A generative diffusion model enables multi-objective on-demand inverse design of piezoelectric metamaterials†

Chun-Yu Lei^ab, Jian Wang^a, Run-Lin Liu^ab, Meng-Jun Zhou^a and Zhong-Hui Shen*^ab
aState Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Center of Smart Materials and Devices, Wuhan University of Technology, Wuhan 430070, China. E-mail: zhshen@whut.edu.cn
^bSchool of Materials and Microelectronics, Wuhan University of Technology, Wuhan 430070, China

First published on 22nd July 2025

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1. Introduction

Piezoelectric materials are multifunctional electronic materials with the capacity to convert mechanical energy into electrical energy,^1,2 which have broad applications in wireless communications,³ ultrasonic sensing,^4,5 wearable sensing devices,^6,7 and robotics.^8,9 In addition to the piezoelectric coefficient, d₃₃, other parameters must be specifically tailored to meet the multiple performance requirements of different applications. For example, acoustic applications such as hydroacoustic transducers and medical ultrasonic probes require acoustic impedance (, where ρ is the relative density, ρ₀ is the intrinsic density and E is Young's modulus) that matches with that of water and biological tissues,¹⁰ respectively. Mechanically, different sensing components within a robot demand tailored mechanical responses, while wearable devices must adapt to the body's ergonomic characteristics.¹¹ In sensing applications, a high piezoelectric voltage coefficient (, where d₃₃ is the piezoelectric coefficient, ε^T₃₃ is the relative permittivity and ε₀ is the vacuum permittivity) is highly desirable for generating higher output voltage, while a high figure of merit is critical for maximizing power output.¹² As illustrated in Fig. 1a, the above performance metrics are basically governed by four fundamental parameters: the piezoelectric coefficient d₃₃, relative permittivity ε^T₃₃, Young's modulus E and relative density ρ. However, these four parameters exhibit a robust interdependence, whereby any one property exerts a constraining influence on the others.¹³ As a result, traditional dense piezoelectric ceramics are unable to achieve on-demand tuning of multiple performance metrics by balancing the piezoelectric response and mechanical compatibility.

To meet the multi-performance demands of various application scenarios, introducing pores into piezoelectric materials has proved effective to break the strong coupling relationship between these four parameters.^13–15 Although pores simultaneously reduce both d₃₃ and ε^T₃₃, d₃₃ decreases more slowly, resulting in a higher g₃₃ and FOM₃₃. Moreover, pores lower the Z, expanding the application in the field of ultrasound with a lower ρ and enhancing the tunability of the piezoelectric material's mechanical properties.¹⁰ Therefore, the design of porous structures represents a pivotal aspect in the tailoring of the piezoelectric properties of porous ceramics. However, some common methods, such as the freeze-drying method,¹⁶ have inherent limitations in producing pores with diverse shapes, which in turn restricts structural design flexibility. The advances in additive manufacturing have enabled scientists to fabricate intricate and meticulously manipulable porous structures,^8,17,18 commonly referred to as metamaterials. Metamaterials, often realized through the self-assembly of small-scale blocks, exhibit properties unattainable in natural materials.¹⁹ As illustrated in Fig. 1b, piezoelectric metamaterials exhibit enhanced multi-performance tunability and expanded design flexibility in comparison with bulk ceramics and conventional porous ceramics.¹⁷ To date, researchers have achieved novel piezoelectric properties, including fully non-zero, negative,¹⁸ and twist piezoelectric coefficients,⁸ which show the potential to revolutionise robotics and intelligent sensing applications. However, the vast multidimensional structure–property space of piezoelectric metamaterials renders it nearly impossible for human intuition to precisely guide experiments that meet the demands of diverse scenarios. Furthermore, the structure–performance relationships required for multi-scenario and multi-objective designs are more complex than ever before.^20,21 Tailoring metamaterial architectures to achieve specific multi-objective behaviors across various application scenarios remains a significant challenge.^22,23

The ongoing developments in artificial intelligence (AI) and machine learning (ML)²⁴ are making the on-demand design of piezoelectric metamaterials increasingly attainable.^25–27 Nevertheless, the prevailing machine learning methods mainly achieve optimization of design indirectly by optimizing key parameters for the generation of metamaterial structures.^27–29 These methods often rely on specific configurations of training data and may encounter difficulties when applied to entirely new scenarios. In recent years, there has been growing interest in some generative models such as variational autoencoders (VAEs) and generative adversarial networks (GANs).²⁷ However, these models are largely limited to single-target designs³⁰ and may exhibit issues such as mode collapse and limited diversity in generated outcomes.³¹ The primary challenges in multi-objective design lie in the complex coupling and trade-offs among multiple properties, particularly when simultaneously optimizing the electrical, mechanical, and acoustic performance.¹³ Enhancing one property may significantly impact others, resulting in a high-dimensional, nonlinear optimization problem with potentially conflicting objectives.²⁰ The traditional ML models struggle to efficiently navigate and represent such nonlinear, multi-dimensional trade-offs. To date, there has been no satisfactory approach for addressing the on-demand design of piezoelectric metamaterials with coupled electrical, mechanical, and acoustic multi-objective properties.

The advent of text-to-image large models, such as latent diffusion models,³² has introduced a good approach towards multi-objective on-demand design. In this model, designers only need to input text descriptions containing multiple objective conditions (such as geometry, size, color, surface texture, etc.), and the model can generate images that meet these specifications.³² Compared with traditional generative methods such as VAEs and GANs, as well as optimization-based approaches like Genetic Algorithms (GAs), diffusion models offer superior generation fidelity, higher training stability, better controllability, and significantly improved efficiency, especially in high-dimensional and multi-modal design spaces. Furthermore, when conflicts or incompatibilities arise between multiple design objectives, the model is capable of generating innovative solutions, such as images depicting individuals under extreme conditions (e.g., advanced age incompatible with high athletic ability). This is enabled by the stochastic sampling mechanism and the rich latent space exploration capability of diffusion models, which allow diverse and flexible outputs while satisfying constraints. This emergent ability³³ (defined as novel features and behaviors that manifest as the model's scale increases) opens up new possibilities for solving complex coupling and trade-off problems between multiple features.

In this work, we propose a multi-objective inverse design method, designated as GFV (including a generator, filter, and validator), which is based on latent diffusion models (LDMs), convolutional neural networks (CNNs) and finite element simulations (FEMs). It could rapidly tailor metamaterial structures to achieve multi-objective on-demand designs across different application scenarios, as illustrated in Fig. 1c. This approach does not necessitate the input of expert knowledge, as designers are only required to input the desired performance indicators (d₃₃, ε^T₃₃, E, and ρ) to generate the corresponding metamaterial structure. Furthermore, this strategy is also applicable to other metamaterials with on-demand multi-objective design requirements, including those in mechanical, optical, and electromagnetic domains.

2. Results

As shown in Fig. 1c, we first generated a dataset of 17000 structure–performance pairs (ESI S1 and Fig. S1†) using high-throughput finite element simulations (see Methods for details) for GFV training, taking the piezoelectric ceramic lead zirconate titanate (PZT) as the scaffold material for the metamaterial as an example. Note that this model can be applied not only to ceramics like PZT but also to metamaterial design where polymers and their composites serve as the backbone. The GFV process consists of three main modules: the generator, the filter, and the validator. The generator employs a latent diffusion model to generate initial piezoelectric metamaterial structures from specified inputs (d₃₃, ε^T₃₃, E, ρ). The filter, powered by a convolutional neural network, evaluates the performance of the generated structures and discards those that significantly deviate from the target requirements. Finally, the validator verifies the filtered structures with the most probable desired piezoelectric properties using finite element models. A detailed description of each module follows below.

2.1 Generation of the structure database of piezoelectric metamaterials

As the latent diffusion model relies on a data-driven approach, a comprehensive dataset of metamaterial structures and their corresponding properties is essential. The design space for metamaterials is nearly infinite, encompassing a wide range of options,³⁴ including trusses, 2D shell structures, and 3D topological surfaces.^35,36 This study focuses on truss structures due to their superior mechanical flexibility and enhanced piezoelectric response under dynamic loading.³⁷ This flexibility makes truss-based designs particularly suitable for advanced applications such as precision actuators and robotics, compared to other structural types, and has been demonstrated to enable unprecedented performance.⁸

To achieve a rich and diverse performance space, we employed a virtual growth program³⁸ to generate piezoelectric metamaterial structures. The virtual growth program consists of four steps: (i) constructing the underlying network topology, (ii) designing the geometry of the building blocks that can be placed in each grid, (iii) defining the adjacency rules between the building blocks, and (iv) specifying the probability parameters for the building blocks. First, a 20 × 20 underlying network topology was employed. As shown in Fig. 2a, we selected several building block geometries, including the “L”-shaped building block, “−”-shaped building block, “T”-shaped building block, and “+”-shaped building block, along with their various rotations. We also considered the building blocks with multiple angles (53°, 60°, 75°, 90°, 105°, and 120°) to create a rich variety of structure types. Fig. 2b defines the adjacency rules, permitting only node-to-node connections to prevent unconnected paths. Fig. 2c illustrates how structure growth can be regulated using the input building block probability parameters. Based on the Wave Function Collapse algorithm, at each step, the node with the lowest entropy among all unconnected nodes is selected for growth. The formula for node entropy is as follows:

	(1)

where n_i refers to the type of building block that can be connected to the node, and P_j denotes the normalized probability of the building block j∈n_i to be chosen. The type of building block is then randomly chosen based on the provided probability parameters, with blocks having higher probability values being more likely to be selected.

To systematically investigate these effects and provide theoretical support for subsequent inverse design, we examine the effect of rod angle orientation by dividing the building blocks in Fig. 2a into six distinct groups based on their angles. Each group includes the same “−”-shaped building block, “+”-shaped building block (without angle variation), the “L”-shaped building block and the “T”-shaped building block at angles of 53°, 60°, 75°, 90°, 105°, and 120°. To minimize the influence of inconsistencies in building block content on the properties of the piezoelectric metamaterials, 20 structures were generated for each angular group with the same probabilistic parameters, ensuring that the overall connectivity topology and building block content were consistent. The properties of these structures were then calculated using the two-step finite element method (see Methods).

Fig. 2d illustrates the relationship between ρ, E, ε^T₃₃, d₃₃, g₃₃ and angle for six groups of building blocks with similar contents (“+”-shaped 40%, “T”-shaped 40%, “L”-shaped 10%, “−”-shaped 10%). The density of the metamaterials decreases as the angle of the basic building blocks increases, although the overall decrease is modest, with the difference between the highest and lowest values being only 0.06. This difference primarily arises from variations in the ρ of the basic building blocks, which decreases as the angle increases. In application scenarios where lower density metamaterials are preferred, obtuse-angle building blocks yield better results. In contrast to ρ, E value of the metamaterials does not follow a simple decreasing trend with increasing building block angles. It first increases and then decreases, with a peak at 90°. This behavior is attributed to the fact that at 90°, the building block tends to distribute forces more uniformly along its vertical orientation, leading to a relatively small deformation. Similarly, ε^T₃₃, d₃₃, and g₃₃ also exhibit a trend of first increasing and then decreasing with an increase in angles, reaching a peak at 90°. ε^T₃₃ increases by up to 26%, while d₃₃ can increase by as much as 57%, leading to a similar trend in g₃₃. Therefore, despite the analogous variation trends exhibited by the three parameters namely ε^T₃₃, d₃₃, and g₃₃, their disparate amplitude of variation ranges afford a substantial design space for optimizing different performance parameters.

To elucidate the relationship between structure and piezoelectric properties, we introduce a structural descriptor to describe the electric-force transmission path, namely the average shortest path (see Fig. 2e). The shortest paths from all top endpoints of the metamaterials to the bottom were computed using the A* algorithm³⁹ (ESI S6†) with averaged length. This descriptor reflects the main stress transfer path within the metamaterial. It is our intention to ascertain the correlation law between the metamaterial structure and properties from the stress and electric field distribution.⁴⁰ As shown in Fig. S2,† the stress field distribution is concentrated on the shortest path of the metamaterial. It can be demonstrated that the shorter the shortest path, the higher the electric field distribution (as well as polarization distribution) along the shortest path, as proven by Gauss's theorem. The concentration of a high polarization distribution in the stress transfer path results in the piezoelectric material exhibiting a high piezoelectric response. Fig. 2e illustrates the relationship between the average shortest path and the angle. When the angle is 53°, the average shortest path reaches its maximum at 1.41. At 90°, the shortest path is minimized to 1.16. As the angle approaches 90°, the average shortest path decreases, leading to a 23% increase in g₃₃. The shorter path facilitates smoother transmission of stress and electric field, which explains why the 90° group enhances piezoelectric performance. Depending on the specific application requirements, different types of building blocks can be selected as the primary focus of investigation.

2.2 Generator

To achieve optimal piezoelectric signals, this study takes 90° building blocks as a representative example to illustrate how to achieve the multi-objective on-demand inverse design of piezoelectric metamaterials. Here, we used a 256 × 256 pixel space to fully capture the microstructural morphology of the complex metamaterial structures. To reduce training costs and time without compromising the accuracy of the generated results, our generator utilizes the latent diffusion model (LDM), which reduces the high-dimensional pixel space to a lower-dimensional latent space for training. The LDM is a generative model designed to learn the diffusion process underlying the probability distribution of a given training dataset. This process involves iterative denoising of noisy samples to produce high-quality outputs. Due to the ability of the LDM to conditionally direct and regulate its generation process, it has become the method of choice for generative applications in areas such as image generation,⁴¹ video generation⁴² and protein structure prediction.⁴³ In our model, the visual depiction of our LDM is shown in Fig. 3, which involves the collaboration of two deep learning models: the variational autoencoder (VAE) and the traditional diffusion model (DM).⁴⁴

First, we create a variational autoencoder (VAE), which is composed of two main components: the encoder and the decoder. The encoder compresses the original 256 × 256 image into a 32 × 32 low-dimensional latent space, while the decoder reconstructs the low-dimensional latent space back into the original 256 × 256 image. Next, to train the diffusion model (DM), the encoder of the trained VAE encodes the metamaterial into a 32 × 32 low-dimensional latent space and then corrupts the encoded samples with varying levels of Gaussian noise. The model learns to predict the noise present in each sample, subtracting it from the input samples to effectively denoise them. The noise prediction model uses a U-Net architecture, which consists of an encoder–decoder structure with skip connections. We built it by combining a residual convolutional layer with a cross-attention layer, in which the latter allows the model to incorporate target conditions (ρ, E, ε^T₃₃, d₃₃) into the denoising process, similar to how transformers are used in natural language processing.⁴⁵ After DM denoising, the image remains in the 32 × 32 latent space, and the VAE's decoder reconstructs this latent space back into the original 256 × 256 image.

For our specific cases, as shown in Fig. 3, we make the following adjustments to the model: (i) there is no need to train additional neural networks (e.g., BERT⁴⁶) to understand the semantics of the condition when the mathematical form of the condition sufficiently conveys its meaning. Our condition consists of only four numbers, which are directly and strongly related to the structural information of the metamaterial. Therefore, using models to encode these simple numbers (e.g., mapping them to a high-dimensional space) would result in the loss or blurring of information (Fig. S3†). (ii) As shown in Fig. S4,† metamaterial structures generated by LDM still contain some ambiguous regions, where it is unclear whether they are part of the actual metamaterial structure. To address this issue, we employ a trained VAE as an image restoration tool to further enhance the quality of the generated images without compromising the accuracy of the model (Fig. S4 and S5†). Since the VAE encoder has learned from pre-existing training data how to extract key features from the original input and encode them into latent space, it can effectively reconstruct clear images, even when the LDM-generated images contain minor blurring, as the overall morphological features in the LDM output remain clear. As illustrated in Fig. S5,† this approach yields more accurate metamaterial structures compared to manual restoration methods based on intuition or simple rules.

To evaluate the performance of the VAE, the intersection over union (IoU) metric was used to calculate the overlap between the real microstructure pixels (P_n) and the reconstructed microstructure pixels (P_n^re). The formula for IoU is as follows:

	(2)

In this study, the IoU score on the test set of 2000 microstructures was 93%, and the detailed reconstruction result is shown in Fig. S6.† This demonstrates an almost perfect match between P_n and P_n^re, indicating that the microstructure can be reconstructed with high quality. The accuracy of the LDM model was measured by calculating the mean absolute percentage error (MAPE) between the resulting properties and the input conditions. The expression for MAPE is given below:

	(3)

where d^gen₃₃, ε^{T gen}₃₃, E^gen, and ρ^gen are the piezoelectric coefficient, permittivity, Young's modulus and relative density of the piezoelectric metamaterial generated by the generator, respectively, whereas d^cond₃₃, ε^{T cond}₃₃, E^cond, and ρ^cond denote the input conditions, respectively.

To evaluate the accuracy of the LDM model, we tested it on a set of 3000 samples from the test set. The results are shown in Fig. 4a as a parity plot, where the x-axis represents the target performance conditions and the y-axis corresponds to the actual performance of the generated structures. Four performance metrics (d₃₃, ε^T₃₃, E, and ρ) are displayed, with the coefficient of determination R² values of 87.5%, 89.8%, 92.2%, and 93.4%, respectively. In addition, we calculated the average MAPE across these 3000 test samples, which was 10.11%. These results indicate that the generator achieves a high degree of text-image alignment. Moreover, we also evaluated the ability of our model to generate new metamaterial structures. The metamaterial structure images were encoded into latent space using a VAE and then downscaled to 2D space using t-SNE (ESI S7 and Fig. S8†). No complete overlap was observed between the newly generated data and the original dataset in the 2D t-SNE space, suggesting that our model has great potential to generate new metamaterial structures.

2.3 Filter and validator

It is important to note that not all piezoelectric metamaterials generated by the model satisfy the specified input conditions. To identify structures that are closer to the desired properties, we employ a filter (a convolutional neural network, CNN) and a validator (a two-step finite element method) to screen optimal structures. The filter predicts d₃₃, ε^T₃₃ and E and filters out the metamaterials that do not meet the required properties. The filter consists of a five-layer CNN followed by a three-layer fully connected neural network. The dataset of 17000 images was split into a training set (80%) and a test set (20%), with the training set used to train the CNN. By training with the CNN, a surrogate model was developed to predict d₃₃, ε^T₃₃ and E directly from the metamaterial structure images. The prediction errors of the model on the test set were evaluated. Fig. S9† shows a comparison between the effective d₃₃, ε^T₃₃, and E obtained from the FEM and the predictions obtained using the machine learning model. Within the range covered by the dataset, the predicted values closely match the FEM results, indicating the accuracy of the model. For d₃₃, the model achieved a coefficient of determination (R²) of 91.7% with a mean absolute error (MAE) of 23 pC N⁻¹. For ε^T₃₃, the model's R² was 96.8% with a MAE of 10.82, and for E, R² was 93.1% with a MAE of 0.177 GPa. These results indicate that the CNN model exhibits high accuracy and reliability in predicting the key performance indicators of piezoelectric metamaterials, enabling effective screening of structures that align with the target performance.

Following the filtering step, a validator was employed to assess the performance of the filtered samples and to identify those closest to the target values. Fig. 4b illustrates the performance alignment of the GFV model in the text-driven structure generation task, evaluated using the MAPE. We sampled 100 sets of data uniformly from the test set to validate the GFV model's ability to generate piezoelectric metamaterial structures on demand. Uniform sampling improves the model's adaptability under different data distributions. The horizontal coordinates of the scatter plot are the input targets and the vertical coordinates are the analog simulation values of the piezoelectric metamaterial properties generated by GFV. In order to further demonstrate the ability of the model in multi-objective simultaneous inverse on-demand design, eight cases of typical conditions (Cond1–Cond8) were selected and in Fig. 4b, these data points are marked with different colors, and the MAPEs of Cond1–Cond8 are 0.6%, 1.2%, 0.9%, 0.8%, 0.5%, 1.5%, 0.6%, and 0.5% respectively. The results show that the generated structures are highly consistent with the target inputs in terms of various performance metrics. The average MAPE of the 100 sets of test data in Fig. 4b is 1.06%, which is much lower than the generation error of the LDM model alone. This is because in comparison with the scheme using only the LDM model, the GFV, by introducing the filter and validator mechanism, leads to a substantial reduction in inverse design error while the computational overhead increases only slightly.

This improvement is achieved by our two-stage generate-and-filter (GFV) strategy: first, the latent diffusion model (LDM) produces a large set of candidate structures conditioned on target properties; then, a CNN-based predictor filters out any samples whose predicted performance deviates from tolerance thresholds. This approach reduces the inverse design error from 10.11% (LDM alone) to 1.06% (GFV), which confirms that the GFV strategy significantly improves the likelihood of obtaining structures closer to the target values, further underscoring the high accuracy of our inverse on-demand generation approach.

2.4 Model comparison and ablation experiments

In addition to employing the LDM as the generator, we investigated a pixel-based manual encoding strategy, where each building block is mapped to a specific pixel value, which is then used to replace the VAE-based latent representation in the LDM framework for subsequent diffusion training. For example, we encode each “+”-shaped building block as pixel value 0, each “L”-shaped building block as 240, and defects as 255 in the grayscale image. The specific pixel values are assigned based on the relative contribution of each building block type to the overall piezoelectric performance of the metamaterial. The complete encoding scheme is shown in ESI S11 and visualized in Fig. S14b.† Specifically, instead of using the original 256 × 256 structural images, we represent each metamaterial sample as a compact 20 × 20 grayscale image, corresponding to a 20 × 20 grid of building blocks. This significantly reduces the dimensionality of the input while retaining essential structural semantics, thereby replacing the VAE-compressed latent space in the LDM for downstream diffusion-based generation, which helps reduce the overall training burden of the model. This encoding approach not only provides a compact and informative representation but also mitigates the boundary blurring issue discussed in section 2.2 by enforcing discrete and interpretable structure-level semantics.

We systematically compared the performance metrics of a manually encoded diffusion model (manual coding + DM) with those of a latent diffusion model (LDM) based on a VAE encoder and the results are listed in Table 1. The results indicate that the manual coding + DM approach achieves a slightly lower generation error (MAPE) of 9.75% compared to 10.11%. In terms of training time, the manual coding approach requires only 48 hours, which is 24 hours shorter than the LDM method. This reduction is primarily attributed to the elimination of the VAE pretraining stage, highlighting the time-saving advantage of manual encoding. Although LDM offers advantages in generalization and representational capacity, this comparison suggests that manual encoding remains a viable and efficient strategy for certain task-specific applications. This strategy is feasible for modular metamaterials composed of discrete building blocks. However, its representational capacity is limited in cases involving non-modular structures, continuous media, or irregular layouts (e.g., Voronoi-based designs). The reliance on prior knowledge limits the scalability of this encoding approach, but it offers practical guidance for targeted inverse design problems.

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We also compared the performance of the proposed GFV strategy with the genetic algorithm (GA) across key metrics, including generation error, generation time, adaptability, and training time and the details are listed in Table 1. The GFV framework, denoted as LDM (CNN + FEM), integrates a latent diffusion model (LDM) as the generator, a convolutional neural network (CNN) as the filter, and a finite element method (FEM) as the validator. This configuration achieves a low generation error (MAPE) of 1.06% and a total generation time of approximately 116 minutes, demonstrating superior precision and robustness in generating high-performance structures. In contrast, the GA method achieves a slightly higher error (MAPE) of 2.58% (see ESI S10† for details), and due to its iterative nature, it requires significantly more computational resources, with a total design time of around 360 minutes. Furthermore, GA relies on task-specific manual encoding for each individual in the population, which inherently restricts its scalability and adaptability when applied to complex design problems. Nevertheless, GA offers an advantage in training efficiency, requiring only about 5 minutes to train a predictive model.

To systematically evaluate the contribution of each functional module within the GFV framework, we conducted ablation studies comparing four configurations: LDM only, LDM (CNN), LDM (FEM), and the full LDM(CNN + FEM) architecture. As shown in Table 1, the experimental results indicate that removing either the CNN filter or the FEM validator leads to a decrease in the accuracy, confirming the critical role of module synergy in overall performance. Notably, the FEM validator reduces the error to 2.57%, outperforming the CNN filter's 4.28%, albeit at a higher computational cost, as the validation of each structure requires approximately one minute. The full configuration achieves the lowest error (MAPE) of 1.06%.

Regarding generation time, the GFV framework requires a longer duration due to the necessity of generating a large number of candidate structures for filtering and subsequent validation. In contrast, other methods demand less time but at the expense of reduced accuracy. This ablation study demonstrates the essential roles and complementary effects of each module within the overall framework, providing strong support for the design rationale of our approach.

In our GFV model, the filter and validator are introduced after the generator, reducing the generation error from 10.11% to 1.06% with minimal computational overhead. Considering that the filter offers faster inference but lower accuracy than the validator, it is employed for large-scale preliminary screening, while the validator is applied in the final stage to ensure the reliability of the generated structures. This strategy effectively eliminates suboptimal candidates resulting from conflicts among competing objectives, retaining only those structures that simultaneously satisfy multiple target criteria. It is particularly well-suited for solving complex multi-objective optimization problems.

2.5 Multi-scenario design on demand

This study aims to achieve multi-objective inverse on-demand design of piezoelectric metamaterials based on different application scenarios. In these scenarios, the model not only supports fixed-point optimization but also handles open-ended objectives (such as minimizing or maximizing specific attributes). Below, we present the application of our model in three typical use cases to illustrate its effectiveness in real-world tasks. Fig. 5a illustrates three typical application scenarios: wearable devices,⁴⁷ robots with self-powered systems, and medical ultrasound probes.⁴⁸ These application requirements can be classified into three design categories: low E and high g₃₃, high E and high FOM₃₃, and Z matching with high g₃₃. The first category is suitable for applications requiring high flexibility and comfort^49,50 (e.g., insoles, knee pads, and smart soft grippers); the second category applies to self-powered structural components in high-strength working environments^20,51 (e.g., robot legs and feet); and the third category is for scenarios requiring efficient acoustic wave transmission,^52,53 such as ultrasound probes.

To meet these design requirements, we set the following conditions on the basis of meeting the mechanical or acoustic impedance requirements while maintaining as high piezoelectric response as possible. For low E applications, the input conditions are marked by the blue pentagon in Fig. 5b (E = 0.2 GPa, ). The E value in these conditions is comparable to that of PVDF-based piezoelectric materials commonly used in existing wearable devices, typically ranging from 0.1 to 3 GPa.^54,55 For high-strength self-powered structural components, the input conditions are marked by the blue pentagon in Fig. 5c (E = 7 GPa, ), where E is in the highest region of the dataset. For ultrasound probe applications, the input conditions are shown by the blue pentagon in Fig. 5d (, ), where Z is similar to that of human tissue (1.5 M Rayl). The details of the selection methods for all parameters are included in ESI S8.† When the Z or E thresholds are satisfied, the values of d₃₃ and ε^T₃₃ for all input conditions fall outside the Pareto frontier within the dataset to optimize g₃₃ and FOM₃₃. The generated MAPE values for the optimal structures closest to the target are 4.2%, 2.3%, and 4.5% for the above three conditions, respectively. The difficulty of inverse generation increases as the input design conditions exceed the performance range of the original dataset,⁵⁶ resulting in higher errors compared to previous results. To select the piezoelectric metamaterial structures that meet the requirements, we propose a weighted evaluation formula based on target features (see ESI S9†). This formula supports the handling of open-ended objectives (such as minimizing or maximizing specific attributes), thereby better meeting the requirements of practical applications. The 100 optimal structures selected are located in the yellow regions of Fig. 5b–d and the optimal piezoelectric metamaterial structures are shown as red pentagons, all of which exceed the dataset boundaries. The performance metrics of the optimized metamaterials are as follows: (b) g₃₃ = 470 × 10⁻³ V m N⁻¹, E = 0.26 GPa (c) FOM₃₃ = 84.5 × 10⁻¹² m² N⁻¹, E = 7.1 GPa and (d) g₃₃ = 258 × 10⁻³ V m N⁻¹, Z = 1.56 M Rayl. Fig. 5e–g displays the optimal structures generated in three distinct application scenarios. In the case of the low E, high g₃₃ structure, a substantial void region is incorporated into the topological network, resulting in a notable increase in porosity. This porosity effectively reduces E while significantly increasing g₃₃.⁵⁷ For high E and high FOM₃₃, a fully vertical connected rod design is adopted, achieving the maximum E and limiting the expansion of the design space,⁵⁸ making it difficult to break the dataset boundaries. In the case of Z matching and high g₃₃ structures, the material's structural characteristics do not undergo extreme changes, as both Z and g₃₃ values remain within an intermediate range.

This study demonstrates that the proposed model is capable of effectively learning the intricate relationship between the structural and functional properties of piezoelectric metamaterials. It successfully captures local features related to electrical, mechanical, and acoustical properties, such as porosity and vertically connected rods, thereby overcoming the complex coupling and trade-offs between multiple objectives (as shown in Fig. S11† and Fig. 5b–d). Based on these insights, the model generates high-performance piezoelectric metamaterial structures, fulfilling multi-objective design requirements for diverse application scenarios. Notably, the piezoelectric metamaterial structures used in this study are composed of the piezoelectric ceramic PZT and 90° building blocks. If other application requirements arise (such as negative Poisson's ratio or lower density), the application space of inverse on-demand design can be further expanded by adjusting the intrinsic properties of materials or the types of building blocks.

3. Conclusions

To address the challenge of simultaneous multi-objective optimization in piezoelectric metamaterials, this work proposes a scalable and efficient GFV (generator, filter, and validator) multi-objective inverse on-demand design method based on a generative artificial intelligence latent diffusion model. This method is capable of generating on-demand piezoelectric metamaterial structures that simultaneously satisfy electrical, mechanical, and acoustic property requirements, with an error (MAPE) of only 1.06%, without the need for specialist knowledge. Moreover, we present illustrative examples of the GFV method for multi-objective design in diverse application scenarios. These examples demonstrate that the method can generate piezoelectric metamaterial structures with high electromechanical performance, beyond the dataset, in accordance with multi-objective requirements. The complexity of the coupling and trade-off relationships between multiple objectives, where enhancing one property may significantly affect others, can result in high-dimensional, non-linear optimisation problems. This complexity represents a significant challenge to conventional optimisation design methods. The GFV model, inspired by generative image modelling, is particularly well suited to this multi-objective on-demand design strategy and overcomes many of the challenges faced by traditional methods. The model is capable of accurately capturing the structural features that affect the material properties, including those related to d₃₃, ε^T₃₃, E, and ρ. Among others, these include pore structures that effectively reduce E while enhancing g₃₃, as well as vertically connected truss structures that reinforce E and combine the emergent abilities that may exist in large models to break the strong coupling relationship between multiple objectives. While satisfying the mechanical and acoustic properties, high d₃₃, low ε^T₃₃ piezoelectric metamaterial structures beyond the Pareto boundary of the dataset are generated.

Moreover, we made several improvements to each module of the GFV model. In the generator, a VAE was introduced to enhance the image clarity of LDM outputs without compromising performance accuracy. In the filter, both fixed-target and open-ended objectives (e.g., minimizing or maximizing specific properties) were supported. In the validator, a self-developed two-step FEM method was employed to accurately evaluate the effective piezoelectric performance of complex structures (see Methods). Therefore, the current work not only reveals the influence of electro-mechanical field coupling on the multifunctional performance of piezoelectric metamaterials but also establishes a simple and efficient multi-objective inverse design method, providing valuable guidance for experimental research. In principle, the current framework can be easily extended by adjusting for multi-objective conditions, different loading scenarios, and kinds of materials. Finally, the proposed framework allows extension to related fields such as mechanical, optical, and electromagnetic applications.

4. Methods

4.1 Training the latent diffusion model

Training of the proposed model involves two stages: (1) training the VAE to learn an expressive latent space and (2) training the diffusion model in the latent space.

4.2 Two-step finite element method

4.3 Poling experiment for PZT-5H

To obtain the experimental relationship between the piezoelectric coefficient d₃₃ and the applied electric field, we conducted a series of controlled poling experiments on PZT-5H ceramics.

Commercial PZT-5H samples with dimensions of 10 mm × 10 mm × 1 mm (purchased from Shenglei, Shaoxing, China) were used in the tests. The samples were subjected to a direct current (DC) poling process in silicone oil maintained at 120 °C to ensure uniform thermal conditions and to prevent electrical breakdown. A constant DC electric field was applied across the thickness of the samples using silver paste electrodes, which were sintered at 580 °C for 15 minutes prior to poling to ensure reliable electrical contact.

The electric field was incrementally increased from 0 kV cm⁻¹ to 20 kV cm⁻¹ in steps of 1 kV cm⁻¹. For each step, the sample was poled for 5 minutes, followed by the measurement of the piezoelectric coefficient d₃₃ using a standard quasi-static d₃₃ meter. In total, 20 field response data points were collected.

These experimentally obtained values served as reference benchmarks for finite element calibration and validation of our inverse design framework.

Author contributions

Chun-Yu Lei: writing original draft, high throughput simulations and machine learning models. Zhong-Hui Shen: writing – review & editing, supervision, and funding acquisition. Jian Wang: formal analysis and data curation. Run-Lin Liu: formal analysis and data curation. Meng-Jun Zhou: writing – review & editing and supervision. All authors have given approval to the final version of the manuscript.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability

All data used are available within this paper and its ESI.† Further information can be acquired from the corresponding authors upon reasonable request.

Acknowledgements

This work was supported by the NSF of China (grant no. 52422206, 52372121 and 92463306, Zhong-Hui Shen, and grant no. 51202141, Meng-Jun Zhou).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5nr01669j

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