Nondestructive sheet resistance prediction of silver nanowire transparent electrode with convolutional neural network

Abstract

Measuring sheet resistance in metal nanowire films without causing damage is critical for advancing transparent conducting electrode technologies, particularly in emerging applications like flexible electronics, displays, and solar cells. Traditional measurement techniques such as four-point probe and Van der Pauw methods often compromise sample integrity and struggle with accurately capturing the electrical homogeneity of nanowire networks. The non-uniform distribution of nanowires significantly impacts electrical performance, with variations in wire density and junction connectivity leading to inconsistent conductivity and potential device failure. This research paper presents a deep learning technique combining Fast Fourier Transform (FFT)-derived and color metric features to predict the sheet resistance of silver nanowire networks. The inputs for the convolutional neural network (CNN) consist of raw high-resolution optical microscopy images, Fast Fourier Transforms of those images, average color representations, and a combination of all three data types, each processed separately. The combination of image, FFT, and average color data yields the best performance. The predictive capacity of the model extends to assessing non-uniformity in nanowire distribution, a crucial parameter for electronic applications. Thus, the integration of image-derived features provides a powerful tool for material property prediction, enhancing quality control, and advancing materials informatics within nanotechnology and device engineering.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-40528-0.

Introduction

Metal nanowires have become a crucial point of research in both industry and academia, given their diverse applications in the promising fields of displays, touch screens, photonics, and green energy harvesting modules like solar cells, where they serve as transparent flexible conducting electrodes (TCEs)^1–5. Considerable efforts have focused on synthesis methods, structural characterization, and device integration^6–19. The sheet resistance in metal nanowires is an important parameter that significantly influences the performance, energy efficiency, and durability of devices, underscoring the necessity for its precise measurement and control in metal nanowire applications^20–22. Among the various transparent conductors available, silver nanowire (AgNW) films have emerged as a leading choice due to their low sheet resistance, high optical transparency, and robust mechanical properties^7,23–26.

Diverse methods have been developed to measure sheet resistance, each with distinct advantages and limitations. Traditional contact methods, such as Van der Pauw and the four-point probe are widely used but risk damaging delicate nanowires and introduce inaccuracies due to contact resistance or surface contamination artifacts^27,28. Non-contact techniques such as terahertz time-domain spectroscopy (THz-TDS), eddy current testing, and electrochemical impedance spectroscopy (EIS) avoid physical contact but require complex calibration and are highly sensitive to optical and material properties^9,29,30.

The electrical performance of nanowire networks is profoundly influenced by their structural uniformity. Non-uniform distributions can lead to variations in local conductivity, which affects the overall sheet resistance of the film. This is because the conductivity in these networks relies on the formation of continuous pathways through overlapping nanowires. When the distribution of nanowires is uneven, it can create areas with insufficient connectivity, leading to increased resistance^31–33. A study by Han et al. highlights that the random distribution of nanowires can make conductive films non-uniform, affecting their electrical conductivity due to the percolation process required for current flow. This research found that the AgNW network formation is governed by two-dimensional diffusion behavior, which resulted in the precise matching of empirical and theoretical critical volume values. Moreover, junction resistance at the points where nanowires overlap plays an essential role in dictating the overall resistance of the network³⁴. High junction resistance can significantly impede electron flow across the network. Research by Choi et al. demonstrates that the wire-wire intersections in AgNWs-TCFs networks are not firmly attached, leading to elevated junction resistance and inconsistent electrical characteristics. The relationship between non-uniform spatial distribution and electrical characteristics is further elucidated in a study by Hwang et al.³⁵ They found that as the AgNWs length shortened from 15.5 to 7.86 μm, the film sheet resistance increased by about 6400%, while the non-uniformity factor rose from 0.29 to 0.52. This indicates that the spreading consistency of AgNWs on the substrate surface significantly impacts the formation of continuous and effective conductive paths³¹.

Understanding and predicting non-uniformity in nanowire networks is crucial for optimizing their performance in various applications. Non-uniform AgNW distribution can result in issues such as uneven heating and the formation of potential hotspots, particularly problematic in flexible heaters, as well as localized failure points in solar cells and touch screens³⁶. Kim et al. demonstrated that nickel-enhanced AgNW transparent films applied on flexible heaters showed good thermal stability and mechanical flexibility, emphasizing the importance of uniform nanowire distribution³⁷.

Given the limitations inherent in current experimental and computational approaches for determining sheet resistance in nanowire thin films, there is a compelling need for alternative methodologies. Deep learning presents a promising opportunity for predicting the sheet resistance of metal nanowires due to its capability to learn complex patterns and relationships from large datasets. This would enable rapid, non-destructive, and scalable assessments of nanowire networks by predicting accurate sheet resistance without direct physical measurements. Deep learning and machine learning have profoundly transformed problem-solving across various scientific domains, including materials science and engineering^38–40.

Deep learning models, especially CNNs, excel at recognizing complex patterns within image data that correlate with material properties like sheet resistance. These models can process large datasets of microscopy images to learn features associated with uniformity and predict variations across samples. The ability to predict non-uniformity using deep learning not only aids in quality control but also informs process optimization during manufacturing. For instance, a study demonstrated how machine learning could be used to enhance optical microscopy for rapid characterization of nanoparticle morphology^40,41. Similarly, applying these techniques to AgNW networks would enable precise predictions about network uniformity and associated electrical properties.

These capabilities position CNNs as an ideal approach for addressing the uniformity prediction challenge in nanowire networks. The broader incorporation of these methods aligns with the trends in the natural sciences, where artificial intelligence (AI) is increasingly utilized in microscopy image analysis to predict material structural properties that were traditionally determined through experimental means.

Deep learning methods have been employed to study microscopy images for structure identification and to explore the connection between microstructure and performance^42–45. Convolutional neural networks (CNNs) have been successfully applied to classify carbon nanostructures from transmission electron microscopy (TEM) images, leveraging hyper-column feature extraction and clustering algorithms for high accuracy. Models like Mask R-CNN and U-Net have proven adept at segmenting and identifying atomic positions and defects in scanning transmission electron microscopy (STEM) images, demonstrating generalization across materials and the ability to predict complex vacancy structures. Additionally, AI techniques have found applications in enhancing the resolution of microscopy images obtained from TEM, scanning electron microscopy (SEM), and scanning tunneling microscopy (STM), unveiling fine details that were previously difficult or impossible to observe^42,43,46,47. This integration of AI with microscopy techniques has significantly advanced image analysis and interpretation, facilitating recognition and prediction of material structural properties such as morphology, distribution, size, and intensity. Despite these advancements, the assessment of physical properties from structural features extracted by using microscopy images remains underexplored.

In this study, we demonstrate the power of deep learning to predict the sheet resistance and non-uniformity of silver nanowire networks, critical parameters that determine their performance in various applications such as transparent electrodes, flexible electronics, and smart textiles. By leveraging a CNN, we integrate high-resolution optical microscopy images with advanced image processing techniques, comprising the Fast Fourier Transform (FFT) and color metric features, to extract both microscopic and macroscopic characteristics that correlate with the electrical properties of the nanowire films. The FFT is a powerful mathematical tool that converts an image from the spatial domain to the frequency domain, revealing valuable information about the periodicity, orientation, and uniformity of the nanowire network that are not immediately visible in the original images. A key advantage of using FFT in this context is its ability to efficiently capture and quantify the spatial frequencies and patterns within the nanowire networks, which are directly related to their electrical properties, while also significantly reducing computational complexity compared to direct spatial analysis.

To further enhance the predictive power of our model, we complement the FFT-derived features with macroscopic characteristics such as the average color of the microscopy images. This holistic approach allows the model to capture both the intricate structural details and the overall material properties, enabling accurate predictions of sheet resistance and non-uniformity based on non-destructive optical imaging. Our methodology not only offers a rapid and cost-effective alternative to traditional electrical characterization techniques but also provides a valuable tool for material property prediction, enhancing quality control, and advancing materials informatics within nanotechnology and device engineering.

Materials and methods

Silver Nanowire (AgNW) Synthesis and Sample Fabrication

The AgNW solutions were characterized by specific morphological properties to ensure high conductivity and transparency: an average diameter of 43 ± 5 nm and an average length of, 19 ± 5 μm, resulting in a high aspect ratio of approximately 440. The commercial AgNW dispersions contained solid content ranging from . For electrode fabrication, the solution was prepared in five distinct weight percentages (, , , , and ) to systematically control final network density. Transparent conductive electrode samples were fabricated by spray-coating the solutions onto standardized glass substrates. To achieve a wide range of sheet resistance values spanning the practical operating window(10–200Ω/sq) the spray-coating process was systematically varied from to . All samples were subsequently annealed at for to reduce inter-wire contact resistance and stabilize the network structure. The effect of solution concentration and coating cycles on AgNW film properties is presented in Supplementary Figure S1.

Data acquisition

Sheet resistance (R_s) measurements

Sheet resistance was measured using a Model M4P205 Four-Point Probe system with a fixed 1000 μm probe pitch, interfaced with a Keithley 2450 digital multimeter. The standard four-point technique was employed, applying current through the outer probes (0.1 to 100 mA) and measuring the voltage drop across the inner probes, with sheet resistance calculated using the geometric correction factor: R_s=4.532×(V/I).To ensure representative electrical characterization of the 4 cm×4 cm substrate, R_s was measured at six systematically selected locations (the center and five points at approximately1cmintervals). Prior to measurements, the system was calibrated weekly using a 100 Ω/sq reference standard. All measurements were conducted at ambient laboratory temperature (20–25 °C)⁴⁸.

Optical microscopy image collection and spatial sampling

High-resolution images of the AgNW networks were collected using a Nikon Eclipse LV100ND optical microscope. Images were captured at a resolution of 1536 × 1024 pixels using a fixed 10× magnification, corresponding to a Field of View (FOV) of approximately 1.28 mm×0.85 mm. Given that this FOV captures only ∼1% of the total substrate area, an effective spatial sampling strategy is critical for model accuracy. For the multi-image prediction study (Fig. 5), ten images were captured from each AgNW sample at randomized spatial locations distributed across the substrate surface: one from the center, four from cardinal directions (∼1 cm from center), and five from intermediate positions. This systematic protocol ensures that the ensemble of images captures the inherent spatial heterogeneity and non-uniformity of the spray-coated network, which is essential for accurate ensemble prediction. Consistent imaging parameters were maintained across all samples: fixed 10× magnification, consistent illumination intensity, exposure time, and focus depth.

Fig. 4.

Scatter plots comparing predicted and true values for sheet resistance and its standard deviation, organized by resistance category: samples with true sheet resistance below 50 Ω/sq (Low Resistance, High Nanowire Concentration, shown in red circles) versus samples with true sheet resistance of 50 Ω/sq or higher (High Resistance, Low Nanowire Concentration, shown in blue circles). (a) Single-image sheet resistance prediction demonstrates an overall of 0.862, with category-specific performance of 0.901 for Low Resistance samples and 0.805 for High Resistance samples. (b) Sheet resistance prediction derived from averaging multiple images achieves an overall of 0.868, with category-specific values of 0.885 for Low Resistance and 0.811 for High Resistance samples. (c) Single-image non-uniformity (standard deviation) prediction yields an overall of 0.720, with segmented results of 0.776 for Low Resistance and 0.636 for High Resistance samples. (d) Multi-image averaging approach for non-uniformity (standard deviation) prediction improves the overall to 0.750, achieving 0.885 for Low Resistance samples and 0.679 for High Resistance samples.

The total dataset used for model training and validation consisted of 1,440 high-resolution optical microscopy images of silver nanowire networks. To ensure robust model development and unbiased performance evaluation, this dataset was randomly divided into training, validation, and testing sets using an 80:15:5 ratio. This resulted in 1,152 samples allocated to the training set, 216 samples dedicated to the validation set (used for hyperparameter tuning and preventing overfitting), and 72 unique samples reserved exclusively for the final, blind performance evaluation in the testing set. This strict separation ensures that all reported performance metrics reflect genuine model generalization capability on completely unseen silver nanowire samples.

FFT implementation details

The practical implementation of the FFT-based feature extraction follows a systematic pipeline designed to capture frequency-domain characteristics of AgNW networks. First, each optical microscopy image is converted to grayscale to obtain single-channel intensity data suitable for frequency analysis. The 2D FFT is computed using NumPy np.fft.fft2() function, which efficiently transforms the spatial image into the frequency domain with computational complexity O(N²log₂N). For a detailed discussion on the computational advantages of FFT over the Discrete Fourier Transform (DFT), please refer to Supplementary Note 1 in Supporting Information. Following the FFT computation, frequency shifting (np.fft.fftshift()) centers the zero-frequency DC component, and the magnitude spectrum is calculated as to compress the dynamic range for visualization and processing.​ This logarithmic scaling is essential because frequency spectra typically span multiple orders of magnitude, and the log transformation enhances the visibility of subtle structural features.​⁴⁹

The resulting FFT magnitude spectrum encodes critical structural information about the AgNW network. Low-frequency components represent large-scale uniformity and overall nanowire density, while high-frequency components capture fine details such as individual nanowire edges and junction points. Directional patterns in the frequency domain reveal preferential nanowire orientations, which directly influence electrical conductivity pathways and sheet resistance. The processed FFT image is then merged with the original grayscale image and a sharpened version into a three-channel RGB composite, enabling the CNN to simultaneously learn from spatial, frequency, and edge-enhanced representations. This multi-domain fusion strategy significantly improves prediction accuracy compared to using any single feature type alone, as demonstrated by the model performance comparison in Fig. 5.

Image preprocessing

Figure 1 presents the comprehensive data preprocessing methodology employed to prepare the optical microscope images of silver nanowires for subsequent analysis. The original high-resolution image captures a complex network of silver nanowires, appearing as bright, interconnected strands against a dark background, and serves as the baseline for extracting both microscopic and macroscopic characteristics, which are crucial for understanding variations in density and distribution. To extract the microscopic characteristics crucial for predicting sheet resistance, a Fast Fourier Transform, an optimized algorithm for computing the two-dimensional Discrete Fourier Transform (2D DFT), is applied to translate spatial information into the frequency domain.

Fig. 1.

Multi-domain feature extraction pipeline integrating spatial, frequency-domain, and macroscopic characteristics. Panel (a) shows the original optical microscopy image of the AgNW network. Complementary feature domains are extracted in (b) the FFT magnitude spectrum (frequency domain), revealing periodicity and alignment patterns, and (c) the average color (macroscopic density). The RGB composite is constructed by assigning the following feature channels: Red (d) = original spatial morphology; Green (e) = FFT magnitude spectrum; and Blue (f) = edge-enhanced features. Panel (g) displays the full-resolution composite (1536 × 1024 × 3). Panel (h) shows the final 32 × 32 × 3 downsampled input used for CNN prediction, achieved through bicubic interpolation to preserve structural information while reducing computational complexity. This multi-domain integration enables robust prediction of sheet resistance and non-uniformity.

The multi-domain feature extraction pipeline transforms raw optical microscopy images into a three-channel input suitable for CNN-based prediction, as illustrated in Fig. 1. The original microscopic image (a) captures the spatial domain, displaying the complex morphology of bright, interconnected strands of silver nanowires against a dark background. When processed through Fast Fourier Transform (FFT), the magnitude spectrum (b) (with zero frequency shifted to the center) reveals structural periodicity not immediately visible in the spatial domain.

Low-Frequency Components and Conductivity: The bright central spot in the FFT image represents the DC (low-frequency) components, capturing gradual brightness variations and overall nanowire distribution. Physically, low-frequency components correspond to large-scale structural features—namely, the overall density and spatial arrangement of nanowires across the substrate. Networks with uniform, well-distributed nanowires produce strong, compact low-frequency patterns, indicating consistent conductivity pathways. Conversely, sparse or irregularly distributed nanowires show diffuse low-frequency patterns, suggesting fragmented conductive pathways and higher sheet resistance. Thus, the characteristics of low-frequency components directly correlate with the bulk electrical properties of the film.

High-Frequency Components and Conductivity: Critically, the cross pattern in the FFT image indicates strong horizontal and vertical frequency components, suggesting prominent alignment and periodicity that provide insights into nanowire uniformity and orientation directly related to sheet resistance. As distance increases from the center, intensity decreases, indicating higher-frequency components that correspond to finer structural details such as sharp edges of individual nanowires and junction points. Physically, high-frequency components encode localized features, the quality and density of nanowire junctions, edge sharpness, and fine-scale variations in wire thickness. Since electrical current must flow through nanowire-nanowire junctions, the quality of these junction regions is critical. High-frequency features that indicate sharp, well-defined junction structures suggest low contact resistance at wire intersections, enhancing overall film conductivity. Conversely, poorly defined or sparse junctions (represented by weak high-frequency components) imply elevated junction resistance and degraded conductivity.

The third information domain, macroscopic characteristics, is captured by the average color (c). Represented as a neutral gray tone, this feature is derived from the overall light intensity and color distribution, reflecting the balance between the bright nanowires and the dark background. Average color serves as a quick, yet robust, visual indicator of overall material distribution and density. These three distinct information sources, spatial morphology (large-scale distribution), frequency domain features (periodicity and junction quality), and macroscopic characteristics (overall coverage), are then merged into a three-channel RGB composite. The channel assignment is crucial for preserving the independent data domains: the Red channel (d) carries the original spatial information, the green channel (e) contains the FFT magnitude spectrum encoding frequency-dependent junction and alignment features, and the Blue channel (f) provides enhanced edge features highlighting fine structural boundaries and nanowire junction points. The resulting full-resolution RGB composite (g) preserves complete spectral information before preprocessing. The final step involves downsampling the input from 1536 × 1024 to 32 × 32 × 3 to create the final CNN input image (h). This is a critical preprocessing step, performed via bicubic interpolation, which ensures the preservation of essential structural information while simultaneously achieving substantial computational efficiency. The 32 × 32 resolution is highly advantageous as it significantly reduces computational overhead and input dimensionality, enabling efficient real-time processing for practical deployment without sacrificing the multi-domain structural details necessary for accurate sheet resistance prediction.

While pixel reduction is crucial for achieving high computational efficiency, it introduces a primary drawback: the potential degradation of accuracy due to the loss of high-frequency information. This lost detail includes the fine-scale geometry of individual nanowire edges and junction structures, features vital for accurately modeling electron transport and junction resistance. Our methodology employs two key strategies to mitigate this loss and ensure robust performance: First, we utilized bicubic interpolation for downsampling, a method superior to simpler techniques (e.g., nearest neighbor) as it preserves subtle structural features and edge clarity far more effectively. Second, and more critically, we employed our multi-domain feature encoding strategy. The Edge-Enhanced Blue Channel (f) and the FFT Magnitude Green Channel (e) explicitly encode the critical high-frequency information (junction details and structural periodicity) prior to downsampling. By compressing these essential physical parameters into their own dedicated input channels, we guarantee their retention in the final input, effectively decoupling the computational efficiency gain from severe accuracy degradation. This successful trade-off allows us to achieve high prediction accuracy with minimal computational overhead, making the model suitable for real-time quality control applications.

Deep learning model architecture

The model design for predicting the sheet resistance of AgNW transparent electrodes combines a CNN architecture tailored for image analysis as shown in Fig. 2 The input to the model is a 32 × 32 pixel image with 3 color channels representing the microscopic characteristics of the AgNW network. The model begins with a sequence of convolutional layers, each succeeded by batch normalization and activation functions. The first layer applies 64 filters with a kernel size of 3 × 3 to the input image, extracting low-level features such as edges and textures⁵⁰. Subsequent convolutional layers with 128 and 256 filters analyze progressively more complex features within the images. Each convolutional block is succeeded by a max-pooling layer that halves the spatial dimensions, effectively compressing the feature representation and making the model more robust to changes in the feature location⁵¹. Dropout layers are interspersed among the convolutional blocks to avoid overfitting by randomly deactivating a portion of the input units during training. Following the convolution and pooling layers, the output is flattened and fed into dense layers with ReLU activation, further refining the feature set. The final dense layer has a linear activation function, which correlates the learned features to the continuous value of sheet resistance, the target output. The learning curves during model training and validation demonstrate convergence after approximately 100 epochs, as detailed in Supporting Information, Figure S2.

Fig. 2.

Convolutional neural network architecture for predicting sheet resistance. The figure illustrates the CNN model architecture used to predict the sheet resistance of AgNW transparent electrodes. The model processes 32 × 32 pixel images with 3 color channels through successive convolutional layers (64, 128, and 256 filters) with batch normalization, activation functions, max pooling, and dropout layers. The feature maps are subsequently converted into a one-dimensional array and fed into densely connected layers, concluding with a linear activation to output the sheet resistance.

Results and discussion

Sheet resistance prediction

Equations 3 and 4 illustrate the meaning of RMSE and R². The RMSE, as shown in the Eq. 3, measures the average magnitude of the prediction errors, providing a direct indication of the model’s prediction accuracy. It is computed as the square root of the average squared differences between the predicted () and actual () values. A lower RMSE value indicates better predictive performance. The R² metric quantifies the proportion of variance in the actual values that can be explained by the predicted values. It is defined as one minus the ratio of the residual sum of squares () to the total sum of squares (), and it ranges from 0 to 1. An R² value closer to 1 signifies that a larger portion of the variance is accounted for by the model, indicating higher explanatory power and predictive accuracy.

3

 

4

 

The performance metrics comparing different input feature combinations are summarized in Table 1, which presents RMSE and R² values for sheet resistance prediction of AgNW electrodes using image-only, FFT-only, average-color-only, and the combined Image + FFT + Average approach. The combined approach achieved the best performance with an RMSE of 6.17 Ω/sq and R² of 0.803, demonstrating the effectiveness of multi-domain feature integration. To assess the effectiveness of the recognition method across the full experimental range (13–200 Ω/sq), we evaluated the prediction error across distinct resistance intervals determined by the fabrication conditions, as shown in Table S1. The results indicate that model performance is influenced by the uniformity of the conductive network, which varies with ink concentration and coating cycles. Specifically, samples with higher material loading (< 50 Ω/sq) exhibited the highest precision (R² = 0.885), while the higher-resistance regime showed increased variance consistent with the stochastic nature of percolation networks.

Table 1.

Comparisons of RMSE and R² for sheet resistance prediction of AgNW electrodes.

Sheet Resistance	Image only	FFT only	Average only	Image + FFT + Average
RMSE	6.90	8.56	7.23	6.17
R2	0.754	0.622	0.730	0.803

Non-uniformity prediction

In Fig. 3, the predictive model ability to discern non-uniformity in nanowire networks is showcased by contrasting two samples with similar sheet resistances but differing standard deviations. Sample (a) exhibits a sheet resistance of 26.37 Ω/sq and a standard deviation of 1.59 Ω/sq, suggesting a more uniform network, as visually supported by a relatively even original image and a focused FFT pattern. The average color feature for sample (a) is also notably darker, indicative of the uniform density of nanowires. On the other hand, sample (b) displays a sheet resistance of 26.73 Ω/sq with a higher standard deviation of 5.15 Ω/sq, revealing more irregularity. This is visually apparent in the original image through the more erratic spacing of nanowires, a dispersed FFT pattern, and a significantly lighter average color feature, which corresponds to the less dense areas of nanowires. The ‘Sum’ images integrate these characteristics, further emphasizing the pronounced difference in darkness of the average color between samples (a) and (b), which align with the variance in their uniformity. These observations are essential for the model’s predictive accuracy regarding the uniformity of conductive networks.

Fig. 3.

Non-uniformity prediction (a) Sample with sheet resistance of 26.37 Ω/sq and a standard deviation of 1.59 Ω/sq, indicating uniform nanowire distribution. (b) Sample with sheet resistance of 26.73 Ω/sq and a standard deviation of 5.15 Ω/sq, indicating higher irregularity of nanowire distribution. The comparison includes original images, FFT patterns, average color features, and ‘Sum’ images to highlight differences in uniformity.

Table 2 showcases the effectiveness of various models in predicting the standard deviation of sheet resistance, which is a vital measure of non-uniformity in nanowire networks. The models compared include those using only image data, FFT data, only average color data, and a combination of all three. The “Image only” model has an RMSE of 2.27 and an R² of 0.594, indicating moderate predictive accuracy. The “FFT only” model shows less accuracy with a higher RMSE of 2.65 and a lower R² of 0.447. The “Average only” model performs slightly better with an RMSE of 2.44 and an R² of 0.533. The best results are observed in the combined “Image + FFT + Average” model, achieving the lowest RMSE of 2.04 and the highest R² of 0.671. While the R² improvement appears modest (0.671 vs. 0.594), the 10.1% RMSE reduction (2.27→2.04 Ω/sq) is highly significant for manufacturing contexts. This multi-domain approach provides essential robustness to optical variations, illumination changes, and microscope drift that spatial features alone cannot capture. Given the negligible computational overhead (< 2% parameters, < 5% training time), the improved generalization capability fully justifies the architectural complexity, making the model highly practical for industrial deployment. The comparison with other sheet resistance measurement methods for transparent conductive films can be found in Supporting Information in Table S2.

Table 2.

Comparisons of RMSE and R² for non-uniformity prediction of AgNW electrodes.

Sheet Resistance	Image only	FFT only	Average only	Image + FFT + Average
RMSE	2.27	2.65	2.44	2.04
R2	0.594	0.447	0.533	0.641

It is important to note that the superior performance of the ‘Image + FFT + Average model is achieved with negligible computational overhead. Since the FFT and average color features are mathematically derived from the same raw image, no additional image acquisition or hardware modification is required. The preprocessing step (FFT computation) adds less than 50 ms per image, and because all models utilize the same 32 × 32 × 3 input dimension, the CNN inference time remains identical across all variants. Thus, the multi-domain approach offers a 10.1% improvement in prediction accuracy without compromising the real-time capability of the system.

Figures 4(a)–(d) present the predicted versus true values for sheet resistance and standard deviation of sheet which serves as a key metric for non-uniformity. The data are organized into two distinct concentration-dependent regimes based on true sheet resistance: Low Resistance (Under 50 Ω/sq, High AgNW Concentration) and High Resistance (Over 50 Ω/sq, Low AgNW Concentration). Predictive performance for each segment is quantified by the R² value, as summarized in Table 3.

Table 3.

Segmented non-uniformity prediction in AgNW networks: single-image vs. multi-image averaging for Low vs. High Resistance groups, showing a marked R² gain in the Low Resistance group.

Figure	Type	Overall R²	Low Resistance R²	High Resistance R²
3a	Single Image - Sheet Resistance	0.862	0.901	0.805
3b	Multiple Image Avg - Sheet Resistance	0.868	0.885	0.811
3b	Single Image - Std Dev	0.720	0.776	0.636
3d	Multiple Image - Std Dev	0.750	0.885	0.679

Sheet resistance prediction demonstrates consistently high accuracy across both concentration regimes (Figs. 4(a) and 4(b)). In single-image analysis (Fig. 4(a)), the model achieves peak performance with an R² of 0.901 for the Low Resistance group, which typically features denser and more uniform nanowire networks. Multi-image averaging (Fig. 4(b)) shows a modest modulation of the Low Resistance performance (R² = 0.885), while maintaining stable accuracy for the High Resistance group (R² ≈ 0.81). The consistently superior accuracy for Low Resistance samples indicates that the model recognition of sheet conductivity is most robust when the nanowire network exhibits high density and strong interconnectivity.

The segmented analysis of non-uniformity prediction (Figs. 4(c) and 4(d)) reveals the profound impact of multi-image data integration. Single-image predictions (Fig. 4(c)) display a substantial performance divergence between concentration regimes, with R² values of 0.776 for Low Resistance versus 0.636 for High Resistance. This differential performance suggests that characterizing the inherently low standard deviation of uniform, high-concentration networks from isolated images is fundamentally limited by image-specific noise and localized measurement artifacts.

The recognition benefit of multi-image averaging becomes especially pronounced in non-uniformity prediction. Figure 4(d) demonstrates that averaging predictions across multiple images dramatically elevates R² for the Low Resistance group from 0.776 to 0.885, a decisive 13.9% improvement. This substantial gain demonstrates that the averaging process effectively attenuates image-specific noise and noise artifacts, yielding a significantly more reliable estimate of the true, intrinsically low standard deviation characteristic of highly uniform networks. The High Resistance group also benefits from averaging, with R² improving to 0.679, although the magnitude of improvement is more modest. The most dramatic performance enhancement and highest absolute accuracy is achieved in the Low Resistance regime, establishing multi-image averaging as a critical methodology for high-fidelity non-uniformity quantification in densely networked samples.

The segmented analysis demonstrates that the deep learning model achieves optimal predictive accuracy in the Low Resistance (High AgNW Concentration) regime and that multi-image averaging is essential for achieving high-fidelity non-uniformity prediction in these highly uniform networks. These findings underscore the importance of concentration-dependent analysis and data integration strategies for developing robust metrology tools for transparent conductive films.

Data comparison

Fig. 5.

Comparison of two AgNW samples: (a) with low sheet resistance (14.67 Ω/sq) and (b) with high sheet resistance (173.52 Ω/sq).

Figure 5 presents a comparative analysis of two distinct samples of AgNW, explained as (a) and (b), showcasing variations in sheet resistance. Sample (a) exhibits a low sheet resistance of 14.67 Ω/sq, while sample (b) has a high sheet resistance of 173.52 Ω/sq. The original images column displays the visual complexity of the nanowire networks, where the density and morphology of the wires are apparent. The corresponding FFT images provide insight into the spatial frequency of the nanowire distribution, with the brightness and sharpness of the central peaks potentially correlating with the uniformity and alignment of the nanowire network, which are critical factors influencing electrical conductivity. The average color columns, through their grayscale tonality, suggest differences in overall wire density and thickness, with sample (a) displaying a lighter tone indicative of less dense or thinner wires compared to sample (b). These macroscopic visual cues might be directly associated with the observed electrical properties, where denser and possibly thicker networks result in higher conductivity.

The Sum images represent a critical diagnostic visualization for the multi-domain approach. They are generated by performing an element-wise summation of the pixel values from the three normalized feature channels (Spatial, FFT Magnitude, and Edge-Enhanced features). This operation creates a single-channel composite that is normalized for display. The “Sum” image is not used as a direct input channel for the CNN model. Instead, it serves to visually synthesize the information from all three channels, accentuating contrasts and allowing observers to clearly differentiate structural complexity. This visualization provides compelling diagnostic evidence for why the multi-domain CNN architecture achieves robust prediction accuracy (low RMSE in Table 2) by demonstrating how the superposition of features highlights structural irregularities crucial for predicting resistance.

This comparative analysis, integrating both microscale FFT-derived patterns and macroscale colorimetric attributes, underscores the multifaceted approach required to accurately predict the sheet resistance of silver nanowires using deep learning techniques.

Generalization capability and future work

Our current study focuses exclusively on silver nanowire (AgNW) networks, which represent one of the most challenging material systems for accurate sheet resistance prediction. AgNW networks inherently exhibit high non-uniformity due to stochastic deposition, high junction resistance variability, and electrical instability. Given that our methodology successfully predicts sheet resistance in this complex system, we expect comparable or improved performance on more uniform metallic nanowire structures, such as those made from copper or gold. The core mechanism of our deep learning approach, translating image-based structural features (including orientation and periodicity extracted via FFT) into electrical properties, is fundamentally designed to handle the structural complexity inherent in nanowire meshes^52,53.

While the current methodology is robust for nanowires, limitations may arise when predicting sheet resistance for structures with fundamentally different geometries, such as non-filamentary nanospheres or continuous nanosheets, where the physical relationship between image features and bulk electrical percolation is distinct. Similarly, substantial variations in the single particle dimensions (diameter or length) or substrate type may necessitate model retraining. Therefore, future work will focus on expanding the model generalizability through comprehensive validation across these different material forms (nanospheres, nanosheets), other metallic nanowires (copper, gold), varying particle dimensions, and diverse substrate types. This effort is necessary to establish the approach full applicability as a universal, high-throughput quality control tool for emerging transparent electrode technologies.

Conclusion

This study presents a non-destructive method for predicting the sheet resistance of silver nanowire networks, which are critical components in transparent conductors. Traditional methods for measuring sheet resistance can be both damaging and inaccurate for delicate nanowire films, making this approach highly valuable for industries focused on process monitoring and quality control. By leveraging high-resolution optical microscopy images integrated with Fast Fourier Transform derived features and average color metrics, the study offers a rapid, efficient alternative to direct physical measurements. The FFT is particularly important because it reveals structural characteristics such as periodicity, orientation, and uniformity in the frequency domain that are not immediately visible in the spatial domain, providing deeper insights into nanowire distribution. This comprehensive approach not only improves predictive accuracy but also enhances understanding of electrical properties like sheet resistance and non-uniformity in nanowire networks. Beyond post-fabrication quality control, this deep learning approach holds significant promise for real-time monitoring and process optimization in additive fabrication techniques. The rapid, non-destructive prediction of sheet resistance from optical microscopy images could be particularly valuable for in-line monitoring during Drop-on-Demand inkjet printing and other additive deposition methods. During such fabrication processes, real-time feedback on nanowire network uniformity and sheet resistance could enable dynamic adjustment of deposition parameters, such as printing speed, drop volume, and substrate temperature, to promote consistent film quality and minimize defects. The integration of this methodology with additive manufacturing systems would facilitate closed-loop control strategies, ultimately improving manufacturing efficiency and product reliability. Future work will focus on adapting the current methodology for integration with such fabrication systems and validating its performance on growing deposits and partially formed networks.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

**Yewon Han and Joel Ndikumana: ** Conceptualization, methodology, silver nanowire sample fabrication, deep learning model development, data acquisition, and original draft preparation. They contributed equally to this work. **Suwon Choi and Donghun Lim** : Experimental measurement of sheet resistance and Optical microscopy image acquisition. **Ildoo Kim and Jun Young Kim: ** Data analysis and interpretation of electrical characteristics. **Junsuk Rho and Kunsik An: ** Conceptualization, supervision, project administration, funding acquisition, and final review and editing of the manuscript. All authors reviewed and approved the final manuscript.

Funding

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1I1A3059714). This work was also supported by a project for Collabo R&D between Industry, University, and Research Institute funded by Korea Ministry of SMEs and Startups in 2025 (RS-2025-02313349). This research was also supported by the Ministry of Trade, Industry & Energy (MOTIE) and the Korea Institute for Advancement of Technology (KIAT) of the Republic of Korea (RS-2025-17022968). This work was also supported by the Ministry of Science and ICT (MSIT) of the Republic of Korea, through the “Digital Innovation Hub Sup-port Project” funded by the National IT Industry Promotion Agency (NIPA) and the Chungbuk Innovation Institute of Science & Technology (CBIST).

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.