Abstract
Timely detection of deformation mechanisms in metallic structural materials is essential for early-warning alerts on potential damages and fractures. Acoustic emission (AE) technologies are commonly used for this purpose due to their non-destructive nature. However, traditional methods often struggle with distinguishing AE signals associated with multiple co-existing deformation mechanisms. To address this challenge, we propose a knowledge-driven unsupervised learning approach. The novel method leverages a family of gradient-driven supervised base learners and integrates them with a knowledge-infused aggregate loss function, effectively transforming the approach into an unsupervised learning framework. Compared to existing methods, our approach excels in identifying co-existing deformation mechanisms associated with AE signals. Experiments on porous 316L stainless steel during tensile process show that the avalanche statistics of the identified dislocation and crack AE signals align closely with classical statistical methods and fracture theory. By integrating with the avalanche theory, our proposed approach can continuously monitor material deformation mechanisms in real-time and provide dynamic early failure warnings. Additionally, the framework demonstrates strong transferability in recognizing multiple co-existing deformation mechanisms in new materials, leveraging its unsupervised learning capability.
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Introduction
Metallic structural materials, like other elastic and inelastic materials, deform under external loads. When the applied load exceeds a critical stress level, defects such as dislocations or cracks can develop, leading to deteriorated performance. Consequently, continuous monitoring of these defects in materials is essential for the timely assessment of material health during service.
Acoustic emission (AE) technologies are a subset of online Non-Destructive Testing methods1,2. These technologies monitor a material’s internal responses by continuously capturing high-frequency elastic waves released due to structural changes or defects1,3,4,5. AE measurements are particularly valuable because they can dynamically and sensitively detect the initiation and growth of defects—such as dislocations and cracks—across a broad range of scales, from nanometers to millimeters1,3,6,7. This capability makes AE technologies highly suitable for monitoring material health in service8,9,10,11. Despite considerable technology advancements of AE measurements, a fundamental challenge arises from materials themselves—the inherently complex signals generated by co-existing deformation mechanisms. These signals are often mixed or closely intertwined, complicating both manual and automatic signal identification efforts.
Conventional manual analysis methods typically examine the three-way correlation between acoustic emission (AE) signals, avalanche statistics, and the associated deformation mechanisms (the microscopic processes or mechanisms through which materials deform under applied stress)12,13. From a statistical perspective, AE signals resulting from material deformations and fractures are usually interconnected, leading to complex avalanche systems12,14,15. Researches on acoustic emissions in brittle materials such as rock16,17, ceramic18, concrete19, and ice20—have shown that the characteristics of AE signals (e.g., energy, amplitude, and duration) during deformation follow power-law distributions21, akin to the avalanche statistics observed in earthquakes. Additionally, different deformation mechanisms produce distinct types of avalanches, as demonstrated through experiments22,23,24, simulations25,26, and physical models27,28. While analyzing materials with a single deformation mechanism is relatively straightforward16,17,18,19,29,30, metallic structural materials present a greater challenge due to their typically complex deformation processes involve multiple co-existing deformation mechanisms24. The ability to accurately distinguish and analyze these co-existing deformation mechanisms in structure materials remains a significant and unresolved issue.
Machine learning methods for analyzing acoustic signals generally fall into three broad categories: supervised31,32,33,34,35, semi-supervised36,37, and unsupervised3,38,39,40,41,42. Supervised methods achieve high recognition accuracy when large amounts of labeled training data are available. Semi-supervised methods, in contrast, require only a small number of expert-labeled samples. Unsupervised methods, on the other hand, operate without labelled samples. In materials design, obtaining training data can be both time-consuming and costly. Consequently, the heavy reliance on labeled training data limits the applicability of supervised methods43,44. Additionally, the sparsity of labeled data can constrain the effectiveness of semi-supervised methods in challenging scenarios. Unsupervised learning methods, such as k-means and Gaussian Mixture Models, are often used for classifying AE signals due to their label-free training advantages3. The current state-of-the-art in unsupervised learning approach includes trainable wavelet scattering network, which automatically extract features from AE waveforms, such as those from earthquakes and icequakes45,46. However, all existing unsupervised learning models face significant challenges in achieving high recognition accuracy, particularly when classifying signals from diverse samples and materials.
To address the limitations of existing methods, we introduce a knowledge-driven unsupervised learning approach. This new approach leverages a family of GDC (base learners) combined with a knowledge-infused aggregate loss function (KIALF). Unlike traditional supervised machine learning approaches that require labeled training samples, our aggregate loss function operates over a collection of AE signals by incorporating domain-specific knowledge, thus eliminating the need for extensive labeled data. By integrating these supervised base learners with the proposed knowledge-infused unsupervised framework, our approach no longer depends on labeled samples. Experiments on porous 316L stainless steel during tensile process demonstrate the superior capability of our method to detect co-existing dislocation and crack AE signals effectively. The avalanche statistics of these identified signals align closely with both statistical methods and the fracture theory. Additionally, our approach, integrated with classical avalanche theory, offers real-time monitoring of material deformation mechanisms and provides dynamic early failure warnings. Compared to various state-of-the-art models, our approach excels in identifying AE signals with co-existing deformation mechanisms, as shown by the consistency of avalanche statistics. It also exhibits strong transferability for recognizing multiple co-existing deformation mechanisms in new materials.
Results
Design of a knowledge-driven unsupervised learning model
Figure 1 illustrates the computational workflow of the proposed unsupervised learning model. The framework evaluates various machine learners as potential base learners, including Gaussian Discriminant Analysis (GDA), Logistical Regression (LR), Support Vector Machine (SVM), Kernel Perceptron, Back-Propagation Neural Network (BPNN), Gradient Boosting Decision Tree47 (GBDT), and Random Forests48. Each base learner processes the power spectrum densities (PSD) of raw AE waveforms (Fig. 1a (1–3)) (see “Methods”). In this framework, each base learner functions as a Gradient-driven Supervised Classifier (GDC), where hyperparameters are adjusted based on gradients of an externally evaluated loss function through the classical back-propagation. For a given GDC configured with specific hyperparameters, it analyzes a collection of PSD data to identify corresponding deformation mechanisms. A KIALF (Fig. 1a (4)) is introduced for autonomous model training and evolution. Unlike traditional supervised learning loss functions, this new function computes aggregate loss based on domain knowledge, enabling unsupervised learning without labeled training samples. After training all base learners, the optimal model is selected based on the overall performance (Fig. 1a (5)). Following the principle of Ockham’s razor49,50, simpler models with competitive performance are preferred. The base learner that best balances these criteria is ultimately chosen as the final GDC model. Once trained, the model can be readily applied to monitor AE signals from new samples of the same material without further tuning. Integrated with the avalanche theory51,52, the approach can dynamically identify deformation mechanisms and provide early failure warnings (Fig. 1a (6–8)).
a The proposed unsupervised learning strategy framework is illustrated as follows: (1) The Acoustic Emission (AE) spectrum derived from experimental input is divided into three regions based on their statistical features: Region 1 (R1) is dislocation dominant, Region 3 (R3) is crack dominant, and Region 2 (R2) is a mix of both mechanisms. (2) Corresponding acoustic waveforms of the AE spectrum. (3) Power spectrum densities (PSD) are extracted from AE waveforms. (4) The Knowledge-Infused Aggregate Loss Function (KIALF) dispatches base learners (GDC) in an unsupervised fashion. (5) The best overall performing base learner is elected as the ultimate choice for the backbone model. (6) New AE waveforms from subsequent experiments are input into the system. (7) The signals are separated and further analyzed using the proposed approach. (8) Early failure warning is conducted based on the separated signals. b Essential steps of the proposed approach. A set of unlabeled AE signals Z is applied to a base learner GDCn, yielding a probabilistic classification output COn. The ratios of crack-related signals in two randomly sampled intervals ri = [si, ei] and rj = [sj, ej] are calculated as \({{{\mathcal{T}}}}_{n}({r}_{i})\) and \({{{\mathcal{T}}}}_{n}({r}_{j})\), respectively. The aggregate trend metric \({{{\mathcal{L}}}}_{{\mbox{Trend}},n,i,j}\) is computed for the interval pair. The full-period loss KIALF (\({{{\mathcal{L}}}}_{n}\)) is obtained via repeated sampling and backpropagated to optimize the base learner parameters θn. The optimized learner GDCnopt is produced by minimizing \({{{\mathcal{L}}}}_{n}\).
Figure 1b illustrates the essential steps internally executed by the proposed approach. Specifically, if a base learner, GDCn, is applied onto an unlabeled AE signal zm, a probabilistic classification output, COn,m, is obtained, indicating the likelihood that the signal corresponds to the dislocation or cracking mechanism as recognized by the learner. Each base learner (GDCn) undergoes independent training to minimize its respective loss function, yielding an optimized version denoted as GDCnopt.
The tailor-designed aggregate loss function is constructed based on our domain knowledge, the ratio of crack signals during deformation should gradually increase from 0 to 1. To quantify this expected trend, we define \({{{\mathcal{T}}}}_{n}(r)\) as the ratio of crack-related signals identified by learner GDCn over time interval r, given by:
To quantify alignment with domain knowledge, we define an aggregate trend metric \({{\mathcal{L}}}_{{{\rm{Trend}}},n,i,j}\) based on two randomly sampled intervals (ri = [si, ei] and rj = [sj, ej]), computed as
in which δ is an auxiliary function, defined as follows:
The loss score decreases as the signal ratio trend aligns more consistently with the forecast according to the aforementioned domain knowledge.
Third, we aggregate interval-specific loss terms into a comprehensive loss function for the full observation period through repeated sampling. A temporally-weighted scheme is applied to account for the greater influence of longer intervals on the overall trend. The tailor-defined loss function for the training stage, \({{{\mathcal{L}}}}_{n}\), can be specified in a normalized form:
Here, N represents the total number of sampling iterations performed across the observation period.
We note: (1) Our framework employs a domain-informed loss function derived entirely from mathematical encodings of materials science principles governing acoustic emission (AE) signal distributions. Operating without labeled data, the system utilizes only AE timestamp sequences acquired during standard monitoring, enabling unsupervised signal sequence reconstruction and base learner optimization. (2) The architecture is intentionally modular—any machine learning model capable of self-optimization through gradient-based updates can be incorporated. This design creates an adaptive ecosystem where the continuous integration of new compatible algorithms systematically enhances predictive performance, while maintaining rigorous adherence to physical constraints encoded in the loss function. The approach uniquely bridges domain knowledge with flexible machine learning, avoiding the data hunger of purely statistical methods while preserving interpretability.
Furthermore, our framework leverages the intrinsic diversity of base learners, where each specialized variant emphasizes distinct signal features. This design intentionally amplifies discriminative capabilities for identifying characteristic patterns among different deformation mechanisms. To systematically evaluate and harness these complementary strengths, we implement a lightweight scoring model that quantifies the predictive performance of each optimized base learner (GDCnopt). The scoring metric prioritizes detection fidelity for the most physically significant signal differentiators, ensuring the ensemble focuses on mechanistically relevant features rather than incidental correlations. The overall performance of a model, denoted as Perf (where higher values indicate better performance), can be evaluated as follows:
The objective function comprises two key components: (1) a performance term evaluating the domain-informed loss \({{{\mathcal{L}}}}_{n}\)’s effectiveness, and (2) a complexity penalty enforcing parsimony through Ockham’s razor principle. This dual structure ensures optimal balance between physical fidelity (encoded in \({{{\mathcal{L}}}}_{n}\)) and model simplicity53. Here, Numn represents the number of trainable parameters in the base learner. The coefficient k is introduced as a tradeoff parameter. Since modern neuro-computing models typically entail a large number of hyperparameters, resulting in a relatively large lg(Numn) term, the value of k is expected to properly balance the two terms in Eq. (4). In this study, k is empirically optimized to 0.1. Through this optimization framework, the system automatically selects the highest-performing base learner to serve as the backbone model, ensuring optimal representation of the underlying physical processes. i.e., \({{\mathcal{L}}}={{\arg }}{\max }_{n}{{\rm{Perf}}}\).
In summary, as illustrated in Supplementary Fig. S1, the key distinction between the proposed framework and traditional supervised learning methods is its ability to train the model without requiring any labeled data. This capability is achieved by analyzing trends and properties derived from a collection of samples and comparing them with expected trends and properties based on established domain knowledge. By incorporating this knowledge into the training process, the new framework enables effective task learning in an unsupervised manner.
Performance of the proposed approach
Figure 2 shows the results of applying the proposed approach to the experimental dataset for porous 316L stainless steel under uniaxial tension13. Figure 2a illustrates three distinct stages identified using a peer statistical method13. The percentages of AE signals in these stages are 70.48%, 27.96%, and 1.56%, respectively. The figure also shows that the maximum stress of the porous material reaches 45 MPa at the 11,800 s, after which the material exhibits continuous fractures. Figure 2b, c present the results obtained using the proposed approach with the optimally identified classifier (a BPNN, details provided later). The model successfully identified 12,867 dislocation signals (Fig. 2b) and 413 crack signals (Fig. 2c) from a total of 13,280 AE signals. Both deformation mechanisms are observed to co-exist throughout the entire tensile process. Even in the early stage (region 1), where dislocation signals predominate, the presence of crack signals is significant. Similarly, in the later stage (region 3), where crack signals dominate, dislocation signals are also present and non-negligible.
a Time series of all AE signals (colored lines) overlaid with the stress-time curve (black). Blue, green, and orange indicate dislocation-dominant (Region 1), mixed (Region 2), and crack-dominant (Region 3), respectively. b, c AE signals separated into dislocation (b) and crack (c) spectra, with 21 and 14 superjerks (dashed lines), respectively. d, e Temporal evolution of power-law energy exponents (ε) for dislocation (d) and crack (e) signals, calculated by maximum likelihood estimation between adjacent superjerks. The sample size between consecutive superjerks in (d) is 147, 287, 313, 2185, 266, 1129, 1705, 123, 21, and 9, respectively. The sample size between consecutive superjerks in (e) is 88, 71, 57, 22, 9, 17, 20, 29, and 5, respectively. The red dashed line indicates the mean field value (ε = 1.33). Data in (d and e) are shown as mean ± standard deviation. Source data are provided as a Source Data file.
To explore the performance impact of base learners on the proposed approach, we compared top 5 best-performing machine learning algorithms as our base learner. These include LR, GDA, SVM, GBDT, and BPNN. The outcomes for each classifier of all AE signals are collectively analyzed using maximum likelihood estimation (MLE)54 to construct a maximum likelihood (ML) curve. MLE is a method that directly determines whether a group of identified AE signals follows only one deformation mechanism. A single deformation mechanism leads to a horizontal ML plateau extending from the lowest energy kink to a higher energy kink in the ML curve. In contrast, co-occurrence of multiple deformation mechanisms leads to a decrease of the ML curve at high energy54. As a result, by measuring the harmonic mean of the plateau lengths in the curves (HMLE, detail in “Methods”), we quantitatively assess the avalanche statistics of each recognized deformation mechanism. The left panel of Fig. 3a illustrates the performance of each classifier evaluated by overall performance and HMLE for the candidate classifiers explored. The classifier’s overall performance, evaluated by Pref (Eq. (4)), is consistent with the statistical performance of HMLE.. For both criteria, the BPNN with 237 trainable parameters achieves the best overall performance and highest HMLE amongst all candidates.
a Performance of candidate classifiers and peer methods evaluated by four metrics: HMLE (purple, harmonic mean of plateau lengths by maximum likelihood estimation), number of trainable parameters (green), aggregate loss impact \({{{\mathcal{L}}}}_{n}\) (blue), and overall performance (orange), where \({{{\mathcal{L}}}}_{n}\) is the domain-informed loss, Numn is the number of trainable parameters for candidate classifiers, and k is a tradeoff parameter. Candidate classifiers for comparison include Gaussian Discriminant Analysis (GDA), Logistical Regression (LR), Support Vector Machine (SVM), Gradient Boosting Decision Tree (GBDT), and Back-Propagation Neural Network (BPNN). Peer methods include k-means, ScaNet45 and an Imbalanced Discriminative Neural Network (IDNN) with focal loss55. Circles and error bars indicate mean and standard deviation from 100 resamples. b Temporal evolution of crack signal ratios identified by different methods during tensile loading. Regions are defined by the statistical method13. Curves are computed using a 1200 s sliding window, with shaded areas denoting 90% confidence intervals. c Maximum likelihood estimation (MLE) curves of dislocation (dark blue) and crack (orange) signals in Regions 1 and 3 classified by the statistical method (+) and the proposed method (○). Sample sizes: statistical method—9360 (dislocations), 204 (cracks); proposed approach—9198 (dislocations), 93 (cracks). d MLE curves for Region 2 (green) and signals identified as dislocations and cracks by the proposed method. Sample size: 3636 for dislocation, 80 for cracks, and 3716 for mixture. Data in (d and e) are shown as the mean ± standard deviation. e Normalized cumulative AE amplitude of separated crack signals versus ideal fracture model (black line), demonstrating physical consistency. Source data are provided as a Source Data file.
Figure 3b illustrates the ratios of crack signals as a function of load time for different separation methods. The proposed approach identifies less than 10% of crack signals before 10,000 s, with a significant increase in ratio as the load progresses towards the fracture point. Since dislocations dominate the early stages of the tensile process and the ratio of initial crack signals should not exceed 50%, the results from the proposed approach align more closely with the domain knowledge encoded in Eqs. (1)–(3).
Ablation study of the proposed approach
We performed quantitative ablation studies to evaluate three key components of our framework: classifier selection criteria, loss function formulation, and optimization strategy. Our proposed classifier selection criterion Eq. (4), combines model performance (first term) with complexity measured by trainable parameters (second term). Individual evaluation of these terms revealed opposing trends: while GBDT showed superior raw performance (GBDT > BPNN > LR > SVM > GDA), LR exhibited optimal simplicity (LR > BPNN > GDA > SVM > GBDT). Crucially, only the combined evaluation yielded results (BPNN > GBDT > LR > SVM > GDA) consistent with HMLE validation. Supplementary analyses confirmed BPNN’s crack signal identification most closely matched domain knowledge across loading conditions (Supplementary Table S1), validating our dual-term selection approach.
To evaluate the robustness of our loss function formulation, we conducted additional ablation studies comparing alternative formulations: (1) a non-logarithmic version \(\left(\frac{{\sum}_{m:{z}_{m}\in r}{{{\rm{CO}}}}_{n,m}}{{\sum}_{m:{z}_{m}\in r}1}\right)\) and (2) a count-based version using absolute numbers of Type II signals \(\left({\sum}_{m:{z}_{m}\in r,{{{\rm{CO}}}}_{n,m} > 0.5}1\right)\). Using synthetic datasets with known labels (20,000 PSD spectra per dataset, containing two signal types with distinct average spectra and temporally-evolving Type II signal ratios following power-law distributions; see Supplementary Note-02 and Fig. S2), we quantified classification accuracy across all candidate classifiers (BPNN, LR, GDA, SVM, GBDT). Results (Supplementary Tables S2–S6) demonstrate that removing the logarithmic transformation consistently degraded performance (reduced accuracy, F1 scores, and increased RMSE and Area across all datasets). Notably, the count-based approach failed to produce meaningful classification, confirming the importance of ratio-based evaluation in our formulation.
To assess the effectiveness of our optimization approach, we compared gradient-based optimization against a non-gradient driven optimization method, specifically a genetic algorithm (GA). Performance metrics (Supplementary Fig. S3 and Tables S7–S10) demonstrate that gradient-optimized classifiers (BPNN, LR, GDA, and SVM) consistently surpass their GA-trained counterparts in accuracy, F1 score, RMSE, and Area across all evaluated conditions. This performance advantage is particularly notable given the substantially longer training times required by the GA approach, highlighting both the efficiency and effectiveness of gradient-based optimization for this application.
Comparison with other machine learning methods
The latest unsupervised method for identifying deformation mechanisms from AE signals is ScaNet45, a deep learnable scattering network with Gaussian mixture model clustering. The most efficient supervised method for this task is the neural network with Focal loss55, which is trained using labeled data and a loss function designed to handle imbalanced data56 (denoted as IDNN). IDNN is particularly well-suited for metallic structural materials where the distribution of AE signals is highly skewed (e.g., in the dataset reported in Fig. 2a, 70.48% of pseudo-labels correspond to dislocations, leaving only 1.56% for cracks). In this study, we compare the proposed approach with three top-performing peer methods: k-means, ScaNet, and IDNN.
Using three characteristic synthetic datasets with known labels (Supplementary Note-02 and Fig. S2), we quantitatively evaluated our method against established approaches (k-means, ScaNet, and IDNN). The results (Supplementary Fig. S4 and Tables S11–S14) demonstrate that our framework consistently achieves superior performance across all evaluation metrics, regardless of temporal evolution patterns in Type II signal ratios. Notably, it simultaneously: (1) attains the highest accuracy and F1 scores, and (2) yields the lowest RMSE and Area measures. This robust performance advantage holds across all three synthetic datasets with differing trend characteristics and different regions.
We then compare the proposed approach with the above 3 best-performing existing methods using the dataset shown in Fig. 2a. In this case, the number of clusters in k-means and ScaNet is set to 2, in order to recognize the two deformation mechanisms shown in Fig. 2. Moreover, the IDNN is the same as the network used in the proposed approach, and are trained via supervised learning with pseudo-labels. The pseudo-labels are derived from previous statistical results13, and assume that all the AE signals in region 1 are from dislocation movement, whereas all the AE signals in region 3 are from cracks. The right panel of Fig. 3a and Supplementary Table S1 illustrate that the performance of the proposed approach (HMLE) shows better performance than the above 3 existing methods. Figure 3b shows that the ratio of crack signals evolving temporally. The IDNN method shows similar trend with the proposed approach, as the present IDNN method share same neural network with the present approach and using a large number of pseudo-labels according to the knowledge of previously known statistical results13. However, the performance of the proposed approach significantly surpasses that of k-means and ScaNet. Both of these methods indicate approximately 50% crack AE signals at the beginning of the tensile process, which contradicts established material science knowledge. Furthermore, the AE signals identified by ScaNet for the entire AE spectrum in Fig. 2a are further analyzed using MLE, with results shown in Supplementary Fig. S5. The MLE curves for dislocation and crack signals identified by ScaNet exhibit clear mixing behavior and do not reach a plateau. These characteristics suggest that the clustering-based methods, such as k-means and ScaNet, result in greater misclassification errors for mixed AE signals compared to the proposed approach.
Comparison with statistical theory and classic fracture theory
We compared the avalanche statistics of the classified dislocation and crack AE signals from the proposed approach with the results obtained using statistical method via pseudo-labels. As depicted in Fig. 3c, for region 3 of the stress-time curve (Fig. 2a) where the crack mechanism dominates, the MLE curve identified by the proposed approach exhibits a comparable over more than 3 decades plateau length. The energy exponent ε = 1.25 ± 0.03 from the proposed approach compared to ε = 1.31 ± 0.03 by the statistical method. For region 1 of the stress-time curve (Fig. 2a), when the dislocation mechanism dominates, the MLE curve from the proposed approach shows a comparable energy exponent by the statistical method. These results indicate that for cases where only one deformation mechanism dominates the deformation process, the AE signals do not exhibit noticeably enhanced avalanches statistics compared to the peer statistical methods. However, in regions where both mechanisms co-exist with non-negligible fractions (region 2 of the stress-time curve in Fig. 2a), the proposed approach demonstrates superior. As shown in Fig. 3d, the MLE curve from the proposed approach shows a meaningful plateau length with the energy exponent ε = 2.18 ± 0.03 for dislocation AE signals and ε = 1.48 ± 0.07 for crack AE signals. In contrast, the statistical method cannot distinguish the mixed AE signals as there is no plateau in the MLE curve.
We further compare the AE signals with classic fracture theory57, where a log-periodic correction to scaling can effectively fit the cumulative amplitude of crack AE signals. This relationship is derived from the predictions of seismic energy release through a renormalization group approach for the regional fault network with a discrete hierarchy. The relationship between cumulative amplitude of AE events and the fracture evolution can be expressed as follows:
where A, B, and C represent constants, λ describes the wavelength of the oscillations in the log space, and ψ denotes a phase shift. This model has found applications in predicting the failure of large earthquakes and cyclic loading of thermal barrier coatings57,58. Figure 3e compares the cumulative amplitude and calculated ideal curve of crack signals, and the crack signals identified by the proposed approach align closely with the fracture theory.
Discussion
The phenomenon of crack AE signals in brittle materials approaching critical phenomena before failure, where the energy exponent ε decreases as the failure point is approached, has been observed in previous studies51,52,59,60,61. This behavior serves as an indicator for early warning of failure. However, it remains unclear whether ductile materials, such as most metal materials with plastic deformation, exhibit similar critical phenomena due to the challenge of recognizing possible co-existing AE signals during deformation. With the AE signals for dislocations and cracks identified by the proposed approach, we are now able to investigate this phenomenon.
The superjerk-based analysis is commonly used to examine the evolution of the energy exponent over time. Superjerks refer to avalanche signals with energies exceeding those of all previous events51, including the final major collapse and other “near collapse” events. By detecting superjerks, a full AE spectrum can be segmented into smaller sections, with superjerks serving as transition points. The temporal evolution of the energy exponent within each segment can then be modeled individually using the power-law distribution with an exponent ε.
For the material studied in Fig. 2, the ranks of the superjerks are indicated, with 21 superjerks in dislocation signals and 14 in crack signals (shown in Fig. 2b, c). The energy exponents of the dislocation signals remain stable in Fig. 2d, hovering around 2.2 throughout the deformation process. However, the energy exponents of crack signals initially drop from about 2.0 to 1.5 at superjerk 7, then continue to decrease to around 1.33 at superjerk 8, corresponding to the mean field value. Subsequently, the energy exponents stabilize at this level until fracture (shown in Fig. 2e).
The region between superjerk 7 and 8 suggests that fracture is imminent, while fracture has occurred beyond superjerk 8. This indicates that crack AE signals exhibit clear critical phenomena close to failure, whereas dislocation signals do not show this trend. Superjerk analysis can be conducted during AE testing, allowing for real-time failure warnings by monitoring the energy exponents of crack signals. Early failure warnings can be generated when the exponent begins to decrease towards the threshold value of 1.5–1.6, with a safety margin.
The experimental data presented supports the validity of the proposed approach. To further explore its efficacy for online monitoring of material health, we apply the approach to a new material with different porosity (30%) compared to the original training data (porosity = 35%). Simultaneous AE measurement and in-situ optical observation are carried out during the tensile process.
Figure 4a illustrates the in-situ optical observation and longitudinal two-dimensions Digital Image Correlation (2D-DIC) strain patterns near the final fracture at different times. Surface cracks are observed at 5270 s (DIC image at T4) with a noticeable change in the strain pattern. In contrast, the stress-time curve during tension (shown in Fig. 4b) indicates that the stress reaches its maximum, and macroscopic fracture begins at 4607 s (T2). This comparison highlights that while the maximum stress can serve as a fracture indicator in some cases, it may be even more sensitive than in-situ optical observations. However, relying solely on the maximum stress as the indicator of failure can be too late for practical applications. Once the stress reaches its maximum, irreversible fracture of the material occurs spontaneously, rendering it impractical for real-time monitoring.
a Optical microscope images overlaid with strain maps from digital image correlation. b Stress-time curve (black) and AE signal sequence (green). c Identified crack signals (orange) with 11 superjerks (dashed lines). d Temporal evolution of Power-law energy exponents (ε) for crack signals between consecutive superjerks, calculated via maximum likelihood estimation (MLE). Sample sizes between consecutive superjerks are 29, 14, 58, 78, 8, 9, and 35, respectively. Red dashed line marks the mean field value ε = 1.33. T1: early warning by proposed method; T2: fracture strain from stress curve; T3: early warning by ScaNet45; T4: fracture point from optical observation. Data in (d) are presented as mean ± standard deviation. Source data are provided as a Source Data file.
The crack AE signals identified for the new dataset are depicted in Fig. 4c. Superjerk analysis reveals that the energy exponent decreases from approximately 2.0 at the onset of deformation to 1.6 at superjerk 8 (T1), and then further decreases to a mean field value of 1.33 at superjerk 10. This suggests that crack propagation begins to occur in the region between superjerk 8 and 10, with fracture initiation occurring immediately after superjerk 10. Given that no cracks are observed on the sample surface at superjerk 8 (DIC image at T1) and the tensile strength (DIC image at T2), superjerk 8 provides an early warning of failure within a safety margin. Thus, by comparing the early warning of failure based on the stress-time curve and in-situ optical observation in the new dataset, online monitoring of AE signals associated with crack propagation using the proposed approach proves to be more effective in monitoring early failures of metals with multiple deformation mechanisms.
The crack AE signals identified by the ScaNet model for the new dataset are illustrated in Supplementary Fig. S6. However, due to misidentification of crack signals, the early failure warning moment generated by the ScaNet (DIC image at T3) is observed to be later compared to the early failure warning moment generated by the proposed approach (T1) and the fracture strain moment estimated according to the stress-time curve (T2). This underscores the importance of accurate signal identification in AE analysis for early detection of failure in metallic materials.
In addition, in Supplementary Note-03 and Fig. S7, we further demonstrate that the proposed approach can be easily extended to identify other mixed deformation mechanisms, e.g., dislocation entanglement and dislocation movement signals in cold rolling 316L stainless steel. Furthermore, when independently applied to distinguish signals related to dislocations movement versus those related to dislocation entanglement, both the proposed approach and the statistical method produced highly similar results. The Kolmogorov–Smirnov test is employed to determine whether the AE energies of same mechanism follows the same power-law distribution. The lack of a statistically significant difference between the outcomes generated by the two approaches, as indicated by the Kolmogorov–Smirnov test with a derived p-value, p > 0.99, underscores the robustness and effectiveness of the proposed approach in accurately identifying deformation mechanisms from acoustic emission signals.
To verify that our classifier learns generalized features rather than simply memorizing training data, we performed a critical comparison using a new dataset (Fig. 4a). We evaluated: (1) our full approach versus (2) direct interpolation from known data (simulating pure memorization). MLE analysis (Supplementary Fig. S8) revealed that while both methods produced similar energy exponents for crack signals (ε = 1.46 ± 0.06 for memorization vs 1.43 ± 0.07 for our approach, consistent with mean field value12,28), they diverged dramatically for dislocation signals. The memorization approach failed to produce the characteristic plateau in MLE curves, indicating fundamental misidentification. In contrast, our method consistently achieved the expected plateau, demonstrating its ability to correctly identify underlying physical mechanisms rather than simply recalling training patterns. This clear differentiation confirms our classifier’s genuine learning capability beyond basic pattern memorization.
While our method enables detection of co-existing deformation mechanisms from AE signals, several challenges remain for real-world structural applications. Key limitations include sensor degradation, environmental variability, baseline drift, and structural complexity. Future research should focus on: (1) developing sensors with enhanced long-term stability, (2) creating robust environmental compensation methods, and (3) improving noise rejection algorithms. Promising directions include optimized sensor selection62,63, environment-specific calibration protocols64,65, prognostic health monitoring systems66, and advanced denoising techniques67,68. These complementary approaches will be essential for transitioning AE monitoring from laboratory settings to field applications.
In conclusion, we propose a knowledge-driven unsupervised learning approach, which utilizes supervised base learners as its backbone, and further incorporated with tailor-designed KIALF. Compared with peer learning methods, the approach demonstrates superior capability for identifying co-existing deformation mechanisms from AE signals, and a strong transferability in recognizing multiple co-existing deformation mechanisms from varying materials. The proposed approach can continuously monitor material deformations in real-time and dynamically providing early failure warnings.
Methods
Sample preparation
Porous 316L stainless steel samples are produced by powder metallurgy. The average grain diameter of the powder is ca. 20 μm. Porosity of the sample is about 35%, which is determined by relative density measurements. The composition of the porous 316L stainless steel and cold rolled 316L stainless steel is 8.7C-15.7Cr-11.1Ni-0.1Mn-2.5Mo-0.8Si (wt.%). Dog-bone shaped samples with a gauge range of 25 mm × 4 mm × 1.37 mm was cut using an electro-discharge machine.
Experiments
Tensile experiment
The room temperature uniaxial tensile experiments on tensile samples are performed using an Instron 5969 Universal Testing System. The testing system applies a tensile rate of 0.01 mm/min to the samples, ensuring controlled and consistent loading conditions. During the experiment, the time sequences of various parameters, including stress, strain, force, and displacement, are collected. These parameters provide essential information about the mechanical behavior of the material under tension. The data is recorded with a time resolution of 1 Hz, allowing for detailed analysis of the material’s response to the applied load over time. The tensile test configuration and specimen geometry are presented in Supplementary Fig. S9, with panel (a) and (c) showing the experimental setup and panel (b) detailing the dog-bone specimen dimensions.
Digital image correlation (DIC) analysis
During the tensile process of porous 316L stainless steel, optical images are captured at a rate of 2 frames per second. Given that the sample is porous, we conduct DIC analysis without the need for speckle, as the surface pores naturally provide contrast differences in the images. The DIC analysis is conducted in 2 dimensions using Ncorr69, which is an open-source MATLAB program specifically designed for 2D-DIC analysis. Ncorr enables precise measurement of deformation and strain fields by tracking the displacement of features within the images over time. By analyzing the displacement of features between successive images, DIC allows for the calculation of strain fields and deformation behavior of the material under loading conditions. This information is crucial for understanding the mechanical response of the material and correlating it with other experimental data, such as acoustic emission signals.
Detecting AE signals
AE measurements were conducted using a VS600-Z1 piezoelectric sensor (Vallen-Systeme GmbH; 4.75 mm diameter × 5.8 mm height) with a 200–800 kHz frequency range optimized for detecting both dislocation movement and crack propagation32,70, The sensor, mounted directly on the specimen surface, fed signals through a 49 dB pre-amplifier to an AMSY-6 acquisition system (Vallen-Systeme GmbH) operating at 20 MHz. System parameters were carefully calibrated through control experiments: a 100 μs duration discrimination time (DDT) was selected to capture dislocation events, while a 21.1 dB threshold was established using rubber specimens to isolate instrumentation noise from material-specific AE signals32. This background noise experiment allowed us to measure the noise without the AE effect of the metal sample. During the tensile test, the sensor continuously records the waveform stream. The equipment stores the signals segmented based on the amplitude threshold and DDT. This segmentation process allows for the isolation of individual AE events for further analysis. For each detected AE event, various statistical features are calculated, including energy, amplitude, duration, and others. These features provide crucial information about the characteristics of each AE signal, which are subsequently used for further analysis and classification of the deformation mechanisms.
Power spectral density extraction from AE waveform
In this work, the waveform of an AE signal is initially captured and transformed into its Power Spectral Density (PSD) spectrum, denoted as G(f), where f represents frequency ranging from 95 to 850 kHz. The PSD spectrum is discretized into 51 key frequency sampling points spanning the range of 0 to 1000 kHz, with a frequency gap of 20 kHz between every pair of adjacent sampling point. Each PSD estimate is normalized to unity by dividing G(f) by the total energy in the working frequency range, resulting in normalized PSD estimates denoted as \(\widetilde{G}\left(f\right)=G(f)/{\int }_{{f}_{\min }}^{{f}_{\max }}G(f){df}\), which are expressed in the logarithmic decibel (dB) scale. To reduce dimensionality and facilitate downstream processing, particularly for small data learning, a Principal Component Analysis based dimensionality reduction procedure is applied. This compression effectively reduces the initial 51-dimensional representation vector to a more compact 16-dimensional version while retaining the essential information. The compact representation vector is subsequently fed into a tunable classifier with adjustable hyperparameters.
Key implementation steps of the knowledge-driven learning framework with aggregate loss optimization
The learning framework initializes each base learner GDCn with random weights and iteratively optimizes its hyperparameters θn. During training, GDCn processes signal set Z = {zm | m = 1,2,…,M} with timestamps t = [t1,t2,…,tM], generating corresponding classification outputs COn =[COn,1,COn,2,…, COn,m]. These outputs are evaluated through a specialized temporal loss function \({{\mathcal{L}}}({{{\bf{CO}}}}_{{{\boldsymbol{n}}}},{{\boldsymbol{t}}})\) that operates on time-windowed signal subsets rather than individual data points. This architecture enables the model to learn from continuous temporal patterns characteristic of deformation mechanisms. Each GDCn undergoes independent optimization to minimize \({{\mathcal{L}}}({{{\bf{CO}}}}_{{{\boldsymbol{n}}}},{{\boldsymbol{t}}})\), producing an optimized learner GDCnopt.
Evaluation on the performance of a classifier
If AE signals follow the avalanche behavior, their AE energy distributions follow a power-law P(E) ~ E-ε. The MLE method is used to fit the energy exponent ε in greater detail. A plateau in MLE curve indicates that the AE signals represent one single deformation mechanism. Mixing of two deformation mechanisms leads to a decrease of the ML curve at higher energies. To quantitatively evaluate the performance of a classifier, the harmonic mean of the plateau length (HMLE) of dislocations (ld) and cracks (lc) in MLE curves (\({H}_{{\mbox{MLE}}}=\frac{2{l}_{d}{l}_{c}}{{l}_{d}+{l}_{c}+\xi }\)), is used to identify the performance of a classifier. ξ is a small constant to make sure the calculation is legitimate in case that both plateau length for dislocations and cracks (ld and lc) are zero. The plateau lengths of the MLE curves, ld and lc (unit: lg scale), are determined by human experts through majority voting. Any statistical bias of each mechanism will result in a low score.
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Data availability
The processed results data, together with synthetic and experimental AE datasets are available at https://github.com/XJTU-BYGou/AE_KDUL71. The statistical results generated in this study are provided in the Source Data file. Source data are provided with this paper.
Code availability
The code under GPL-3.0 license is available on GitHub at https://github.com/XJTU-BYGou/AE_KDUL71.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. W2411048 - X.D., No. 52401166 - Y.C.), 111 Project 2.0 (BP2018008 - J.S.), and the National Key Research and Development Program in China (No. 2022YFB3707600 - X.D.), E.K.H.S. is grateful to EPSRC (EP/P024904/1 - E.K.H.S.) and H2020 Marie-Sklodowska-Curie Actions (861153 - E.K.H.S.).
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X.D. and S.X. were the lead scientists of the study and proposed the core concept. Y.C. performed sample preparation and tensile experiments as well as acoustic emission and DIC measurement. B.G. implemented the codes, performed the training, and analyzed the data. B.G., S.X., and X.D. wrote the paper, with contributions from T.L. and E.K.H.S. X.D. and J.S. supervised the research. All authors contributed to the discussion.
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Gou, B., Chen, Y., Xu, S. et al. Detecting deformation mechanisms of metals from acoustic emission signals through knowledge-driven unsupervised learning. Nat Commun 16, 6877 (2025). https://doi.org/10.1038/s41467-025-61707-z
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DOI: https://doi.org/10.1038/s41467-025-61707-z