test accuracy of EIS data classification as analyzed by a ML model.

[0036] FIG. 10 illustrates a summary of test accuracy of ML model classification of EIS data subjected to different preprocessing methods.

[0037] FIG. 11A to FIG. 11F illustrate exemplary confusion matrices depicting individual ResNet models' performance with six distinct data preprocessing strategies.

[0038] FIG. 12 illustrates training accuracy and model performance of ML model classification of EIS data based on different preprocessing strategies.

[0039] FIG. 13A and FIG. 13B illustrate the confusion matrix of the supporting vector classifier models.

[0040] FIG. 14A to FIG. 14C illustrate the receiver operating characteristic curves of different preprocessing strategies.

[0041] FIG. 15 illustrates a robustness assessment of models with magnitude-vs- phase representation and Method III normalization.

[0042] FIG. 16A to FIG. 16F illustrates simulated EIS spectra with different levels of Gaussian noises and corresponding Kramers-Kronig transforming results.

[0043] FIG. 17A illustrates the confusion matrix of ResNet models with magnitude-vs- phase representation, Method III normalization, and training noise a = 0.01.

[0044] FIG. 17B illustrates the receiver operating characteristic curves of ResNet models with magnitude-vs-phase representation, Method III normalization, and training noise o = 0.01.

[0045] FIG. 18A to FIG. 18D illustrate the “importance” analysis of ML model when classifying EIS data.

[0046] FIG. 19A to FIG. 19F illustrates the “importance” analysis of ML model when classifying EIS data based on different preprocessing strategies.

[0047] FIG. 20A to FIG. 20E illustrate the mathematical measurement parameters used in an exemplary embodiment.DETAILED DESCRIPTION OF THE INVENTION

[0048] It will be understood that the components of the embodiments, as generally described herein and illustrated in the appended figures, may be arranged and designed in a variety of different configurations. Thus, the following more detailed description of various embodiments, as represented in the figures, is not intended to limit the scope of the present disclosure but is merely representative of various embodiments. While various aspects of the embodiments are presented in drawings, the drawings are not necessarily drawn to scale unless specifically indicated.

[0049] The present invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The described embodiments are to be considered in all respects only as illustrative and not restrictive.

[0050] Reference throughout this specification to features, advantages, or similar language does not imply that all of the features and advantages that may be realized with the present invention should be or are in any single embodiment of the invention. Rather, language referring to the features and advantages is understood to mean that a specific feature, advantage, or characteristic described in connection with an embodiment is included in at least one embodiment of the present invention. Thus, discussions of the features and advantages and similar language throughout this specification may, but do not necessarily, refer to the same embodiment.

[0051] Furthermore, the described features, advantages, and characteristics of the invention may be combined in any suitable manner in one or more embodiments. One skilled in the relevant art will recognize, in light of the description herein, that the invention can be practiced without one or more of the specific features or advantages of a particular embodiment. In other instances, additional features and advantages may be recognized in certain embodiments that may not be present in all embodiments of the invention.

[0052] Reference throughout this specification to “one embodiment,” “an embodiment,” or similar language means that a particular feature, structure, or characteristic described in connection with the indicated embodiment is included in at least one embodiment. Thus, the phrases “in one embodiment,” “in an embodiment,” and similar language throughout this specification may but do not necessarily, all refer to the same embodiment.

[0053] Many embodiments relate to the proper preprocessing of electrochemical impedance spectroscopy (EIS) data for ML-based analysis for a classification task of different equivalent circuit models and / or mechanisms. Some embodiments, within a defined parameter space, demonstrate that preprocessing methods significantly impact the classification accuracy of ML models. In several embodiments, representing Z(ES, f) in Bode-type polar coordinates with normalization at each Esvalue yields optimal performance. Several embodiments underscore the importance of proper data preprocessing and provide a foundational framework for future ML-based EIS research across diverse electrochemical applications such as batteries and sensors.

[0054] Typically, EIS entails monitoring sinusoidal current ( / (f)) in response to sinusoidal voltage (E(f)) around a central potential (Es) across varying frequencies (f). While impedance (Z(f)) is straightforwardly computed as the ratio of output to input signals under Es, diverse visualization methods exist in the current EIS analysis offering distinct advantages, with Nyquist and Bode plots prevalent. With the rise of ML, rethinking data representation in EIS for ML models becomes imperative. Recent advancements showcase ML’s capability in automatically analyzing EIS data, specifically for the classification and parameter regression of equivalent circuit models. However, the lack of discussion on proper data representation for ML-based EIS analysis persists, particularly regarding the primary use of Nyquist plots as input format for ML models. Besides the intricacy of EIS data, ML architectures necessitate normalized input data for general applicability, which adds complexity to EIS data preprocessing. Despite EIS's significance and the burgeoning interest in ML-based EIS analysis, the preprocessing of Z(ES, f) remains underexplored, posing a critical challenge for next-generation ML-based analyses.

[0055] In many embodiments the strategy of Z(ES, f) preprocessing is evaluated based on the classification accuracy of a residual neural network (ResNet) model trained by the same Z(ES, f) dataset following corresponding preprocessing methods, incorporating both data representation and normalization methods. In several embodiments, a Z(ES, f) dataset numerically simulated by finite-element method, following the practice and mechanistic definitions of ResNet-based voltammogram analysis. Although a limited dataset does not encompass all EIS application scenarios, the findings of various embodiments are potentially applicable to general ML models for EIS analysis, such as predicting equivalent circuit models, providing a general protocol evaluating data preprocessing methods.

[0056] FIG. 1 shows preprocessing and ML analysis in accordance with many embodiments. EIS data can be provided (101) for analysis. The EIS data can be represented (102) on either a Cartesian Nyquist plot (102A) or a magnitude-vs-phase derived from a polar Bode plot (102B). The represented data may undergo normalization (103) according to three methods: no normalization (Method I) (103A), layer normalization (Method II) (103B) where the EIS data is normalized as a whole, or instance normalization (Method III) (103C), where each Esis normalized. The represented and normalized EIS data can be transformed (104) into a tensor array for input (105) into the ResNet (ML) architecture. The EIS data can be analyzed by the ResNet architecture, and an output can be received (106) classifying the EIS data.

[0057] FIG. 2 shows EIS data is collected by applying a sinusoidal voltage E(f) 201, consisting of potential perturbation and the bias potential Es, over a series of frequencies f, and monitoring the corresponding sinusoidal response current / (f) of the system 202. Impedance Z203 is defined as the ratio of output signal E(f) and input signal / (f), yielding a complex quantity. Following validation of impedimetric data, analysis can be conducted through various plotting formats among different components. The most widely used formats are the Nyquist and Bode plots. The Nyquist-like Cartesian coordinate represents each Z(Es, f) data as a two-dimensional matrix {Zj’(Es), Z”(ES. / )}, where Z’ and Z” are the real and imaginary components of the impedance, respectively. The Bode-like polar coordinate represents it as {<py(Es, / ), |Z|y(Es. / )}, is the phase angle and |Z| is themagnitude of the impedance. Each plot, in accordance with various embodiments, offers distinct insights into the data as shown in FIGS. 3A and 3B: Nyquist plot 301 displays impedance in the complex plane (the absolute value of imaginary part Z" vs. the real part Z’) while Bode plots illustrate the system’s response amplitude of impedance |Z| 302 and phase angle shift |<p| 303 as functions logarithmic value of frequency logio( - The Magnitude vs. Phase 304 of the system can be derived from the Bode plots.

[0058] Diverse visualizations can be employed supplementarily to extract comprehensive insights from data obtained in a single experiment. However, the current trend particularly in mechanistic analysis leans towards the Nyquist plot, attributed to its straightforward distinctive semi-circle visualization. Taking analyzing aluminum corrosion in concentrated KOH as an example, a series of EIS data can be obtained across various potentials and visualized by Nyquist plot, and the interpretation involves identifying the electrochemical process that best aligns with the loci of experimental impedance spectra, utilizing a set of optimized model parameters. Determining the superior data representation in EIS remains an open question, given the diverse features of various formats, and the challenge of statistically evaluating the impact of data representation on manual EIS interpretation further complicates this inquiry.

[0059] With the advent of artificial intelligent transforming every corner of our society and research community, the issue of data representation in EIS analysis for a ML model warrants another reckoning, if the ML-based analysis is considered to augment the human researchers in the future. ML presents a promising solution for advancing mechanistic analysis in electrochemical systems, leveraging ML’s proficiency in uncovering hidden patterns and providing data-driven insights with minimal human interference. Recent advancements showcase ML’s capability in automatically analyzing EIS data, specifically for the classification and parameter regression of equivalent circuit models. Yet there is a dearth of discussion about the proper data representation of EIS data for ML-based analysis, with the seemingly ill-justified consensus of deploying Nyquist plot, namely the Cartesian coordinate of Z(ES, f), as the EIS input format for ML models.

[0060] The unique nature of ML architecture, which requires input data normalized to unity for model’s general applicability, poses additional challenges for inputting EIS data into a ML model. Different normalization strategies of input data can lead to different levels of performance even for the same ML architecture. In the context of EIS analysis, Z(ES, f)’s sensitivity towards Escan indicate Z(ES, f)’s dependence under different Esvalues in a comparative analysis. Hence, in some embodiments, it is critical to properly normalize multiple Z(f) traces under various Esvalues as the input data for a ML-based comparative analysis. To date, proper normalization has not been investigated, at least in the public domain. Despite EIS’s significance in electrochemistry and the growing interests in ML- based EIS analysis, the topic of Z(ES, f) preprocessing that includes proper representation and normalization remains surprisingly under-explored. It seems imprudent to ignore such a critical issue in the context of developing next-generation ML-based EIS analysis.

[0061] To determine the optimal normalization methods, given the large variations of magnitude in EIS simulated with different electrochemical parameters and applied potentials, different strategies based on min-max scaling are tested, in accordance with many embodiments, for each data format: original data without normalization (Method I), layer normalization (Method II), or instance normalization (Method III). FIG. 4A shows three normalization protocols that may be applied, in accordance with various embodiments: no normalization (Method I) 401, layer normalization across all Es,i values within the dataset (EIS data normalized across the various channels of applied potentials for a single example) (Method II) 402, and instance normalization for each Es, value (EIS data independently normalized for each applied potential channel for a single example) (Method III). Methods II and III scale values between -1 and 1, maintaining relative magnitudes (Method II) 402 or introducing variations (Method III) 403. In many embodiments, layer normalization (Method II) can keep the same relative magnitude between different applied potentials as the original data (Method I), while instance normalization (Method III) can introduce greater variations of magnitude across instances or channels. FIG. 4A shows the exemplified visualizations of EIS spectra for three different normalization methods per representation. All values are scaled into a specific range between -1 and 1 for both normalization strategies. FIG. 4B shows the inputvectors for normalization Method II and Method III in accordance with various embodiments. FIG. 4C shows the Esapplied bias potentials and the horizontal and vertical axis of the impedance representation, Z’ and -Z” respectively for the real and imaginary part of the impedance of the Nyquist-type Cartesian coordinate and |Z| and (p respectively for the magnitude and phase angle of the impedance.

[0062] The impact of different preprocessing methods of various embodiments is compared by testing the classification of EIS data by ResNet of various embodiments. The training set for ResNet consists of numerically simulated EIS spectra using the finite- element method of many embodiments based on five targeted homogeneous electrochemical mechanisms (FIGS. 5A to 5E) for classification: a single-electron transfer with any level of reversibility (E) (FIG. 5A), an E step followed by a C step with any level of reversibility (EC) (FIG. 5B), an E step preceded by a C step (CE) (FIG. 5C), a system of two E steps connected by an irreversible rate-limiting C step with the second E step being more thermodynamically facile than the first one (ECE) (FIG. 5D), and a two- electron transfer that is similar to ECE yet the second E step is replaced by a solution disproportionation reaction (DISP1) (FIG. 5E). In some embodiments the E mechanism is a reversible / quasi-reversible single-electron transfer with a redox potential Eo. In several embodiments the EC mechanism is the aforementioned E step followed by a homogenous reversible chemical reaction (C step). In many embodiments, the CE mechanism is an E step preceded by a reversible C step. In various embodiments, the ECE mechanism is a system of two E steps connected by an irreversible rate-limiting irreversible C step with the second E step being more thermodynamically facile than the first E step. In some embodiments the DISP1 mechanism is a net two-electron transfer that is similar to the ECE mechanism, yet the second E step is replaced by a solution disproportionation reaction. In FIGS. 5A to 5E, graphs 511, 521, 531, 541, 551 show the original simulated data with original scales represented in Nyquist-type Cartesian coordinates. Graphs 512, 522, 532, 542, 552 show the data normalized with Method III and represented in Nyquist-type Cartesian coordinates. Graphs 531, 532, 533, 534, 535 show the data normalized with Method III and represented in Bode-type polarcoordinates. For Method III normalization, the x and y axis’s scale is from -1 to 1 in FIGS.5A to 5E.

[0063] The construction of numerical models of partial differential equations (PDEs), boundary conditions, and initial conditions of several embodiments follows the mechanisms’ definitions in textbooks. The parameters defining the reaction systems include but are not limited to, concentrations, interfacial charge transfer rate constant in the electrochemical process, and thermodynamic equilibrium constants of the chemical process. The constraints and sampling approaches of these parameters ensure physical realism and class diversity. For the parameters incorporated in the impedance technique of many embodiments, the number of measured frequencies may be set to 500 points, distributed logarithmically in the frequency range (f) from 1 Hz to 100 kHz. Inspired by the reaction mechanism analysis (RMA) based on E IS as a function of applied potentials, EIS spectra collected under different fixed values of applied potentials Eapp, from Eo -0.4 to Eo +0.6 V at 50 or 100 mV intervals for each reaction system, providing more information to potentially distinguish between candidate mechanisms. Approximately 5,000 simulated data points, with fourteen impedance spectra for each data point, are generated for each mechanism by randomly sampling electrochemical parameters within constrained ranges, ensuring a sufficient, balanced and diverse training dataset in accordance with some embodiments. Dynamic Gaussian noise can be stochastically added on the preprocessed raw data during the training and test processes in accordance with various embodiments.

[0064] In some embodiments, the ML-model can be based on a Faster regional convolutional neural network (R-CNN) architecture for mechanistic investigation of cyclic voltammograms. Some embodiments utilize a ResNet architecture for EIS analysis due to its ability to handle complex, high-dimensional data. In several embodiments, its deep architecture with residual skip connections can prevent vanishing gradients, ensuring more efficient and robust training than basic neural networks or convolutional neural networks. In contrast, simpler linear models may struggle with high dimensionality and non-linear relationships in EIS data. In many embodiments, a series of ML models based on a ResNet architecture, more specifically ResNet with 18 residual learning layers(hence ResNet-18) can be used. In several embodiments, a ML-model has been trained by simulated voltammograms to classify electrochemical mechanisms among five of the most classic mechanisms, with an input data structure of three-dimensional tensor of 6- scan single redox voltammograms ({v, / (E)n, n = 1 to 6). Many embodiments utilize a ResNet architecture for EIS analysis due to its ability to handle complex, high-dimensional data. In many embodiments, a series of ML models based on a ResNet-18 can be trained and validated by the same initial {Z(Es,i, fj)} dataset, as described above, yet preprocessed under different combinations of data representation and normalization, largely following the protocol for ResNet-based analysis of cyclic voltammograms. In several embodiments, ResNet is an advanced evolution of convolutional neural network, chosen for ML model construction and evaluation, encouraged by previous successful application in extracting features from high-dimensional cyclic voltammograms. Several embodiments of ResNet architecture’s unique skip connections within the convolutional layers can effectively handle vanishing and exploding gradient problems during the training process, leading to effective learning and fast convergence. To mitigate the effects of randomness stemming from data splitting and parameter initialization in the training process, 8 replicates of ResNet models (n = 8) can be trained with about 3500 Z(ES, f) data points, randomly sampled for every training process, for each mechanisms through 1000-epoch training in accordance with various embodiments. The size of 3500 Z(ES, f) data points for the training set is chosen because the ResNet model’s accuracy, in accordance with several embodiments, under the representation of |Z| and cp, steadily increases with increasing Z(ES, f) numbers in the training set before plateauing beyond 3500, which indicates the model has achieved a stable performance level, suggesting that further increases in training set size may not yield significant improvements in accuracy, as shown in FIG. 6A. The ResNet model of many embodiments is considered sufficiently trained after 1000 epochs, because the cross-entropy loss that surrogates the ResNet’s accuracy in the training process has asymptotically approaching zero after 1000 epochs of training, as shown in FIG. 6B.

[0065] In many embodiments, EIS data requires complex representations and optimal normalization techniques due to significant variations in impedance amplitude acrossdifferent potentials. Within the same 5-mechanism parameter space that were defined in prior ResNet-based voltammogram analysis, many embodiments are directed to the classification performance of a ResNet model trained by simulated Z(ES, f) data, dependent on the preprocessing methods of Z(ES, f) in the training date (FIG. 7). In many embodiments, the preprocessing methods include both data representation — Nyquist- type Cartesian coordinate of Z’ and Z” versus Bode-type polar coordinate of \Z\ and cp — as well as data normalization — without any normalization (Method I), one normalization for each Z(ES, f) data point (Method II), and independent normalization at each Esvalue (Method III) — for 14 Z(f) traces under different Esvalues. Among different combinations of Z(ES, f)’s representation and normalization methods, in many embodiments, the preprocessing method can strongly affect the classification accuracy of the correspondingly trained ResNet model for ML-based EIS analysis (FIG. 10). Some embodiments show that representing Z(ES, ) in the Bode-type polar coordinate of |Z| and cp with normalization at each Esvalue (Method III) can yield the highest accuracy and robustness in the correspondingly established ResNet model.

[0066] In several embodiments, the trained ResNet model can yield an output vector y = {yi , / 2, s, y4, ys} as shown in FIG. 7, in which each y quantitatively represents the propensity as a surrogate for the statistical probability associated with the respective mechanisms of E (FIG. 5A), EC (FIG. 5B), CE (FIG. 5C), ECE (FIG. 5D), and DISP1 (FIG.5E). The classification process designates the mechanism with the largest y component as the most probable one for the electrochemical system.

[0067] As shown in FIG. 3B, the impedance spectra of various embodiments can be represented in two ways, Nyquist plot 301 , or magnitude vs phase angle of impedance 304, both covering the identical whole information collected by EIS. Similar to the case of cyclic voltammogram, a two-dimensional matrix can be deployed compatible with ResNet architecture as the ML model’s input — {f, (Z’(E), Z”(E))}n(n, the number of applied potentials; f, frequency; Z’ and Z”, the real and imaginary component of impedance respectively) for Nyquist-based representation, and {f, (p(E), Z(E)}n(n, the number of applied potentials; f, frequency; (p, phase angle; and Z, the magnitude of impedance) for magnitude-vs-phase-based representation.

[0068] The two-dimensional input matrix {Z(ES. / , fj)} can be transformed into a three- dimensional tensor {n, y, m}. In some embodiments, n - 14 corresponding to the number of simulated applied potentials, y = 3, storing Esvalues and either Cartesian or polar coordinates, and m = 500 correlating to the number of simulated frequencies. In several embodiments, when y processes a dimension of 3: the first two channels represent EIS features, (Z’(E), Z”(E)) for Nyquist-based representation and (<p(E), Z(E)) for magnitudephase-based representation; the third channel contains the values of applied potentials. The yielded output is a vector y = {yi , yi, ys, y4, ys} (FIG. 7) representing a quantitative distribution of propensity as a surrogate for the statistical probability associated with the respective mechanisms of E, EC, CE, ECE, and DISP1. The classification process designates the mechanism with the largest y component as the most probable one for the electrochemical system.

[0069] In several embodiments, the effectiveness of the combination of the data representation and normalization preprocessing is determined by the accuracy of the ML analysis. FIG. 8 shows the EIS data input and architecture of the ML model used to test the classification of the ResNet of various embodiments. The output of the ResNet architecture to characterize the EIS data is compared to the human analyzed EIS data characterization. FIG. 9 shows the box plot of test accuracies of individual ResNet models utilizing the 6 different preprocessing methods of many embodiments with different data preprocessing methods (n = 8). FIG. 10 shows a summary of the average classification accuracies form ResNet based on different Z(ES, f) preprocessing strategies, n - 8. Among these six data preprocessing strategies, the magnitude-vs-phase representation with Method III normalization yielded the best test accuracies (FIG. 11 A), with an average accuracy of 89.2% (standard deviation: 1.7%) on the test set. In comparison, the other two magnitude-vs-phase-based models yield less satisfactory results, with an average accuracy of 77.2% (standard deviation: 10.1 %) for Method II (FIG. 11 B) and 70.9% (standard deviation: 11.8 %) for Method III (FIG. 11C). On the contrary, the ResNet models trained with Nyquist representation yields a 71.8% average accuracy (standard deviation: 1.4%) with Method III normalization (FIG. 11 D), while Method II (FIG. 11 E) and Method I (FIG. 11 F) resulted in lower averages: 68.9% (standard deviation: 6.7%) and63.7% (standard deviation: 10.2%), respectively. The exemplary confusion matrices of individual models with six preprocessing strategies are illustrated in FIGS. 11 A to 11 F.

[0070] In many embodiments, data preprocessing methods significantly affect the resultant ResNet model’s accuracy of mechanistic classification as shown in FIG. 9, FIG.10, and FIG. 12. When tested on unseen EIS data, the classification accuracy of ResNet model under a certain preprocessing method (“Test accuracy” in FIG. 12), averaged from 8 replicates for statistical validity (n = 8), ranges from the 63.7±10.8% (Cartesian coordinate + Method I) to 89.2±1.7% (polar coordinate + Method III). In many embodiments, models trained with the Cartesian representation (63.7% to 71.8%) are less accurate than those with the polar representation (70.9% to 89.2%). For the same representation, accuracy increases from Method I (63.7% and 70.9% for Cartesian and polar coordinates, respectively, same below) to Method II (68.9% and 77.2%) to Method III (71.8% and 89.2%). Within the defined space of electrochemical mechanisms and the corresponding training set of several embodiments, the optimal data preprocessing method involves representing {Z(Es,i, f)} in the polar coordinate of |Z| and p and individually normalizing {Z(ES. / , fj)} data for each Esvalue (Method III). These findings affirm the critical role of data preprocessing in ML-based EIS analysis and hint that the commonly used Nyquist plot (Cartesian coordinate) may not be optimal, in contrast to strong preference of Nyquist plot in human-based EIS analysis.

[0071] Utilizing supporting vector classifier (SVC) models with magnitude-phase and Nyquist-based data representations and the same Method III normalization, in accordance with several embodiments, further confirms the enhanced accuracy of the magnitude-phase approach. FIGS. 13A and 13B show the confusion matrix of the SVC models trained by EIS data {Z(ES, , } under different preprocessing approaches. FIG.13A represents training data with polar coordinate of |Z| and cp while normalizing {Z(ESj / , fj)} individually at each Es(Method III). FIG. 13B represents training data with Cartesian coordinate of Z’ and Z” while normalizing {Z(ES, / , } following Method III. The results of various embodiments show consistent, albeit less satisfactory, performances compared to ResNet models. The magnitude-phase SVC model reached an overall test accuracy of 82.4%, while the Nyquist SVC model scored 66.8%. The consistent improvement inaccuracy with the magnitude-phase representation indicates an inherent advantage of such representation, particularly suited for the specific mechanistic analysis task, instead of model bias.

[0072] Additional analysis of the accuracy differences among ResNet models with various data preprocessing methods of several embodiments suggests that the models of higher classification accuracy are better at discerning the mechanisms that are also prone to be misclassified based on human analysis. In ML, it is common to deploy the plot of confusion matrix as a performance evaluation tool that represents the accuracy of a classification model. Each preprocessing method of various embodiments is plotted corresponding to the trained ResNet model’s confusion matrix using the medium model from the 8 replicates (FIGS. 11A to 11 F). Each row shows the percentage of EIS data, based on a specific designed mechanism ("true label"), classified into another mechanism ("predicted label"). The EC mechanism seems to be the most difficult to classify, evident by the marginal accuracy improvement for EC mechanism across all ResNet variants of some embodiments. Yet the difference in preprocessing strategy has significant influence over other mechanisms, most notably for E and DISP1 mechanism and to a lesser extent for ECE mechanism. The ResNet models of various embodiments often confuse E, EC, and DISP1. This is similar to observations in ML-based voltammogram analysis, and consistent with human analysis detailed in textbooks. Because the same {Z(Es, fj)} dataset is used and only the preprocessing strategy varies, such that different preprocessing strategy can affect ML model’s classification capability unevenly across different mechanisms in accordance with several embodiments. In a similar vein, some additional mechanistic information could be easier to uncover under a suitable preprocessing strategy of various embodiments.

[0073] To better evaluate models’ performance of various embodiments, the receiver operating characteristic (ROC) curve is utilized to assess the model's ability to distinguish one electrochemical mechanism from the others, where its corresponding area under the curve (AUC) value quantifies the model's overall discriminatory power. Higher AUC values indicate better discrimination, with a value of 1 indicating perfect performance. In FIG. 14A, in accordance with an embodiment, the model, with magnitude vs phase datarepresentation and Method III normalization, shows an excellent classification performance with AUC values above 0.98 for five mechanisms, with an average of 0.99. As a comparison, two suboptimal models, in accordance with several embodiments, show lower AUC values of DISP1, E, and EC mechanisms, corresponding to their confusion matrices (FIGS. 11B and 11 D) with noticeable misclassification these three mechanisms (FIGS. 14B and 14C). The superior accuracy with magnitude-vs-phase based representation and Method III normalization, compared to the others, implies ResNet models’ more efficient information extraction from the input, in accordance with many embodiments.

[0074] In several embodiments, the robustness of the models against noise can be evaluated, to access its applicability in classifying real-world experimental data. Some embodiments introduce random dynamic Gaussian noise with varying standard deviation o (0, 0.01, 0.1) relative to the original data scale into the simulated spectra for the noise- added training set. Subsequently, these models were tested with unseen data containing varying gaussian noise (o from 0 to 0.3) (FIG. 15). The model trained with o = 0.01 maintained an overall accuracy of over 87% until o > 0.1 , and this high accuracy persisted above 85% until o > 0.25. Even at training noise o = 0.1, the model exhibited an overall accuracy of around 85%, remaining relatively robust up to o = 0.3. In contrast, the zeronoise model saw an immediate drop in accuracy, declining from 92.5% to 77% upon the introduction of 0.01 test noise and performed below 60% as noise levels increased beyond 0.1. This demonstrates the remarkable capacity of the optimal models trained with noise-added data to maintain high accuracy levels even when noise levels exceed the training standard.

[0075] As presented previously, introducing training noise enhances robustness with a minor accuracy tradeoff for the magnitude-vs-phase model of various embodiments. To determine the most acceptable noise level for reliable, real-world impedance data, Kramers-Kronig (K-K) transforms can be employed, a mathematical approach to validate the consistency of experimental EIS spectra. FIGS. 16A to 16F shows the exemplary simulated EIS spectra with different noise levels (0, 0.01 , 0.1 ) and their corresponding K- K transform fitting results. A noisy experimental impedance data can empirically pass theK-K test if the residual errors between the fitted and measured values are similar in size to the measurement noise. As demonstrated in FIG. 16B, noise o = 0.01 results in residual errors x2at a similar order, with a value of 0.024, suggesting that the expected noise level of validated EIS spectra should be approximately 0.01 of the magnitude. Thus, the ResNet model trained with o = 0.01 is theoretically applicable for experimental data validated by K-K transforms, with the confusion matrix and ROC curves plotted in FIGS.17A and 17B.

[0076] The significant dependence of ML model’s classification accuracy on training data’s preprocessing strategy investigation into its origins. In many embodiments, "importance" analysis, a valuable tool in machine learning, identifies the regions or features in the input data that can have the most significance influence on the model's decision-making process, by visualizing the relative magnitudes of gradient values of the model’s output with respect to the input data. Hence, the “importance” analysis can be used to see where the ResNet model focuses within the EIS data for mechanistic classification in a multi-potential comparative analysis. As shown in FIGS. 18A to 18D, the gradient values can be collectively normalized based on the highest value achieved among all the ResNet models and plotted in the gray scale. Additionally, for easier visualization of “importance” on each simulated spectrum with varying magnitudes, all simulated spectra can be individually normalized and plotted on the Cartesian coordinate of Z’ and Z” (Nyquist plot), irrespective of the preprocessing methods used for the input data in accordance with many embodiments. Although the training dataset includes ~10,000 data points in accordance with an embodiment, a prevailing trend is observed from manual inspections, with the example provided below. As detailed below, such an “importance” analysis provides some insights towards the difference created by preprocessing approaches for ML-based EIS analysis.

[0077] In some embodiments, the preprocessing approach of the EIS input data notably impacts the resultant ResNet model’s capability of extracting mechanistically relevant information. FIGS. 18A and 19A display the example for a {Z(ES, / , fj)} data point of simulated based on the CE mechanism. While the ML model, in accordance with an embodiment, trained by EIS data represented by polar coordinate and normalized viaMethod III correctly classifies the mechanism as CE as 100% propensity (FIG. 18B), the models from other preprocessing methods generate inaccurate classification: for example, 82% propensity of DISP1 mechanism from the model with representation of polar coordinate and normalization via Method II (FIG. 18C), and 81% propensity of ECE mechanism from the model with representation of Cartesian coordinate and normalization via Method III (FIG. 18D). Concurrent with such discrepancy in classification, the “importance” analysis for those three models of various embodiments reveals drastic differences. The model, in accordance with an embodiment, can be trained from representation of polar coordinate and normalized via Method III (FIG. 18B) to display a distribution of “importance” across both low and high frequency regions, capturing information related to both kinetics and mass transfer, with a particular emphasis on the mass transfer-controlled low-frequency region.

[0078] In contrast, both models of various embodiments with incorrect classification (FIGS. 18C and 18D) only focus the model’s “importance” on the high frequency region with little if any “importance” on the low-frequency part. Models of correct mechanistic assignment, in accordance with several embodiments, tend to spread the “attention” across the whole frequency domains that cover both (electro)chemical kinetics and mass transport, while models of erroneous classification usually fail to spread the model’s “attention” across a wide range of frequencies. The suboptimal model with magnitude-vs- phase representation and Method II normalization misclassified it as DISP1, with “importance” plot primarily emphasized on the high frequency region (FIG. 19B). Another suboptimal model, the Nyquist-based and Method-ll-normalized one, can predict the system as ECE with a relatively higher confidence, whose “importance” exhibits a higher focus on the high frequency region (FIG. 19E) predominantly associated with kinetics information. The gradient regions in FIG. 19C and FIG. 19F illustrate the specific different areas where the optimal model puts more emphasize on than the suboptimal one: red sections indicate higher "importance" in the optimal model, while blue regions signify higher "importance" in the other model. FIG. 19C shows the “importance” discrepancy between models, in accordance with some embodiments, trained with magnitude-vs- phase (FIG. 19D) and Nyquist (FIG. 19B) representations, both using Method IIInormalization. FIG. 19F shows the “importance” discrepancy between models, in accordance with some embodiments, with Method II (FIG. 19E) and Method III (FIG. 19D) normalization, both using magnitude-vs-phase representation. Red sections indicate higher “importance” in the optimal model in accordance with an embodiment, while blue regions signify higher “importance” in the other models of various embodiments. In many embodiments, these differences in the models' attention towards various parts of the EIS pattern suggest that the optimal model extracts information and hidden features more effectively, providing an explanation for the significantly improved performance observed. More exemplary results of the “importance” analysis of two models can be found in the supporting information. The preprocessing of EIS data seems to strongly affect how the correspondingly trained ResNet model uncover features and extract mechanistic information across a wide range of frequencies for a balanced evaluation that includes both kinetics and mass transport.Exemplary Embodiment

[0079] Finite-element simulations of electrochemical impedance spectroscopy (EIS) spectra are conducted using COMSOL Multiphysics v5.6. The Electrochemistry Module and electroanalysis physics are employed to establish a one-dimensional model under the inert supporting electrolyte assumption. A time-dependent solver specialized for alternating current (AC) impedance is incorporated, using an adaptive mesh with a maximal mesh size of one twenty-fifth of the diffusion layer thickness at the maximum frequency. For impedance spectrum simulation, a harmonic perturbation is applied to an assigned applied potential Eappover a range of frequencies from 1 Hz to 100 kHz. Simulations are iterated using COMSOL Multiphysics 5.6 with MATLAB R2022b, with model parameters sampled by Python 3 scripts and communicated to COMSOL through MATLAB. Additional sanitization is implemented after COMSOL simulation to ensure electrochemical feasibility. Several embodiments conduct approximately 30,000 valid simulations, each with 14 different applied bias potential values (Eapp) in the EIS spectra.

[0080] In many embodiments, two data representation strategies and three normalization approaches are employed for data preprocessing before theimplementation of machine learning. For each data point comprising simulated EIS spectra at n number of Eappvalues, EIS spectra are preprocessed and translated into the two-dimensional matrix {f, (Z E), Z’ E))}nor {f, ((p(E), Z(E))}nfor two different representations, Nyquist and magnitude-vs-phase-angle plot, respectively. For layer normalization (Method II), some embodiments initially normalize the simulated values (Z'(E), Z"(E)) or (<p(E), Z(E)) by subtracting the minimum value across different applied potentials and dividing it by the range (maximum - minimum values) among those potentials. Then, the values are scaled from the range of 0 to 1 to the range of-1 to 1 by multiplying the normalized value by 2 and subtracting 1. In the instance normalization (Method III), a similar procedure is applied, but the values are independently normalized using the maximum and minimum values at each applied potential. Original values, without any normalization strategies (Method I), are utilized as input to the ML algorithms in the non-normalization case for comparative analysis. Dynamic Gaussian noise was stochastically added on the preprocessed raw data during the training and test processes.

[0081] In many embodiments, the COMSOL-based finite-element numerical simulation is used for EIS, following the partial differential equations and initial conditions under the supporting electrolyte assumption for each mechanism. The models, in accordance with various embodiments, contain a one-dimensional domain. The length of the domain, Ld is adaptively chosen based on the diffusion coefficient (D) and the lowest frequency (m / n), to be sufficiently large compared to the time scale of diffusion. Diffusion layer assumption is implemented in the simulation, in accordance with various embodiments. In many embodiments, x = 0 denotes the electrode and x = Ld denotes the boundary diffusion layer.

[0082] In some embodiments the presence of a large quantity of inert supporting electrolyte in the designed domain is assumed. Under these conditions, the resistance of the solution is sufficiently low that the electric field is negligible.

[0083] The diffusion equation, also known as Fick’s 2nd law, (EQ. 2) describes the chemical transport of the electroactive species / . Here, in many embodiments, f, denotesthe mechanism specific function that describes any possible C step in the solution. In some embodiments, fi - 0 denotes the absence of any homogenous C steps.<

[0084] The diffusion coefficients Di, sampled logarithmically, are assumed to be same D for all the molecular redox species in the solution. The hypothesis of constant D values is reasonable for two reasons: (1) the assumed reversible and quasi-reversible E step suggest a small reorganization energy and the resultant a small change of the molecular structure; and (2) the value of D is relatively insensitive to the changes of chemical identities since the scaling relationship between D and molecular weight is relatively weak based on the Stokes-Einstein relationship (EQ. 3).(D OC MW' (EQ. 3)

[0085] The initial concentration of the reduced species CR,I is linearly sampled from 0.1 mM to 100 mM.

[0086] The AC Impedance study is used in some embodiments to model a harmonic perturbation applied to a fix center electrode potential Es, retaining terms up to the first order in perturbations of EQ. 4.

[0087] Here, in many embodiments, Ape1^ is the perturbation amplitude, assigned as 0.005 V, which is small with respect to RT / F, so that the Butler-Volmer equation (EQ. 5) can be linearized. Therefore, the system responds linearly to the perturbation, the current and the concentrations are also subject to a sinusoidal perturbation at the same frequency.<>*7 = <Ps,ext ~ E0 / R(EQ. 6)

[0088] Here, in many embodiments, the overpotential (EQ. 6), is the difference between the applied harmonic perturbation and the equilibrium potential. The equilibrium potential is assigned as a zero during simulation, a = - and 0.5 x CR idenotes the equilibrium difference between the applied harmonic perturbation and the equilibriumconcentration when E = Eo / Rand CR= Co. The exchange current density io is logarithmically sampled with the upper-bound io, upper and lower-bound i0. lower.

[0089] The AC Impedance Study makes the approximation that the dependent concentration variables a (including the oxidant, the reactant, and species involved in the following chemical steps if any) can be expressed as the sum of a stationary a,o due to the center electrode potential voltage Eapp, and a sinusoidal perturbation a,i to the concentration resulting from the harmonic perturbation of EQ. 7.>

[0090] If a,i is complex, it implies that the response of the concentration profile is out- of-phase with the applied waveform.

[0091] The domain equation with the harmonic perturbation follows the frequency domain form of Fick’s 2nd Law from EQ. 2).

[0092] For the mesh setting, in some embodiments, the default mesh is modified by specifying the maximum element size on the electrode surface as one twenty-fifth of the minimal diffusion layer thickness at the highest frequency, to ensure an adaptive and fine mesh based on diffusion coefficient D, as shown in EQ. 8.>

[0093] The study consists of two study steps: First, a stationary study is conducted to get the steady-state solution of ci,0 influenced by the center electrode potential voltage Eapp. Notably, in the case of the ECE and DISP1 mechanism, achieving convergence in the stationary study proves elusive. Consequently, a time-dependent step extending 30 s is employed and the solution at the final time step serves as the initial values for the subsequent study. Second, the frequency domain perturbation study solves the perturbation around the initial concentration values obtained from the preceding study, for a range of applied frequencies. Specifically, 500 frequency values are selected logarithmically equally distributed from 1 Hz to 100 kHz. The outputs are the real and imaginary values over the specified frequency range. A parametric sweep is used to investigate up to 14 different values of the center electrode potential voltage Eapp. -0.3, -0.2, -0.15, -0.1, -0.05, 0, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, and 0.6 V.

[0094] The impedance results are saved in the simulation solution of various embodiments, expressed as conj(tcd.Zvsgrnd_es1), describing the impedance with respect to ground. The real part Re(Z) and imaginary part lm(Z) are derived inside COMSOL software using real(conj(tcd.Zvsgrnd_es1)) and imag(conj(tcd.Zvsgrnd_es1)), respectively. During data preprocessing, the Nyquist-type Cartesian coordinate of Z’ and Z” are calculated via Z’ = Re(Z) and -Z” - -lm(Z). The magnitude (|Z|) is calculated using EQ. 9.

[0095] The phase angle (<p) is calculated using EQ. 10.<EQ 10>

[0096] In the simulation of various embodiments, the transient current is not taken into account. In many embodiments, a stationary study is used to obtain steady-state concentration gradients and then these gradients are used as the initial values for frequency domain perturbation to obtain impedance responses for E, EC, and CE mechanisms, which inherently does not account for transient effects. For ECE and DISP1 mechanisms, a time-dependent study is employed due to the lack of a steady-state solution for the concentration gradient given the electrochemical and chemical processes involved. For the time-dependent simulations, in several embodiments the duration is set to 30s, and the concentration gradient at t = 30 s is used as the initial values for frequency domain perturbation. Any transient currents resulting from the applied potentials are expected to become negligible at steady state or after the first 30 seconds. Consequently, both the stationary and time-dependent studies are designed to capture steady-state or near-steady-state impedance responses, making transient currents insignificant in many embodiments.

[0097] A list of groups of randomly sampled parameters are first generated using Python 3 scripts, where the sampling method for each parameter (linearly or logarithmically) are listed in FIGS. 20A to 20E. Additional sanitization is then conducted before COMSOL simulation to ensure the resulting electrochemical impedancespectroscopy spectra represented the corresponding mechanism and are electrochemically accessible.

[0098] Machine-learning code is implemented on PyCharm IDE using Python 3 scripts. PyTorch machine-learning frameworks are used to implement residual neural networks (ResNet). Scikit-learn frameworks are used to implement supporting vector classifier (SVC). Graphs are generated using the Matplotlib library and the Pyplot module. Each data point is processed using python API OpenCV into a three-dimensional tensor {n, y, m} with a size of {14 x 3 x 500} as an input. In this tensor, n = 14 as it corresponds to the number of Es values; m = 500 as it correlates to the number of f values; y = 3 with the channels store the Es values, as well as {Zy (Es, / ) Z”y(Es, / )}\Z\j (Esj)} for the Cartesian or polar (magnitude-vs-phase change) coordinates, respectively. EIS features were sequenced from the lowest to highest frequency, though the information of the frequency values is not provided to the machine learning model. The trained ResNet model yields an output vector y = {yi, y2, ys, y4, ys} (FIG. 7), in which each yzquantitatively represents the propensity as a surrogate for the statistical probability associated with the respective mechanisms of E, EC, CE, ECE, and DISP1. The input tensor is evaluated using a kernel / filter of size {14 x 3 x 7}. The kernel only views the data present in the tensor, and the filter is not changed with different values of n. In terms of the hyperparameters of the ResNet-18 model, the standard learning rate of 1 x 1Q-3and standard weight decay of 1 x 10-5is used for all training. Training is performed on a machine using an Intel i7-12700H 14-Core CPU with 64 GB of RAM and an Nvidia GeForce RTX 3060 GPU. It takes approximately 24 h to train the ResNet model with 1000 epochs and just seconds to predict the class of one sample. Each individual ResNet model undergoes 1,000 training epochs to ensure enhanced accuracy, with the training dataset consisting of 3,500 shuffled data points for each mechanism type. The test dataset comprises about 1,500 randomly sampled data points for each mechanism category. To establish the ensemble models with instance normalization, some embodiments employ a majority voting approach with eight individual ResNet-18 models by aggregating their results. To evaluate the ensemble model's performance, several embodiments utilize a test dataset consisting of 1 ,000 previously unseen data points foreach mechanism. Each simulated EIS spectra are evaluated by the eight individually trained ResNet-18 models, yielding a set of prediction values (y vectors). These y vectors are subsequently averaged, and the final predictions are determined based on the maximal component in the y vector.DOCTRINE OF EQUIVALENTS

[0099] This description of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form described, and many modifications and variations are possible in light of the teaching above. The embodiments were chosen and described in order to best explain the principles of the invention and its practical applications. This description will enable others skilled in the art to best utilize and practice the invention in various embodiments and with various modifications as are suited to a particular use. The scope of the invention is defined by the following claims.

[0100] As used herein, the singular terms “a,” “an,” and “the,” may include plural referents unless the context clearly dictates otherwise. Reference to an object in the singular is not intended to mean “one and only one” unless explicitly so stated, but rather “one or more.”

[0101] As used herein, the terms “approximately” and “about” are used to describe and account for small variations. When used in conjunction with an event or circumstance, the terms can refer to instances in which the event or circumstance occurs precisely as well as instances in which the event or circumstance occurs to a close approximation. When used in conjunction with a numerical value, the terms can refer to a range of variation of less than or equal to ± 10% of that numerical value, such as less than or equal to ±5%, less than or equal to ±4%, less than or equal to ±3%, less than or equal to ±2%, less than or equal to ±1 %, less than or equal to ±0.5%, less than or equal to ±0.1 %, or less than or equal to ±0.05%.

[0102] Additionally, amounts, ratios, and other numerical values may sometimes be presented herein in a range format. It is to be understood that such range format is used for convenience and brevity and should be understood flexibly to include numerical valuesexplicitly specified as limits of a range, but also to include all individual numerical values or sub-ranges encompassed within that range as if each numerical value and sub-range is explicitly specified. Where ranges are described, the range should be understood to include the endpoints of the ranges, and the endpoints of such ranges are also contemplated to stand on their own as inventive, individual data points and to form the endpoints of other ranges. For example, a ratio in the range of about 1 to about 200 should be understood to include the explicitly recited limits of about 1 and about 200, but also to include individual ratios such as about 2, about 3, and about 4, sub-ranges such as about 1 to about 10, about 10 to about 50, about 20 to about 100, about 100 to about 200, and so forth, and related ranges such as greater than about 1 or less than about 200.

Claims

WHAT IS CLAIMED IS:

1. A method for analyzing electrochemical impedance spectroscopy (EIS) data comprising:providing an EIS data set;representing the EIS data set;transforming the EIS data set into a tensor array;inputting the tensor array into a residual neural network; andreceiving an output from the residual neural network.

2. The method of claim 1, wherein the step of representing the EIS data set further comprises representing the EIS dataset as a Cartesian plot.

3. The method of claim 2, wherein the Cartesian plot comprises a Nyquist plot.

4. The method of claim 1 , wherein the step of representing the EIS data set further comprises representing the EIS data set as a polar plot.

5. The method of claim 4, wherein the polar plot is derived into a magnitude-vs-phase plot.

6. The method of claim 1 , further comprises normalizing the EIS data set.

7. The method of claim 6, wherein the step of normalizing the EIS data set comprises layer normalization.

8. The method of claim 6, wherein the step of normalizing the EIS data set comprises instance normalization.

9. The method of claim 1 , wherein the tensor array comprises 3 elements;wherein a first tensor array element is defined by a number of applied potentials of the EIS data set;wherein a second tensor array element is defined by the representation of the EIS dataset and a value of applied potentials of the EIS data set; and wherein a third tensor array element is defined by a number of frequencies of the EIS data set.

10. The method of claim 1, wherein the residual neural network is trained to classify EIS data based on a mechanism category.

11. The method of claim 10, wherein the mechanism categories comprise of E, EC, CE, ECE, and DISP1.

12. The method of claim 10, wherein the output received yields an output vector based on the mechanism categories.

13. A system for classifying electrochemical impedance spectroscopy (EIS) data, comprising:an EIS data set;a processor programed to condition the EIS data set, wherein conditioning comprises:representing the EIS data set under at least one of a Nyquist Plot and a Bode Plot;normalizing the EIS data set; andtransforming the EIS data set into a tensor array; anda processor programmed to:receive the tensor array;receive a training data set comprising simulated EIS data; andinput the tensor array and the training data set into a machine learning algorithm, wherein the machine learning algorithm is configured to output a classified EIS data set.

14. The system of claim 13, wherein normalizing the EIS data set comprises layer normalizing.

15. The system of claim 13, wherein normalizing the EIS data set comprises instance normalization.

16. The system of claim 13, wherein the tensor array comprises three elements; wherein a first tensor array element is defined by a number of applied potentials of the EIS data set;wherein a second tensor array element is defined by the representation of the EIS data set and a value of applied potentials of the EIS data set; and wherein a third tensor array element is defined by a number of frequencies of the EIS data set.

17. The system of claim 13, wherein the training data set is classified simulated EIS data.

18. The system of claim 17, wherein the classified simulated EIS data is classified into mechanism categories comprising at least one of: E, EC, CE, ECE, and DISP1.

19. The system of claim 13, wherein the classified EIS data set is classified into mechanism categories comprising at least one of: E, EC, CE, ECE, and DISP1.

20. The system of claim 13, wherein the machine learning algorithm is a residual neural network.

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