Method for collaborative verification of material formula sampling distribution quality

Abstract

This invention relates to the fields of materials informatics and data quality control, specifically a collaborative verification method for the quality of chemical formula sampling distributions in materials. Addressing the problems in existing methods for verifying the quality of chemical formula sampling distributions in materials—such as one-sided evaluation dimensions, decoupling from intrinsic performance characteristics, insufficient utilization of historical experience, lack of operability and interpretability of verification results, and neglect of multi-model prediction consistency—this invention achieves comprehensive collaborative evaluation of chemical and performance spaces through a multi-dimensional collaborative evaluation and adaptive intelligent verification framework. This includes adaptive matching of verification standards to performance sensitivity, full activation of historical data value, effective output of transparent diagnostics and precise optimization guidance, and quantitative improvement of multi-model prediction consistency. This provides reliable quality assurance for high-throughput material screening and improves R&D efficiency.

Description

Technical Field

[0001] This invention relates to the fields of materials informatics and data quality control, specifically a collaborative verification method for the quality of material chemical formula sampling distribution. Background Technology

[0002] In the data-driven materials research and development paradigm, high-throughput virtual screening has become a key means to accelerate the discovery of new materials. In this process, the intelligent generation and efficient sampling of material chemical formulas are the starting point for computation. The quality of their distribution directly determines the accuracy of subsequent performance predictions, screening success rate, and R&D costs, playing a central role in materials innovation across multiple fields such as new energy, aerospace, electronic information, and biomedicine. The distribution characteristics of the generated large-scale sample set directly determine the training effect of subsequent performance prediction models, screening success rate, and the efficiency and cost of new material discovery.

[0003] However, current post-hoc quality assessments of sampling distributions are still in their early stages and have the following limitations:

[0004] The evaluation dimensions are singular: it only focuses on the number of samples or simple statistical characteristics, without taking into account the diversity of chemical composition and the quality of performance distribution coverage;

[0005] Disconnected from performance characteristics: The verification standards are static and general, and the evaluation focus is not dynamically adjusted based on performance sensitivity;

[0006] Insufficient use of historical data: The rationality of sampling was not effectively assessed by prior evaluation of historical typical value ranges and high-performance distributions;

[0007] The results are not very instructive: they only output a single judgment or score, and cannot clearly define the dimensions of the problem or the direction of optimization;

[0008] Ignoring multi-model consistency: There is a lack of quantitative assessment of the consistency of the distribution of prediction results from multiple models, making it impossible to warn of model bias.

[0009] Therefore, there is an urgent need for a comprehensive quality verification method that deeply integrates performance sensitivity priors, historical typical value range knowledge, chemical space diversity measurement and multi-model consistency analysis, and can provide intelligent diagnosis and optimization suggestions. Summary of the Invention

[0010] The purpose of this invention is to provide a collaborative verification method for the quality of material chemical formula sampling distribution, so as to solve the problems of existing technologies such as incomplete evaluation system, rigid standards, insufficient knowledge utilization, weak guidance of results, and insufficient consideration of model uncertainty.

[0011] To achieve the above objectives, the present invention provides the following technical solution: a collaborative verification method for the sampling distribution quality of material chemical formulas, comprising the following steps:

[0012] S1. Input data acquisition and preprocessing: Receive performance prediction data of chemical formula sampling results of one or more prediction models for the same batch of materials, and obtain the target performance index and its corresponding performance sensitivity information and historical typical value range [min_val, max_val];

[0013] S2. Multi-dimensional quality score calculation:

[0014] Based on the performance prediction data and the historical typical value range, calculate the historical typical value range coverage score S_range;

[0015] Based on the list of elements used for sampling and the sampling results of the chemical formula of the material, the chemical composition spatial sampling quality score S_sampling is calculated;

[0016] If multiple prediction models exist, the multi-model prediction consistency score S_consistency is calculated based on the performance prediction data of each model.

[0017] S3. Calculation and judgment of the comprehensive quality score of sensitivity internalization: Based on the sensitivity level of the target performance index, dynamically adjust the fusion weight of the chemical composition spatial sampling quality score (S_sampling) and the historical typical value range coverage score (S_range), calculate the comprehensive quality score (Score_combined), compare it with the preset acceptance threshold (T_accept), and output the judgment result.

[0018] S4. Generate a structured verification report and intelligent diagnosis: Based on the scores of each dimension in step S2 and the judgment results in step S3, output a structured report containing core judgment results, details of scores for each dimension, a list of risk factors, confidence levels, and targeted optimization suggestions.

[0019] Preferably, the target performance indicators are quantifiable core characteristic parameters in material development, including but not limited to thermoelectric figure of merit (ZT), power factor (PF), Seebeck coefficient, oxygen evolution onset potential, Tafel slope, melting point (Tm), permittivity, contact resistance (Rc), superconducting critical temperature (SC_copper / SC_traditional), and thermal conductivity.

[0020] The sensitivity information and historical typical value range of the target performance index in step S1 are derived from a pre-set structured knowledge base of material performance sensitivity, which is constructed through the following steps:

[0021] Input a dataset containing the chemical formulas of materials and their historical performance data;

[0022] For each performance metric in the dataset, calculate the multidimensional statistical characteristics of its data;

[0023] By incorporating and correcting the statistical characteristics based on the variability of material composition, a quantitative sensitivity score for the performance index is generated.

[0024] Based on the preset threshold range in which the quantified sensitivity score falls, the sensitivity level of the performance index is determined, including high sensitivity, medium sensitivity, and low sensitivity.

[0025] By statistically analyzing the distribution of historical data for this performance indicator, we can obtain its historical typical value range.

[0026] The output includes structured records containing performance indicator names, quantified sensitivity scores, sensitivity levels, and historical typical value ranges, which are then compiled to form the knowledge base.

[0027] Preferably, the calculation process of the historical typical value range coverage score (S_range) in step S2 includes:

[0028] Baseline Proportional Score: Calculate the proportion of samples whose predicted values ​​fall within the historical typical value range [min_val, max_val] corresponding to the target performance index (P_in_range).

[0029] Uniformity score of distribution within the range: Extract the subset of predicted values ​​(in_range_values) that fall within the range of the historical typical values, and calculate the uniformity score by equal weighting the normalized range, normalized standard deviation and interquartile range expansion.

[0030] Range boundary utilization score: Calculate the proportion of the intersection between the predicted value and the historical typical value range corresponding to the target performance index within the historical typical range;

[0031] Triple weighted fusion: The basic proportion score, uniformity score and boundary utilization score are weighted and fused with a weight of 0.5:0.3:0.2 to obtain the historical typical value range coverage score S_range, that is, S_range=min(1.0,0.5×basic proportion score+0.3×uniformity score within the range+0.2×boundary utilization score within the range).

[0032] Preferably, the formulas for calculating the uniformity score of the normalized range, normalized standard deviation, interquartile range expansion, and their equally weighted average are as follows:

[0033] The formula for calculating the normalized range is:

[0034] `range_norm = min(1.0, value_range / (max_val - min_val))`, where `value_range` is the actual range of the predicted value subset, and its calculation formula is:

[0035] value_range = max(in_range_values) - min(in_range_values), where max(in_range_values) is the maximum value of in_range_values ​​and min(in_range_values) is the minimum value of in_range_values.

[0036] The formula for calculating the normalized standard deviation is:

[0037] std_norm=min(1.0,std_dev / (max_val-min_val)×2.0), where std_dev is the standard deviation of in_range_values ​​and 2.0 is the empirical scaling factor;

[0038] The formula for calculating interquartile range extension is:

[0039] quantile_spread=(Q3-Q1) / (max(in_range_values)-min(in_range_values)), where Q1 and Q3 are the 25th and 75th quantiles of in_range_values, respectively;

[0040] The formula for calculating the uniformity score using equal-weighted average is as follows:

[0041] uniformity_score=(range_norm+std_norm+quantile_spread) / 3.

[0042] Preferably, the calculation process of the chemical composition spatial sampling quality score (S_sampling) in S2 includes:

[0043] Uniformity score of element proportion distribution (S_uniformity): For each element or composite element, analyze its proportion value sequence {r_i} in all chemical formulas, calculate the empirical cumulative distribution function (eCDF) of the sequence, and obtain the uniformity score (1-D) of the element by comparing the empirical cumulative distribution function (eCDF) with the Kolmogorov-Smirnov (KS) statistic D of the uniform distribution theory CDF in the [0,1] interval. The average of the scores of all elements is used to obtain the uniformity score of element proportion distribution S_uniformity.

[0044] Chemical formula pattern coverage score (S_coverage): Set the discretization precision p, discretize the element proportion of each chemical formula into integer coordinate vectors, count the number of unique integer coordinate vectors. For a system with n elements, at precision p, the total number of distinguishable patterns is approximately M_theoretical = (1 / p)^(n-1), calculated as S_coverage=min(1.0, number of unique integer coordinate vectors / (α* M_theoretical)), where α is the expected coverage coefficient less than 1, and n is the number of elements in the element list used for sampling;

[0045] The overall score (S_sampling) is obtained by combining S_uniformity and S_coverage with a weight of 0.7:0.3. The overall score is S_sampling = 0.7 × S_uniformity + 0.3 × S_coverage.

[0046] Preferably, the calculation process of the multi-model prediction consistency score (S_consistency) in step S2 includes:

[0047] Data alignment: Extract the predicted value sequences from each model, unify their length, and perform Z-score standardization;

[0048] Correlation coefficient matrix calculation: Calculate the Pearson correlation coefficients between all pairs of models to form an M×M symmetric matrix, where M is the number of models;

[0049] Consistency score: The average absolute value of the correlation coefficients of the upper triangular part of the matrix (excluding the diagonal) is taken to obtain S_consistency.

[0050] Preferably, the dynamic weight allocation rule in step S3 is as follows:

[0051] When the target performance is high-sensitivity performance, the weight of the historical typical value range coverage score is W_range=0.7, and the weight of the chemical composition spatial sampling quality score is W_sampling=0.3;

[0052] When the target performance is of medium sensitivity, W_range=0.5, W_sampling=0.5;

[0053] When the target performance is low-sensitivity or unknown, W_range=0.3, W_sampling=0.7;

[0054] The formula for calculating the overall quality score is: Score_combined = W_sampling × S_sampling + W_range × S_range;

[0055] The preset acceptance threshold T_accept is set to 0.75.

[0056] Preferably, the confidence level calculation process in step S4 is as follows:

[0057] Confidence score = Score_combined × 0.6 + S_consistency × 0.3 + min(1.0, number of models / 5) × 0.1;

[0058] Confidence level is determined by confidence score range: ≥0.8 is high, [0.6, 0.8) is medium, and <0.6 is low.

[0059] Preferably, the targeted optimization suggestions in step S4 are generated based on risk factors, including:

[0060] If S_range < 0.6 and the target performance index is a highly sensitive performance, it is recommended to adopt a hierarchical clustering sampling strategy and increase the sampling density in the historical high value range corresponding to the target performance index.

[0061] If S_sampling < 0.5, it is recommended to expand the range of element ratio variation or switch to the stratified_clustering sampling strategy;

[0062] If S_consistency < 0.7, it is recommended to check the model differences and use weighted averaging or stacked ensemble to fuse the outputs of multiple models.

[0063] Compared with the prior art, the beneficial effects of the present invention are:

[0064] This invention and method construct a multi-dimensional collaborative, adaptively weighted intelligent verification framework through input data preprocessing, multi-dimensional quality score calculation, performance-sensitivity-driven dynamic weight fusion and quality judgment, and structured verification reports and intelligent diagnostic outputs. This invention achieves comprehensive collaborative evaluation of chemical and performance spaces, adaptive matching of verification standards with performance sensitivity, full activation of historical data value, effective output of transparent diagnostics and precise optimization guidance, and quantitative improvement of multi-model prediction consistency. It deeply integrates with upstream performance quantification and intelligent sampling methods to form a generation-verification-optimization closed loop, significantly improving the reliability and R&D efficiency of high-throughput material screening. It is applicable to chemical sampling quality verification scenarios for multi-component systems and materials containing composite elements. Attached Figure Description

[0065] Figure 1 This is a schematic diagram of the overall process of the collaborative verification method for the chemical formula sampling distribution quality of the materials of the present invention;

[0066] Figure 2This is a schematic diagram of the sub-process for calculating the historical typical value range coverage score (S_range) of this invention;

[0067] Figure 3 This is a schematic diagram of the sub-process for calculating the chemical composition spatial sampling quality score (S_sampling) of the present invention;

[0068] Figure 4 This is a schematic diagram illustrating the sensitivity-driven dynamic weight allocation and comprehensive score fusion of the present invention.

[0069] Figure 5 This is a schematic diagram illustrating the calculation of the multi-model prediction consistency score (S_consistency) of the present invention;

[0070] Figure 6 This is a schematic diagram illustrating the composition of the structured verification report of this invention. Detailed Implementation

[0071] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0072] Currently, the post-hoc quality assessment of sampling distributions is still in its early stages, and mainly suffers from the following limitations:

[0073] The evaluation dimensions are too narrow: existing methods mostly focus on the sufficiency of the number of sampling points or simple statistical characteristics of chemical spatial distribution (such as mean and variance), lacking a systematic framework for synergistic evaluation from the two dimensions of "input space" (chemical composition diversity) and "output space" (predicted performance distribution). The breadth of the sampling results in chemical formula model coverage, the uniformity of elemental proportion distribution, and the coverage quality in historically known high-performance ranges after mapping through downstream models have not been uniformly quantified and weighed.

[0074] Disconnected from performance characteristics: Quality verification standards are typically static and generic, failing to correlate with the "sensitivity" or "robustness" of the target material's performance indicators. For highly sensitive properties such as thermoelectric figure of merit (ZT) and power factor (PF), even minor compositional changes can lead to drastic fluctuations in performance values. Their effective component windows are narrow, thus requiring more concentrated coverage of sampling within the critical performance range. However, existing methods cannot dynamically adjust the evaluation focus based on performance sensitivity, resulting in insufficient quality control of sampling for highly sensitive properties or excessive constraints on sampling for low-sensitivity properties.

[0075] Insufficient utilization of historical data: Materials research and development has accumulated a wealth of historical performance data, which contains common value ranges (typical value ranges) for various performance indicators and prior knowledge of the distribution of high-performance materials. Current validation methods fail to effectively utilize this valuable historical experience as a benchmark to evaluate whether the distribution of performance predictions from new sampling results is reasonable, whether the known high-value ranges have been sufficiently explored, and whether there are omissions of range boundaries.

[0076] The results offer weak guidance: most validation methods output a simple pass / fail judgment or a single score, much like a "black box." Researchers cannot know the specific dimensions of the quality problem (such as whether it is insufficient chemical diversity or inadequate performance coverage), and it is even more difficult to obtain clear and actionable optimization guidance, hindering the rapid iteration and optimization of sampling strategies.

[0077] Ignoring multi-model consistency: To improve reliability, high-throughput computing often employs multiple prediction models for cross-validation or ensemble prediction. If different models show significant distributional differences or contradictory trends in their predictions for the same sample set, it will severely impact the confidence level of the overall result. Current technologies lack quantitative evaluation metrics for the distributional consistency of multi-model prediction results, making it impossible to identify and warn of systematic biases between models.

[0078] This invention proposes a comprehensive quality verification method for materials chemical formula sampling results. This method deeply integrates prior knowledge of performance sensitivity, historical typical value ranges, chemical spatial diversity measurements, and multi-model consistency analysis, and provides intelligent diagnostic and optimization suggestions. Its core breakthrough lies in transforming the previously isolated and subjective quality inspection into a quantitative evaluation system deeply integrated with the intrinsic properties of materials (sensitivity), historical experience (typical value ranges), and the sampling process itself (chemical diversity). Through close integration with upstream "performance sensitivity quantification methods" and "intelligent sampling generation methods," an automated, adaptive, and interpretable verification closed loop is constructed, providing crucial quality assurance and decision support for high-throughput materials computing processes.

[0079] Please see Figure 1-6 This invention provides a technical solution: a collaborative verification method for the sampling distribution quality of material chemical formulas, comprising the following steps:

[0080] S1. Input Data Acquisition and Preprocessing: Receive performance prediction data from one or more prediction models for the same batch of material chemical formula sampling results (generated by the "Method for Generating Material Chemical Formulas Based on Performance Sensitivity Adaptive Hierarchical Sampling" applied for on 2026-03-13 with application number 202610303873.2). This data is typically stored in DataFrame format, containing a `formula` column and at least one performance prediction value column. Also, acquire and input the target performance index and its corresponding performance sensitivity information, historical typical value range [min_val, max_val], i.e., the list of elements used for sampling. The target performance index in step S1 is a quantifiable core characteristic parameter in material development, including but not limited to thermoelectric figure of merit (ZT), power factor (PF), Seebeck coefficient, oxygen evolution onset potential (onset), Tafel slope, melting point (Tm), dielectric constant (permittivity), contact resistance (Rc), superconducting critical temperature (SC_copper / SC_traditional), and thermal conductivity (Thermal_Conductivity).

[0081] The sensitivity information and historical typical value range of the target performance index are derived from a pre-defined structured knowledge base for material performance sensitivity. This knowledge base (generated by the application "Quantification and Grading Method of Material Performance Sensitivity Based on Multidimensional Statistical Analysis" filed on 2026-01-23 with application number 2026100902412) is constructed through the following steps:

[0082] Input a dataset containing the chemical formulas of materials and their historical performance data;

[0083] For each performance metric in the dataset, calculate the multidimensional statistical characteristics of the data, including the coefficient of variation, quantile range ratio, absolute skewness, absolute kurtosis, and outliers of the performance data.

[0084] By incorporating and correcting the statistical characteristics based on the variability of material composition, a quantitative sensitivity score for the performance index is generated.

[0085] Based on the preset threshold range in which the quantified sensitivity score falls, the sensitivity level of the performance index is determined, including high sensitivity, medium sensitivity, and low sensitivity.

[0086] By statistically analyzing the distribution of historical data for this performance indicator, we can obtain its historical typical value range.

[0087] The output includes structured records containing performance indicator names, quantified sensitivity scores, sensitivity levels, and historical typical value ranges, which are then compiled to form the knowledge base.

[0088] A list of elements used for sampling, such as ["Cu","Fe","AlO3"], is used to assess the diversity of the chemical composition space.

[0089] S2. Multi-dimensional quality score calculation: Parallel calculation of quality scores for three core dimensions, including historical typical value range coverage score (S_range), chemical composition spatial sampling quality score (S_sampling), and multi-model prediction consistency score (S_consistency).

[0090] The evaluation sampling results of the historical typical value range coverage score (S_range) are obtained by predicting the performance value distribution through downstream models. It assesses the reasonableness and distribution quality of coverage within the known historical typical range, rather than simply the "proportion within the range." The calculation process includes:

[0091] Basic Proportion Score: Calculates the proportion (P_in_range) of samples whose predicted values ​​fall within the historical typical value range [min_val, max_val] corresponding to the target performance index. This directly reflects the degree of overlap between the prediction results and the historical experience range, and is the most direct indicator of "whether it hits the experience range". Backtracking experiments were conducted on more than 30 sets of real material datasets covering thermoelectric materials, perovskites, high-entropy alloys, etc. This indicator showed the highest consistency with the distribution of historical high-performance samples within the typical value range (correlation coefficient 0.88).

[0092] Uniformity score within the range: For predicted values ​​falling within the typical value range, a distribution quality score within that range is calculated. Specifically, a subset of predicted values ​​falling within the historical typical value range (in_range_values) is extracted, and a uniformity score is calculated by equally weighted averaging of the normalized range, normalized standard deviation, and interquartile range expansion. This encourages uniform exploration within the effective range and avoids over-concentration of predicted values ​​in a narrow sub-interval within the typical range, ensuring the breadth of exploration within the effective range. In actual sampling, if predicted values ​​are concentrated only in the middle of the typical range, although the basic coverage rate is met, high-value areas at the boundary may be missed. This indicator encourages uniform exploration and improves the robustness of subsequent screening.

[0093] The formulas for calculating the uniformity score using the normalized range, normalized standard deviation, interquartile range expansion, and their equally weighted average are as follows:

[0094] The formula for calculating the normalized range is:

[0095] `range_norm = min(1.0, value_range / (max_val - min_val))`, where `value_range` is the actual range of the predicted value subset, and its calculation formula is:

[0096] value_range = max(in_range_values) - min(in_range_values), where max(in_range_values) is the maximum value of in_range_values ​​and min(in_range_values) is the minimum value of in_range_values.

[0097] The formula for calculating the normalized standard deviation is:

[0098] std_norm=min(1.0,std_dev / (max_val-min_val)×2.0), where std_dev is the standard deviation of in_range_values, and 2.0 is an empirical scaling factor. The scaling factor of 2.0 is obtained through statistical optimization of more than 30 sets of data. This normalization process ensures that the standard deviations of performance indicators of different dimensions are comparable.

[0099] The formula for calculating interquartile range extension is:

[0100] quantile_spread=(Q3-Q1) / (max(in_range_values)-min(in_range_values)), where Q1 and Q3 are the 25th and 75th quantiles of in_range_values, respectively. This value is already in the range [0,1]. The closer the value is to 0.5, the more uniform the distribution. The closer the value is to 0 or 1, the more skewed the distribution is to one side.

[0101] The formula for calculating the uniformity score using equal-weighted average is as follows:

[0102] uniformity_score=(range_norm+std_norm+quantile_spread) / 3;

[0103] The use of equal weights (1:1:1) instead of weighted fusion is because these three metrics characterize distribution uniformity from different dimensions: range_norm focuses on the overall coverage breadth, std_norm focuses on the numerical dispersion, and quantile_spread focuses on the central distribution pattern. In tests on 30 material datasets containing different distribution patterns (skewed, concentrated, and uniform), the equal weight scheme performed best in distinguishing between "surface uniformity" and "true uniformity" distributions (with an F1 score of 0.85), outperforming other weighted schemes (such as the 0.5:0.3:0.2 scheme, which had an F1 score of 0.76).

[0104] This scoring method encourages predicted values ​​to be evenly distributed within a typical range, avoiding excessive concentration in a narrow sub-interval. Through triple normalization, it ensures that the distribution of predicted values ​​for different material systems and different performance dimensions can be compared fairly, significantly improving the robustness and interpretability of the uniformity assessment.

[0105] Boundary Utilization Score: This score calculates the proportion of the intersection between the predicted value and the historical typical value range corresponding to the target performance index within that historical typical range. It encourages full exploration of the complete historical value range and avoids the "range shrinkage" phenomenon caused by model bias or insufficient sampling. This score serves as a supplement, ensuring that potential high-performance materials at the boundary are not overlooked due to excessive focus on the central region. Specifically, it uses the proportion of the intersection between the predicted value and the typical range within the typical range, with an upper limit of 1. The calculation method is as follows:

[0106] Actual coverage ratio = min(1.0, (min(max_val, maximum predicted value) - max(min_val, minimum predicted value)) / (max_val - min_val));

[0107] This scoring system encourages the prediction distribution to fully explore the complete range of historically known values ​​and avoid missing boundary areas. Through testing with 30 sets of synthetic and real data containing different distribution patterns (concentrated, uniform, and missing boundaries), the current weight combination (0.5:0.3:0.2) performs best in terms of the overall score and the matching degree with the preset ideal distribution model (Kappa coefficient reaches 0.82).

[0108] Compared to other weight combinations (such as 0.6:0.2:0.2 or 0.4:0.4:0.2), this combination has the best sensitivity in distinguishing between "surface coverage" and "depth coverage";

[0109] Triple weighted fusion: The basic proportion score, uniformity score, and boundary utilization score are weighted and fused with a weight of 0.5:0.3:0.2 to obtain the historical typical value range coverage score S_range, i.e., S_range=min(1.0,0.5×basic proportion score+0.3×uniformity score within the range+0.2×boundary utilization score within the range). This weight combination was tested with 30 sets of data with different distribution patterns and performed best in the consistency evaluation between the comprehensive score and the prior knowledge of the distribution of historical high-performance materials (Kappa coefficient reached 0.82).

[0110] The calculation process for the spatial sampling quality score (S_sampling) of chemical composition includes:

[0111] Elemental distribution uniformity score (S_uniformity):

[0112] Proportion Extraction: For each input element or composite element, for each basic element or composite element to be evaluated (such as AlO3), the atomic proportion sequence {r_i} of that element is resolved from all N chemical formulas, where i=1,2,...,N (N is the number of samples). To ensure comparability, the sum of the element proportions of each chemical formula has been pre-normalized to 1.

[0113] Ideal distribution reference: "Ideal uniform distribution" is defined as a uniform distribution within the possible proportional value range of an element, usually within [0,1].

[0114] Distribution comparison and scoring:

[0115] 1) Calculation of the empirical cumulative distribution function (eCDF): Arrange the sequence {r_i} in ascending order to obtain the ordered sequence {r_(i)}. For any proportion x, its eCDF value Fn(x) is defined as: Fn(x) = (number of samples in the sequence less than or equal to x) / N.

[0116] 2) Theoretical CDF: The theoretical CDF uniformly distributed in the interval [0,1] is F_uniform(x)=x.

[0117] 3) Goodness-of-fit measure: The complement of the Kolmogorov-Smirnov (KS) statistic is used as the uniformity score. The KS statistic D is the maximum absolute deviation between the eCDF and the theoretical CDF in the domain [0,1], i.e., D = sup_{x∈[0,1]} | Fn(x) - F_uniform(x) |, which measures the maximum deviation between the current distribution and the uniform distribution;

[0118] In actual calculation, it is only necessary to calculate the difference at each sorted data point r_(i) and its left neighbor (r_(i-) = r_(i) - ε, where ε is a very small positive number):

[0119] diff1_i = | (i / N) - r_(i) |

[0120] diff2_i = | ((i-1) / N) - r_(i) |

[0121] Iterate through all i, and take the maximum value among all diff1_i and diff2_i, which is the D value of that element.

[0122] 4) Uniformity score: elem_uniformity_score = 1 - D. The range of D is [0,1], and the smaller the value, the closer it is to a uniform distribution. Therefore, the closer elem_uniformity_score is to 1, the more uniform the proportion of the element.

[0123] Overall uniformity score: The average of the elem_uniformity_scores for all elements is used to obtain the final element proportion distribution uniformity score S_uniformity.

[0124] The uniformity score for elemental distribution directly assesses whether the distribution shape is uniform, rather than the magnitude of the statistical measure. Even if the values ​​of the statistical measure (such as range and standard deviation) are different, a high score can be obtained as long as the distribution shape is close to uniform. This is more in line with the original intention that "high-quality sampling should broadly and uniformly explore the chemical space."

[0125] Chemical formula model coverage score (S_coverage):

[0126] "Pattern" definition: We define a "pattern" as the discrete coordinates of a chemical formula on a simplified scale grid. For example, the scale value of each element is rounded to a specified precision (such as 0.01), and the chemical formula is then represented as an integer coordinate vector at that precision.

[0127] Pattern extraction: For each chemical formula, the proportions of its elements are discretized with the precision described above to generate an integer coordinate vector (i.e., the "pattern").

[0128] Coverage assessment:

[0129] 1) Theoretical coverage space: For a system with n elements, at a precision of p, the total number of distinguishable modes is approximately M_theoretical = (1 / p)^(n-1).

[0130] 2) Actual coverage ratio: The number of unique patterns N_unique_patterns that appear in the statistical sampling results.

[0131] 3) Coverage score:

[0132] S_coverage=min(1.0,N_unique_patterns / (α* M_theoretical))

[0133] Here, α is the expected coverage coefficient less than 1, representing the expectation that the sampling can cover a certain proportion (e.g., 10%) of significantly different patterns in the theoretical space. The default value of α is 0.1; when the number of sampled elements n≥5, α is 0.05; when n=2-4, α is 0.1; when n=1, α is 0.2. The value is determined by the fact that the more elements there are, the higher the dimension of the chemical space, the more theoretical patterns there are, and the lower the expected coverage ratio. This indicates that we do not expect to cover the entire theoretical space (exponentially), but rather a reasonable subset. This score measures the breadth of the sampling's exploration of the discretized chemical space.

[0134] The chemical pattern coverage score assesses the dispersion of sampling points on a discretized grid, and is more indicative of whether the sampling covers a sufficient number of "distinctive" chemical composition regions.

[0135] The overall score (S_sampling) is obtained by fusing the elemental distribution uniformity score (S_uniformity) and the chemical formula pattern coverage score (S_coverage) according to a preset weight (e.g., 0.7:0.3). The calculation formula is as follows:

[0136] S_sampling=0.7×S_uniformity+0.3×S_coverage

[0137] Uniformity (S_uniformity) better reflects the quality of exploration in continuous space, and is therefore given a higher weight (0.7); coverage (S_coverage) assesses the exploration range in discrete space and serves as an important supplement (0.3).

[0138] The calculation process for the multi-model prediction consistency score (S_consistency) includes:

[0139] Data alignment: Extract the performance prediction sequence of each model for the same batch of samples. To ensure fair comparison, all sequences are truncated or padded to the same length (usually the minimum length or the first N samples, such as N=10000), and Z-score normalization is performed to eliminate dimensional differences.

[0140] Correlation coefficient matrix calculation: Calculate the Pearson correlation coefficients between all pairs of models to form an M×M symmetric matrix, where M is the number of models;

[0141] Consistency score: The average absolute value of the correlation coefficients in the upper triangular part of the matrix (excluding the diagonal) is taken as S_consistency, which serves as a quantitative indicator of the consistency of the predicted distributions among models. High consistency indicates that different models have similar response trends to the chemical space, and the results are more reliable;

[0142] This step uses absolute value averaging instead of direct averaging because it focuses on whether the trends are consistent, regardless of whether the correlation is positive or negative (negative correlation may indicate a systematic opposite bias in the model). In comparative experiments involving more than five typical models, including our self-developed material performance prediction model and commonly used models (such as random forests, gradient boosting, and neural networks), this method more sensitively detects inconsistencies between models. Its recall rate for detecting the "model bias group" reached 0.91, higher than the 0.78 of the direct averaging method. In practical applications, this method is suitable for consistency evaluation of two or more models; this embodiment uses only two models (Model_A and Model_B) as an example for illustration.

[0143] S3. Calculation and Judgment of Comprehensive Quality Score for Sensitivity Internalization: Based on the sensitivity level of the target performance index, the fusion weights of the chemical composition spatial sampling quality score (S_sampling) and the historical typical value range coverage score (S_range) are dynamically adjusted to calculate the comprehensive quality score (Score_combined), which is then compared with a preset acceptance threshold (T_accept). The judgment result is output. The multi-model prediction consistency score (S_consistency) is used as an independent confidence index and does not participate in the fusion calculation of the comprehensive quality score (Score_combined) to avoid masking the core contradiction between sampling quality and range coverage. S_consistency will be used specifically for subsequent overall confidence level assessment, and the dynamic weight allocation rule is as follows:

[0144] When the target performance is a high-sensitivity property, the weighting for historical typical value range coverage (W_range) is 0.7, and the weighting for chemical composition spatial sampling quality (W_sampling) is 0.3. This weighting is because high-sensitivity properties are extremely sensitive to compositional changes; even small changes can lead to significant fluctuations in performance values, significantly impacting the success or failure of material development. The effective performance window is also narrow, necessitating that sampling fully cover the historical high-value range. If S_range is insufficient, the accuracy of subsequent screening will significantly decrease. In the case of predicting ZT values ​​for thermoelectric materials, when S_range < 0.6, the subsequent screening accuracy for high-sensitivity properties drops below 40%, significantly lower than that for medium-sensitivity properties (65%).

[0145] When the target performance is of medium sensitivity, W_range=0.5 and W_sampling=0.5. The reason for the weighting is to achieve a balance between the breadth of exploration and the depth of focus, which is applicable to most material performance indicators.

[0146] When the target performance is low-sensitivity or unknown performance, W_range = 0.3, W_sampling = 0.7. The reason for the weight allocation is to prioritize ensuring the diversity of the chemical space. Low-sensitivity performance is insensitive to compositional changes, and chemical diversity is the fundamental guarantee for discovering new materials. Therefore, the breadth of sampling exploration is emphasized first;

[0147] The formula for the comprehensive quality score is: Score_combined = W_sampling × S_sampling + W_range × S_range;

[0148] The value of the preset acceptance threshold T_accept is 0.75, and this threshold represents a general high standard requirement for the quality of the material sampling distribution.

[0149] If Score_combined ≥ T_accept, it is judged as "acceptable".

[0150] If Score_combined < T_accept, it is judged as "needs optimization";

[0151] Among them, the above dynamic weight allocation rule is determined through the following steps:

[0152] Multi-system retrospective experiment: Select dozens of historical R & D data sets built by ourselves, covering metal alloys, oxides, thermoelectric materials, etc.

[0153] Grid search optimization: Test all weight combinations in the interval [0.1, 0.9] with a step size of 0.1. [[ID=2…]] …

[0154] Evaluation index: Take the correlation (Pearson_r) between the comprehensive score and the success rate of subsequent experimental verification as the optimization goal.

[0155] Optimal combination: The current weight combination achieves the highest correlation (r = 0.78) on the test set, significantly better than the fixed weight strategy (r = 0.52).

[0156] Judgment logic rationality: This judgment logic ensures that regardless of the performance, only when its sampling distribution reaches a high standard in the comprehensive evaluation after sensitivity weighting will it be accepted, making the judgment result always closely related to the objective quality of the sampling set.

[0157] S4. Generate a structured verification report and intelligent diagnosis: Based on the scores of each dimension in step S2 and the judgment results in step S3, output including:

[0158] Core judgment results: Final judgment ("acceptable" / "needs optimization"), comprehensive quality score (Score_combined), fixed acceptance threshold (T_accept) used;

[0159] Detailed scoring for each dimension: detailed scores for each dimension (S_range, S_sampling, S_consistency), and the weights used during fusion;

[0160] Risk Factor List: A risk factor list is automatically generated by comparing the scores of each dimension with preset sub-thresholds (e.g., S_range<0.6, S_sampling<0.5, S_consistency<0.7). Examples include: "Low consistency between Model A and Model B prediction results," and "Insufficient coverage of predicted values ​​within the typical ZT range."

[0161] Confidence level: Based on the combined quality score (Score_combined), model consistency score (S_consistency), and number of models, a confidence level (high / medium / low) is calculated to characterize the reliability of the final judgment result.

[0162] The confidence level is calculated as follows:

[0163] Confidence score = Score_combined × 0.6 + S_consistency × 0.3 + min(1.0, number of models / 5) × 0.1;

[0164] This formula emphasizes the dominant role of overall quality (0.6), while also considering the consistency between models (0.3) and the robustness brought by a certain degree of model redundancy (0.1). The "number of models" term in the formula has been normalized and is applicable to cases with two or more models. The confidence level is output based on the interval of the confidence score (≥0.8 is high, [0.6,0.8) is medium, <0.6 is low).

[0165] A structured report with targeted optimization suggestions: Based on identified risk factors, specific and actionable suggestions are generated from a pre-defined optimization strategy library. This optimization strategy library integrates the adaptive sampling mechanism from "An Efficient Generation Method of Material Chemical Formulas Based on Performance Sensitivity Adaptive Hierarchical Sampling" and combines it with the performance sensitivity knowledge output from "A Quantification and Grading Method for Material Performance Sensitivity Based on Multidimensional Statistical Analysis," forming a closed loop of "generation—verification—optimization." For example:

[0166] If the risk is "insufficient sampling diversity":

[0167] We recommend performing one or a combination of the following actions:

[0168] Adjust the sampling strategy: If the current strategy is dirichlet_optimized or exact_adaptive, you can switch to the stratified_clustering strategy (refer to the patent "A method for generating material chemical formulas based on performance-sensitive adaptive hierarchical sampling") to enhance feature space coverage;

[0169] Increase the base sampling quantity: According to the patent "Method for generating material chemical formula based on performance sensitivity adaptive hierarchical sampling", the base_samples value can be appropriately increased, or the complexity factor can be adjusted.

[0170] Expand the range of ratio variations: relax the lower limit of the ratio during the chemical formula generation stage (e.g., adjust from 0.001 to 0.0005) to increase the exploration of extreme ratio combinations.

[0171] If the risk is "low coverage of typical range and high sensitivity of target performance":

[0172] Prioritize hierarchical clustering sampling: ensure the use of stratified_clustering in high-performance sensitivity scenarios (refer to the patent "A method for generating material chemical formulas based on performance-sensitivity adaptive hierarchical sampling").

[0173] Dynamically increase sampling density: Based on the patent "Method for generating material chemical formulas based on performance sensitivity adaptive hierarchical sampling", the target sampling amount is increased by using sensitivity_factor (e.g., factor 1.5 for high sensitivity), and the number of clusters is increased preferentially within the historical high value range (from the typical_range output by the patent "Quantification and Grading Method of Material Performance Sensitivity Based on Multidimensional Statistical Analysis").

[0174] Calibrate the prediction model: Check the deviation of the prediction model within the historical typical value range. Simply rejecting the model results will lead to data waste. Local model calibration or integration of multiple models can be performed for this performance range to improve the prediction reliability of this range.

[0175] Feedback to the sampling strategy: Record the uncovered performance intervals as prior constraints for the next round of sampling (implementation method, based on the patent "Method for Generating Material Chemical Formulas Based on Performance Sensitivity Adaptive Hierarchical Sampling", such as increasing the weight of the performance values ​​corresponding to the uncovered intervals in hierarchical clustering sampling; adjusting the α parameter in Dirichlet optimization sampling to make the generation ratio more inclined to chemical formula combinations that can fall into the uncovered intervals; and dynamically increasing the local sampling density according to the degree of missing information for the uncovered intervals) to achieve iterative optimization.

[0176] If the risk is "poor model consistency":

[0177] Multi-model comparative analysis: Examine the differences between models in terms of training data, feature engineering, hyperparameters, etc.

[0178] 2) Model ensemble recommendations: If the consistency score S_consistency < 0.7, it is recommended to use weighted averaging, stacked ensemble, or Bayesian model averaging to fuse the outputs of multiple models;

[0179] 3) Recalibrate the model: Perform local calibration on models with large deviations, especially aligning performance values ​​within the range of historical typical values;

[0180] 4) Feedback to the data level: If multiple models show consistently poor performance in a specific chemical formula region, it may indicate that the data in that region is scarce. It is recommended to increase the density of that region in subsequent sampling.

[0181] Report formatting: The final output is a structured JSON or similar document, which is easy to integrate into the automated materials R&D pipeline or for R&D personnel to view directly.

[0182] Example

[0183] ZT performance sampling distribution quality verification of Cu-Fe-AlO3 thermoelectric material system

[0184] 1) Input data preparation:

[0185] Sampling result prediction data:

[0186] Two different machine learning models (Model_A, Model_B) predict the ZT values ​​of the same batch of material chemical formulas (approximately 26,982, see Sampling Strategy Patent Implementation Case 1) generated by the "Efficient Generation Method of Material Chemical Formulas Based on Performance Sensitivity Adaptive Hierarchical Sampling". The results are stored as two DataFrames, named df_model_a and df_model_b.

[0187] df_model_a contains the formula column and the ZT_pred_A column (the predicted value of Model_A).

[0188] df_model_b contains the formula column and the ZT_pred_B column (the predicted value of Model_B).

[0189] Example data (first 3 rows):

[0190] # df_model_a

[0191] formula, ZT_pred_A

[0192] Cu 0.250, Fe 0.250, AlO 3 0.500, 0.48

[0193] Cu 0.100, Fe 0.700, AlO3 0.200, 0.15

[0194] Cu0.333Fe0.333AlO30.334, 0.68

[0195] # df_model_b

[0196] formula, ZT_pred_B

[0197] Cu 0.250, Fe 0.250, AlO 3 0.500, 0.42

[0198] Cu 0.100, Fe 0.700, AlO3 0.200, 0.12

[0199] Cu0.333Fe0.333AlO30.334, 0.65

[0200] Performance context information:

[0201] The target performance index is "ZT". This was obtained from the performance sensitivity knowledge base generated by the paper "Quantification and Grading Method of Material Performance Sensitivity Based on Multidimensional Statistical Analysis".

[0202] performance_patterns = {

[0203] 'ZT': {

[0204] 'typical_range': (0.01, 2.5), # Historical typical range

[0205] 'sensitivity': 'high', # Sensitivity level is "high"

[0206] 'score': 0.737 # Quantization sensitivity score

[0207] }

[0208] }

[0209] List of sampled elements:

[0210] ["Cu", "Fe", "AlO3"], which is consistent with the input of Implementation Case 1 in the sampling strategy patent.

[0211] Organize the data into the validation function input format:

[0212] results = {

[0213] "Model_A": df_model_a,

[0214] "Model_B": df_model_b

[0215] }

[0216] elements = ["Cu", "Fe", "AlO3"]

[0217] performance_target = "ZT"

[0218] Detailed explanation of the execution and calculation process:

[0219] Step S1: Input Data Acquisition and Preprocessing

[0220] The system receives results, elements, performance_target, and performance_patterns as input.

[0221] Step S2: Calculation of multi-dimensional quality scores

[0222] S2.1 Calculation of Historical Typical Value Range Coverage Score (S_range)

[0223] For Model_A:

[0224] Baseline Proportion Score: This score calculates the percentage of samples whose values ​​in column ZT_pred_A fall within the range of (0.01, 2.5). Of the 26,982 predicted values, 23,509 fall within this range.

[0225] P_in_range_A = 23509 / 26982 ≈ 0.871

[0226] Uniformity of distribution within the range:

[0227] Extract the 23509 values ​​that fall within the range in_range_values_A.

[0228] Normalized range calculation:

[0229] The historical typical value range width is: max_val - min_val = 2.5 - 0.01 = 2.49.

[0230] The actual range of the predicted value subset: value_range_A = max(in_range_values_A) - min(in_range_values_A) = 2.43

[0231] Normalized range: range_norm_A = min(1.0, value_range_A / (max_val - min_val)) = min(1.0, 2.43 / 2.49) ≈ min(1.0, 0.976) = 0.976

[0232] Calculation of normalized standard deviation:

[0233] The standard deviation of the subset of predicted values, std_dev_A, is 0.58.

[0234] Based on statistical analysis of historical datasets, an empirical scaling factor of 2.0 was determined. Normalization was then performed using this empirical scaling factor: std_norm_A = min(1.0, std_dev_A / (max_val - min_val) * scaling factor) = min(1.0, 0.58 / 2.49 * 2.0) ≈ min(1.0, 0.466) = 0.466

[0235] Interquartile range expansion calculation:

[0236] The calculated Q1 and Q3 are the 25th and 75th quantiles, respectively. Therefore, the interquartile range (quantile_spread_A) = (Q3_A - Q1_A) / value_range_A = 0.42

[0237] Evenness score calculation:

[0238] Take the average of the three indicators with equal weights:

[0239] uniformity_score_A = (range_norm_A + std_norm_A + quantile_spread_A) / 3 = (0.976 + 0.466 + 0.42) / 3 ≈ 1.862 / 3 ≈ 0.621

[0240] Scope boundary utilization score:

[0241] The minimum value of all ZT_pred_A is pred_min_A = 0.005, and the maximum value is pred_max_A = 2.8.

[0242] The lower bound of the intersection with the typical range: max(0.01, 0.005) = 0.01

[0243] Upper bound of the intersection with the typical range: min(2.5, 2.8) = 2.5

[0244] Actual coverage ratio = (2.5 - 0.01) / (2.5 - 0.01) = 1.0

[0245] Boundary utilization score = min(1.0, actual coverage percentage) = 1.0

[0246] Triple-weighted fusion (using preset weights of 0.5 : 0.3 : 0.2):

[0247] S_range_A = min(1.0, 0.5*0.871 + 0.3*0.621 + 0.2*1.0) = min(1.0,0.4355 + 0.1863 + 0.2) = min(1.0, 0.8218) = 0.8218

[0248] Similarly, Model_B is calculated, resulting in S_range_B = 0.762.

[0249] S2.2 Calculation of Spatial Sampling Quality Score (S_sampling) for Chemical Composition

[0250] An evaluation was conducted on the same batch of 26,982 unique chemical formulas (a list of proportional combinations of the three components Cu, Fe, and AlO3) generated by the "Generation Method" patent:

[0251] 1. Analysis of the uniformity of element (component) proportion distribution (S_uniformity):

[0252] Constructing the empirical cumulative distribution function (eCDF): For each component (Cu, Fe, AlO3), construct the eCDF based on its proportion sequence. Let the proportion sequence be {ri}, with a total of N = 26982 samples. The eCDF is defined as:

[0253]

[0254] Calculation steps (taking Cu as an example):

[0255] 1) Sorting: Sort the Cu ratio sequence in ascending order to obtain an ordered array:

[0256] r(1)≤r(2)≤⋯≤r(N)r(1)≤r(2)≤⋯≤r(N)

[0257] 2) Constructing the eCDF point sequence: For the sorted i-th value r(i), the corresponding cumulative probability is:

[0258]

[0259] 3) Obtain the point-pair sequence:

[0260]

[0261] Calculate the Kolmogorov-Smirnov (KS) statistic:

[0262] Compare the maximum absolute deviation of eCDF Fn(x) and the theoretical uniform distribution CDF Funiform(x) = x on the interval [0,1]:

[0263]

[0264] In actual calculations, it is only necessary to examine the difference between the sorting point and its left neighbor:

[0265] At data point r(i):

[0266]

[0267] In the left neighborhood of the data point (i.e., between r(i) and r(i−1), r(i) can be approximated as the point of consideration):

[0268]

[0269] Iterate through all values ​​of i, and take the maximum value among all diff1 and diff2; this is DCu. The calculated value is DCu = 0.12.

[0270] Calculate the uniformity score:

[0271] elem_uniformity_scoreCu = 1 − DCu = 0.88

[0272] Similarly, calculate the other components:

[0273] Fe: DFe=0.15, score=0.85

[0274] AlO3: DAlO3=0.10, score=0.90

[0275] Overall uniformity score:

[0276]

[0277] Chemical formula pattern coverage analysis (S_coverage):

[0278] The discretization precision is set to p = 0.01 (this precision is used to discretize the continuous scale space of chemical formulas (the precision of chemical formulas sampled in the "Generation Method" is 0.001) into a statistically significant grid to evaluate the macroscopic spatial distribution of the sampling points).

[0279] The number of elements (composite elements) in the system is n = 3 (Cu, Fe, AlO3);

[0280] Theoretical model number estimate: M_theoretical = (1 / 0.01)^(3-1) = 100^2 = 10,000

[0281] Discretizing the 26,982 chemical formulas with a precision of 0.01, the number of unique patterns obtained is: N_unique_patterns = 1850

[0282] The expected coverage ratio coefficient α is set to 0.1, representing the theoretically significant pattern with an expected coverage of 10%.

[0283] Coverage score: S_coverage = min(1.0, 1850 / (0.1 * 10000)) = min(1.0,1850 / 1000) = min(1.0, 1.85) = 1.0.

[0284] (Note: In this example, the expected coverage ratio coefficient α = 0.1, representing 10% of the theoretically distinguishable patterns (M_theoretical) expected to be covered, i.e., 0.1 * 10000 = 1000 patterns. The actual number of unique patterns covered (N_unique_patterns) is 1850, exceeding this expected value, therefore the coverage score S_coverage is a perfect score of 1.0.)

[0285] Overall score calculation:

[0286] S_sampling = 0.7 × 0.8767 + 0.3 × 1.0 = 0.6137 + 0.3 = 0.9137 ≈ 0.914.

[0287] S2.3 Calculation of Multi-Model Prediction Consistency Score (S_consistency)

[0288] Data alignment: All samples (N=26982) of the predicted values ​​of Model_A and Model_B are aligned.

[0289] Calculate the correlation coefficient: Calculate the Pearson correlation coefficient between the two sequences ZT_pred_A and ZT_pred_B, both of length 26982, which is r = 0.92.

[0290] Consistency score: Due to the use of absolute value averaging logic, for the two models, S_consistency =|r| = 0.92.

[0291] Step S3: Calculation and determination of the overall quality score for sensitivity internalization

[0292] S3.1 Dynamic Fusion of Sensitivity Perception

[0293] The sensitivity level of the target performance ZT is "high", with the corresponding weights: W_range = 0.7, W_sampling = 0.3.

[0294] Calculate the overall score for each model:

[0295] Model_A: Score_combined_A = 0.3 * 0.914 + 0.7 * 0.8218 = 0.2742 +0.63665 = 0.91085

[0296] Model_B: Score_combined_B = 0.3 * 0.914 + 0.7 * 0.762 = 0.2742 +0.6104 = 0.8846

[0297] Average combined score: Score_combined_avg = (0.91085 + 0.8846) / 2 = 0.897725 ≈ 0.898

[0298] S3.2 Quality Judgment

[0299] Set a fixed high-quality acceptance threshold T_accept = 0.75.

[0300] Judgment: Score_combined_avg (0.898) >= T_accept (0.75) → “Acceptable”.

[0301] Step S4: Generate a structured validation report and intelligent diagnostics

[0302] Core result: Determined as "acceptable", with an overall quality score of 0.898.

[0303] Dimension details:

[0304] Average coverage score: (0.8218 + 0.762) / 2 = 0.7919

[0305] Sampling quality score: 0.914

[0306] Consistency score: 0.92

[0307] Weights used: W_sampling=0.3, W_range=0.7

[0308] Risk factor identification: Compare each score with a preset sub-threshold.

[0309] S_sampling (0.914) >= 0.5 → No risk.

[0310] S_range_avg (0.7919) >= 0.6 → No risk.

[0311] S_consistency (0.92) >= 0.7 → No risk.

[0312] List of risk factors: [] (empty).

[0313] Confidence level assessment:

[0314] Number of models: 2.

[0315] Confidence score = 0.898 * 0.6 + 0.92 * 0.3 + min(1.0, 2 / 5) * 0.1 = 0.5388 + 0.276 + 0.04 = 0.8548

[0316] A confidence score of 0.855 >= 0.8 indicates a "high" confidence level.

[0317] Optimization suggestion generation: Since there are no risk factors, general positive suggestions are generated.

[0318] Structured report output (JSON format summary):

[0319] {

[0320] "overall_verdict": "Acceptable",

[0321] "overall_combined_score": 0.898,

[0322] "acceptance_threshold": 0.75,

[0323] "performance_target": "ZT",

[0324] "sensitivity_level": "high",

[0325] "dimensional_scores": {

[0326] "average_range_coverage": 0.7919,

[0327] "sampling_quality": 0.914,

[0328] "model_consistency": 0.920

[0329] },

[0330] "dynamic_weights_used": {

[0331] "sampling_weight": 0.3,

[0332] "range_coverage_weight": 0.7

[0333] },

[0334] "risk_factors": [],

[0335] "confidence_score": 0.855,

[0336] "confidence_level": "high",

[0337] "optimization_suggestions": [

[0338] "The current sampling distribution for high-sensitivity ZT is of excellent quality, with sufficient coverage within the historical typical value range, good chemical diversity, and high model prediction consistency, and can be directly used for downstream screening." ]

[0340] }

[0341] in conclusion:

[0342] This embodiment fully demonstrates the verification process of the method of the present invention on a real-world sample prediction result. For the high-performance sensitivity index ZT, in a Cu-Fe-AlO3 ternary system containing composite elements, the verification method not only confirmed that the overall quality of the sampling distribution met the standard (0.898 > 0.75), but also revealed its advantages through multi-dimensional scoring: sufficient coverage of historical typical value range (0.7919), excellent chemical space exploration (0.914), and highly consistent prediction trends among different models (0.92). Finally, a high-confidence "acceptable" judgment and a detailed structured report are output, proving that the present invention can perform in-depth, comprehensive, and performance-related quality diagnosis of sampling results, providing a reliable quality control link for high-throughput material calculation processes.

[0343] This invention constructs a complete evaluation system from three core dimensions: "breadth of chemical composition exploration," "performance range coverage quality," and "model prediction reliability," through parallel computation of historical typical value range coverage score (S_range), chemical composition spatial sampling quality score (S_sampling), and multi-model prediction consistency score (S_consistency). S_sampling precisely quantifies the quality of chemical spatial exploration through elemental ratio uniformity analysis and integer coordinate vector coverage analysis, while S_range focuses on the fit between performance distribution and historical experience. These three elements work together to achieve a full-chain quality assessment from input to output, completely overcoming the limitations of traditional methods.

[0344] A performance sensitivity knowledge base is introduced, and the fusion weights of S_sampling and S_range are dynamically adjusted according to the sensitivity level (high / medium / low / unknown) of the target performance: high sensitivity performance focuses on coverage of historical typical value ranges, low sensitivity performance focuses on chemical spatial diversity, and medium sensitivity performance balances both. This dynamic weight allocation mechanism deeply binds the verification standard with the intrinsic characteristics of the performance, enabling precise quality control for high sensitivity performance and ensuring the breadth of exploration for general performance, thus significantly improving the relevance and reliability of the verification results.

[0345] By constructing a triple evaluation mechanism through S_range scoring (basic proportion coverage, uniformity within the range, and boundary utilization), and deeply integrating prior knowledge of historical typical value ranges, this mechanism not only assesses the overlap between sampled predicted values ​​and historical high-value ranges, but also quantifies the uniformity of distribution within the range and the adequacy of boundary exploration. This design ensures that sampling quality verification aligns with actual R&D scenarios, guarantees that new sampling results fully explore historical high-performance ranges, avoids overlooking key ranges, and transforms historical data into effective constraints and guidance for sampling quality.

[0346] This invention generates a structured report containing core judgment results, detailed scores for each dimension, a list of risk factors, confidence level assessments, and targeted optimization suggestions. It accurately identifies specific risk types such as "insufficient sampling diversity," "low coverage of typical ranges," and "poor model consistency" through sub-threshold comparisons. Simultaneously, it matches personalized solutions from a pre-set optimization strategy library, achieving a leap from "black-box judgment" to "transparent diagnosis + executable optimization," directly supporting iterative upgrades of sampling strategies.

[0347] This invention calculates a multi-model prediction consistency score (S_consistency) by first aligning and standardizing the predicted values ​​of each model, and then quantifying the distributional consistency between models using pairwise Pearson correlation coefficient matrices. This score not only serves as one of the core indicators for confidence level assessment but also effectively identifies systematic biases between models. Combined with model ensemble or local calibration schemes in optimization recommendations, it significantly improves the confidence level of sampling quality verification, providing more reliable quality assurance for downstream high-throughput screening.

[0348] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0349] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A collaborative verification method for the sampling distribution quality of material chemical formulas, characterized in that: Includes the following steps: S1. Input data acquisition and preprocessing: Receive performance prediction data of chemical formula sampling results of one or more prediction models for the same batch of materials, and obtain the target performance index and its corresponding performance sensitivity information and historical typical value range [min_val, max_val]; S2. Multi-dimensional quality score calculation: Based on the performance prediction data and the historical typical value range, calculate the historical typical value range coverage score S_range; Based on the list of elements used for sampling and the sampling results of the chemical formula of the material, the chemical composition spatial sampling quality score S_sampling is calculated; If multiple prediction models exist, the multi-model prediction consistency score S_consistency is calculated based on the performance prediction data of each model. S3. Calculation and judgment of comprehensive quality score for sensitivity internalization: Based on the sensitivity level of the target performance index, dynamically adjust the fusion weight of chemical composition spatial sampling quality score S_sampling and historical typical value range coverage score S_range, calculate the comprehensive quality score Score_combined, compare it with the preset acceptance threshold T_accept, and output the judgment result. S4. Generate a structured verification report and intelligent diagnosis: Based on the scores of each dimension in step S2 and the judgment results in step S3, output a structured report containing core judgment results, details of scores for each dimension, a list of risk factors, confidence levels, and targeted optimization suggestions.

2. The collaborative verification method for the sampling distribution quality of material chemical formulas according to claim 1, characterized in that: The target performance indicators in step S1 are quantifiable core characteristic parameters in material development, including but not limited to thermoelectric figure of merit, power factor, Seebeck coefficient, oxygen evolution initiation potential, Tafel slope, melting point, dielectric constant, contact resistance, superconducting critical temperature, and thermal conductivity. The sensitivity information and historical typical value range of the target performance index are derived from a pre-defined structured knowledge base of material performance sensitivity, which is constructed through the following steps: Input a dataset containing the chemical formulas of materials and their historical performance data; For each performance metric in the dataset, calculate the multidimensional statistical characteristics of its data; By incorporating and correcting the statistical characteristics based on the variability of material composition, a quantitative sensitivity score for the performance index is generated. Based on the preset threshold range in which the quantified sensitivity score falls, the sensitivity level of the performance index is determined, including high sensitivity, medium sensitivity, and low sensitivity. By statistically analyzing the distribution of historical data for this performance indicator, we can obtain its historical typical value range. The output includes structured records containing performance indicator names, quantified sensitivity scores, sensitivity levels, and historical typical value ranges, which are then compiled to form the knowledge base.

3. The collaborative verification method for the sampling distribution quality of material chemical formulas according to claim 1, characterized in that: The calculation process of the historical typical value range coverage score S_range in step S2 includes: Baseline Proportional Score: Calculate the proportion of samples whose predicted values ​​fall within the historical typical value range [min_val, max_val] corresponding to the target performance index, P_in_range; Uniformity score of distribution within the range: Extract the subset of predicted values ​​in_range_values ​​that fall within the range of the historical typical values, and calculate the uniformity score by equal weighted average of normalized range, normalized standard deviation and interquartile range expansion. Range boundary utilization score: Calculate the proportion of the intersection between the predicted value and the historical typical value range corresponding to the target performance index within the historical typical range; Triple weighted fusion: The basic proportion score, uniformity score and boundary utilization score are weighted and fused with a weight of 0.5:0.3:0.2 to obtain the historical typical value range coverage score S_range, that is, S_range=min(1.0,0.5×basic proportion score+0.3×uniformity score within the range+0.2×boundary utilization score within the range).

4. The collaborative verification method for the sampling distribution quality of material chemical formulas according to claim 3, characterized in that: The formulas for calculating the uniformity score using the normalized range, normalized standard deviation, interquartile range expansion, and their equally weighted average are as follows: The formula for calculating the normalized range is: `range_norm = min(1.0, value_range / (max_val - min_val))`, where `value_range` is the actual range of the predicted value subset, and its calculation formula is: value_range = max(in_range_values) - min(in_range_values), where max(in_range_values) is the maximum value of in_range_values ​​and min(in_range_values) is the minimum value of in_range_values. The formula for calculating the normalized standard deviation is: std_norm=min(1.0,std_dev / (max_val-min_val)×2.0), where std_dev is the standard deviation of in_range_values ​​and 2.0 is the empirical scaling factor; The formula for calculating interquartile range extension is: quantile_spread=(Q3-Q1) / (max(in_range_values)-min(in_range_values)), where Q1 and Q3 are the 25th and 75th quantiles of in_range_values, respectively; The formula for calculating the uniformity score using equal-weighted average is as follows: uniformity_score=(range_norm+std_norm+quantile_spread) / 3.

5. The collaborative verification method for the sampling distribution quality of material chemical formulas according to claim 1, characterized in that: The calculation process of the chemical composition spatial sampling quality score S_sampling in S2 includes: Element proportion distribution uniformity score S_uniformity: For each element or composite element, analyze its proportion value sequence {r_i} in all chemical formulas, calculate the empirical cumulative distribution function eCDF of the sequence, and obtain the element uniformity score by using the Kolmogorov-Smirnov (KS) statistic D of the empirical cumulative distribution function eCDF and the theoretical CDF of uniform distribution in the [0,1] interval. The element proportion distribution uniformity score S_uniformity is obtained by averaging the scores of all elements. Chemical formula pattern coverage score S_coverage: Set the discretization precision p, discretize the element proportion of each chemical formula into integer coordinate vectors, count the number of unique integer coordinate vectors. For a system with n elements, at precision p, the total number of distinguishable patterns is approximately M_theoretical = (1 / p)^(n-1), calculated as S_coverage=min(1.0, number of unique integer coordinate vectors / (α* M_theoretical)), where α is the expected coverage coefficient less than 1, and n is the number of elements in the element list used for sampling; The overall score S_sampling is obtained by fusing S_uniformity and S_coverage with a weight of 0.7:0.3, resulting in an overall score S_sampling = 0.7 × S_uniformity + 0.3 × S_coverage.

6. The collaborative verification method for the sampling distribution quality of material chemical formulas according to claim 1, characterized in that: The calculation process of the multi-model prediction consistency score S_consistency in step S2 includes: Data alignment: Extract the predicted value sequences from each model, unify their length, and perform Z-score standardization; Correlation coefficient matrix calculation: Calculate the Pearson correlation coefficients between all pairs of models to form an M×M symmetric matrix, where M is the number of models; Consistency score: The average absolute value of the correlation coefficients in the upper triangular part of the matrix is ​​taken to obtain S_consistency.

7. The collaborative verification method for the sampling distribution quality of material chemical formulas according to claim 1, characterized in that: The dynamic weight allocation rule in step S3 is as follows: When the target performance is high-sensitivity performance, the weight of the historical typical value range coverage score is W_range=0.7, and the weight of the chemical composition spatial sampling quality score is W_sampling=0.3; When the target performance is of medium sensitivity, W_range=0.5, W_sampling=0.5; When the target performance is low-sensitivity or unknown, W_range=0.3, W_sampling=0.7; The formula for calculating the overall quality score is: Score_combined = W_sampling × S_sampling + W_range × S_range; The preset acceptance threshold T_accept is set to 0.

75.

8. The collaborative verification method for the sampling distribution quality of material chemical formulas according to claim 1, characterized in that: The confidence level calculation process in step S4 is as follows: Confidence score = Score_combined × 0.6 + S_consistency × 0.3 + min(1.0, number of models / 5) × 0.1; Confidence level is determined by confidence score range: ≥0.8 is high, [0.6, 0.8) is medium, and <0.6 is low.

9. The collaborative verification method for the sampling distribution quality of material chemical formulas according to claim 1, characterized in that: The targeted optimization suggestions in step S4 are generated based on risk factors and include: If S_range < 0.6 and the target performance index is a highly sensitive performance, it is recommended to adopt a hierarchical clustering sampling strategy and increase the sampling density in the historical high value range corresponding to the target performance index. If S_sampling < 0.5, it is recommended to expand the range of element ratio variation or switch to the stratified_clustering sampling strategy; If S_consistency < 0.7, it is recommended to check the model differences and use weighted averaging or stacked ensemble to fuse the outputs of multiple models.

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