Data-driven inverse design of multifunctional bicontinuous multiscale structures

Abstract

Bicontinuous multiscale structures, commonly observed in nature, comprise two interpenetrating networks that are solid and void phases forming a continuous and interconnected system. These unique architectures exhibit superior multi-physical performances and multifunctionalities; however, their design has been limited by the lack of analytical expressions and the computational challenges in multiscale optimization. This study presents a 3D Large-range, Boundary-identical, Bicontinuous, and Open-cell Microstructure (L-BOM) datasets for the fast data-driven inverse design of multifunctional bicontinuous multiscale structures. Each dataset features identical boundaries, bicontinuous open-cell structures, and broad property coverage for performance exploration. These properties are satisfied by active learning techniques developed with a generative artificial intelligence model. The datasets hold significant promise for advancing the design of bicontinuous multiscale structures with a large-range property space, without additional post-processing to ensure the connectivity. This work further demonstrates the potential of the datasets in devising bone implants, chair components, and multifunctional materials with tunable elasticity and permeability.

This study presents a 3D large-range, boundary-identical, and open-cell microstructure datasets for the fast data-driven inverse design of bicontinuous multiscale structures with tunable elasticity and permeability for multifunctional applications.

Introduction

Natural materials such as femur bones^1–4, wood^5–8, and sponges^9–11 typically exhibit hierarchical organization (Fig. 1a) and bicontinuity (Fig. 1b). These characteristics, collectively referred to as bicontinuous multiscale structures, contribute to efficient moisture and nutrient transport while supporting stress distribution, filtration, and adsorption. The importance of bicontinuous multiscale structures¹² extends across various domains such as biomedicine^13–18, mechanical engineering^19–21, sports equipment^22,23, and energy and environmental applications^24,25, where they fulfill critical needs for high stiffness^26,27, extensive surface area²⁸, superior breathability²⁹, and varying levels of permeability^30–32. This study explores bicontinuous multiscale structures and their potential to enhance material performance, starting with their definition.

Fig. 1. The motivation, challenges, and workflow of data-driven multifunctional bicontinuous multiscale structural design.

a Cancellous bone. b Bicontinuous open-cell structures exhibiting continuity in both solid and void phases. c Boundary connectivity: weakly connected interface (left) vs. well-connected interface (right). d The Large-range, Boundary-identical, Bicontinuous Open-cell Microstructure (L-BOM) dataset, constructed using optimization (Opt.) and active learning (AL), shares an identical boundary mask (“X“ shape). Color represents the surface-to-volume ratio (S/V) of the microstructures. Blue arrows indicate microstructures selected from L-BOM dataset. e Multiscale design of a cancellous bone implant. The cancellous bone is forced at the top (red arrows) and fixed at the bottom (yellow triangles). The two microstructures correspond to those shown in (d).

Thus, the bicontinuous multiscale structure necessitates fundamental building blocks (i.e., microstructures), with the overall architecture formed through the hierarchical assembly of these individual components. This configuration creates two continuous channels facilitating the interaction between the solid and void portions. However, generating bicontinuous multiscale structures faces two significant challenges. First, the lack of an analytical representation for bicontinuous structures complicates inverse optimization efforts^33,34. Second, inversely designing 3D multiscale structures requires substantial computational costs as they span multiple length scales, and equipping different microstructures requires dealing with connectivity issues.

Currently, progress has been made in the design of bicontinuous structures. The experience-based design of bionic structures, inspired by natural multiscale systems such as starfish³⁵ and sea urchin spines¹², mimics natural forms to develop analogous engineering materials. However, this approach often results in fixed structures and requires extensive trial and error. Specific patterns represented by parametric structures, such as spinodal structures³⁶ and triply periodic minimal surface (TPMS) structures³⁷, are also utilized as basic building blocks for the design of bicontinuous structures. Porosity and stiffness can be controlled by adjusting parameters; however, the parameter space is limited, and connectivity processing could introduce external computational overhead. Topology optimization methods^16,38,39 have gained attention for their inverse design capabilities, enabling optimal material distribution for superior performance. However, applying topology optimization for multiscale optimization requires significant computational time and storage memory. The inadequate connectivity leads to structural failures leaving lattices unable to bear loads effectively and complicates the additive manufacturing process. To develop manufacturable, load-efficient designs, the connectivity issue between neighboring microstructures must be resolved when the macrostructure incorporates various microstructures^40–42 (Fig. 1c left). Commonly employed techniques to enhance this connectivity include optimization^43,44, interpolation⁴⁵ (Fig. 1c right), and deformation¹⁷. However, these methods also lead to increased computational costs. Therefore, despite progress in this area, existing approaches still struggle to address the above two key challenges effectively.

This paper introduces a novel method to fast inversely design multifunctional bicontinuous multiscale structures. Central to the method is a fundamental consensus that the multiscale structure retains its bicontinuous properties if its building blocks are bicontinuous, open-cell, and have identical boundaries. Accordingly, the design challenge of bicontinuous multiscale structures can be effectively transformed into building block design problems. Various optimized microstructures serve as the initial dataset. Each microstructure employs a region growth method to calculate the connected component. Then, the bicontinuous open-cell structures are retained. To this end, we propose a novel Active Learning (AL) method to generate Large-range, Boundary-identical, Bicontinuous Open-cell Microstructure (L-BOM) datasets for multiscale design (Fig. 1d). In each iteration of the AL method, we first train a generative Artificial Intelligence (AI) model with the existing data, then apply it to generate microstructures matching the specified elastic tensors while fixing the boundaries as the input boundary, and finally add the generated bicontinuous open-cell microstructures into the dataset. This boundary-aware generation ensures effective structural properties while improving manufacturability by reducing interfacial defects and enabling smooth resin drainage.

This study designs four different boundary masks as constraints for AL. Finally, four L-BOM datasets containing 404,355 microstructures are constructed, with each microstructure comprising 128³ elements. After that, these datasets are applied to the multi-scene, multifunctional, multiscale structure design for practical applications, such as femoral implants (Fig. 1e), chair components, and filtering devices. Specifically, the application potential of bicontinuous open-cell structures in fluid-structure coupling problems, such as filter device design, is highly promising. Beyond that, identical boundary characteristics simplify the assembly process and ensure seamless connectivity between microstructures, significantly accelerating the multiscale optimization design process. In comparison, Yang et al.³⁹ required nearly 10 h to design a 20 × 20 × 20 macrostructure with 128³ elements per microstructure, while our method completes the task in 5 min on a desktop with an Intel Core i7-13700K (3.4 GHz), 128 GB RAM. Therefore, this paper effectively addresses the two aforementioned challenges by constructing the L-BOM datasets, which are inherently complex and time-consuming for the inverse optimization of non-analytic representations. Moreover, the designed bicontinuous structures satisfy the open-hole constraint, facilitating efficient waste removal during additive manufacturing, making it possible to fabricate complex structures.

Results

Dataset construction

To construct the L-BOM dataset, an initial dataset is generated using the LIVE3D framework⁴⁶ (Fig. 2a). Each microstructure is discretized into 128³ elements under cubic symmetry constraints to simplify calculations (Supplementary Section 1). This reduces the degrees of freedom in the elastic tensor matrix C from 21 to 3: C₁₁₁₁, C₁₁₂₂, and C₁₂₁₂. Four boundary types (masks 1–4) are determined based on clustering, testing, and quality evaluation (Supplementary Section 2 and Fig. S1). Microstructure datasets are then reconstructed using these masks as constraints (Supplementary Section 3, Figs. S2 and S3, and Tables S1 and S2).

Fig. 2. Illustration of L-BOM datasets under four distinct boundary masks (masks 1–4).

a–c Construction pipeline for the first boundary (mask 1): a Initial dataset generated via optimization for mask 1 (21,620 microstructures). Color represents the surface-to-volume ratio (S/V) of the microstructures. n denotes the number of microstructures in each dataset. b Active learning (AL) framework combining a denoising U-Net and finite element analysis (FEA). Due to cubic symmetry, only the top-left 1/8 subdomain is used. The U-Net employs elasticity and timestep embeddings in an encoder–decoder architecture to generate microstructures; FEA computes their mechanical properties, which are then used to expand the training dataset. Here, x_t is the noisy input at diffusion timestep t, ${\widehat{\mathbf{x}}}_0$ is the current denoised prediction, and ${\widehat{\mathbf{x}}}_0^{\prime }$ is the prediction from the previous iteration; b and c denote the boundary condition and elasticity tensor, respectively. c Final dataset for mask 1 after AL (106,445 microstructures). d–f Initial and final L-BOM datasets for the three additional boundary masks (masks 2–4). Blue arrows indicate multiple rounds of AL. Source data are provided as a Source Data file.

To expand the dataset, we implement a self-conditioning diffusion model^47,48, guided by predefined boundary conditions B (given mask pattern) and the homogeneous elastic tensor C (Supplementary Section 4 and Figs. S4 and S5). The optimized dataset serves as the initial training set, and the L-BOM dataset is generated through an AL strategy (Fig. 2b). The core denoising module refines inputs iteratively to produce the final structures. The elastic tensor C and time embedding t are conditional inputs to guide the model. Generated structures are filtered to remove those lacking bicontinuous properties by connected component labeling algorithm⁴⁹ and are analyzed to evaluate their properties. The original and generated structures are then combined for retraining until a microstructure dataset with extensive property coverage and identical boundaries is achieved (Fig. 2c). This process is repeated for masks 2–4, resulting in four L-BOM datasets (Fig. 2d–f and Supplementary Movies 1–4).

Dataset analysis

Elastic tensor

Figure 3a (Supplementary Section 5 and Table S3) shows the relative values of C₁₁₁₁, C₁₁₂₂, and C₁₂₁₂ for the four datasets. The fourth dataset exhibits higher C₁₁₁₁ values, meaning greater stiffness under uniaxial loads. The first and fourth datasets show a wider range of C₁₁₁₁, indicating more variability in rigidity. Similarly, the first and fourth datasets allow more lateral deformation, with the fourth having higher C₁₁₂₂, suggesting a stronger Poisson effect. For shear resistance, the second dataset has a broader range of C₁₂₁₂, while the fourth dataset has higher values, making it more resistant to shear deformation.

Fig. 3. Statistical analysis of the four proposed datasets.

a Box plots of the relative elastic tensor components (C₁₁₁₁, C₁₁₂₂, and C₁₂₁₂). n represents the number of microstructures per dataset. b Box plots of porosity (1–V), surface-to-volume ratio (S/V, where S is the surface area), and Poisson’s ratio (ν). c Proportion of anisotropic microstructures that satisfy criterion X in each dataset. Here, K, G, and E denote the bulk modulus, shear modulus, and Young’s modulus, respectively; the subscript V indicates the Voigt upper bound. P(X) represents this proportion and masks 1–4 correspond to the four distinct boundaries. d Proportion of isotropic microstructures that satisfy criterion X in each dataset, where the subscript HS refers to the Hashin–Shtrikman upper bound. Source data are provided as a Source Data file.

Porosity, poisson’s ratio, and surface-to-volume ratio

Figure 3b shows the porosity 1-V, surface-to-volume ratio (S/V), and Poisson’s ratio (v) of the four lattice groups. The fourth dataset has lower porosity, while the first dataset offers the widest range of v, allowing greater flexibility in lateral deformation. It also has the broadest S/V range, making it ideal for applications requiring efficient mass and heat transfer. More details on Poisson’s ratios are provided in Supplementary Table S3.

Anisotropic properties

For anisotropic lattices (Z ≤ 0.95 or Z ≥ 1.05), Fig. 3c shows the ratios of their bulk modulus (K), shear modulus (G), and Young’s modulus (E) to their Voigt upper limits. Z represents the Zener ratio. Detailed statistics are shown in Supplementary Table S4. A higher K/K_V indicates greater rigidity under hydrostatic loads. In the fourth group, over 48% of lattices exceed 40% of the Voigt upper limit for bulk modulus, highlighting their superior stiffness. The G/G_V reflects shear resistance, with about 45% of lattices in the third and fourth groups surpassing 40% of the Voigt upper limits. For E/E_V, over 68% of the third-group lattices and nearly 75% of the fourth-group lattices exceed 40% of the Voigt upper limits, demonstrating higher stiffness. The fourth dataset exhibits significantly higher bulk modulus and Young’s modulus than the third, while their shear moduli remain comparable.

Isotropic properties

For isotropic lattices (0.95 < Z < 1.05), the ratios of K, G, and E to their Hashin–Shtrikman (HS) upper limits are shown in Fig. 3d, with detailed data provided in Supplementary Table S5. The results show that over 50% of the K/K_HS values in the first dataset fall below 40%, whereas in the other three datasets, this proportion is under 11%. This suggests that lattices in the latter three groups offer significantly higher stiffness under hydrostatic loads. Notably, about 90% of the data surpass 40% of the HS upper limits, demonstrating their excellent structural efficiency. For isotropic lattices, those with mask 4 as the boundary have lower Young’s modulus and shear modulus than those with mask 3, indicating that boundary conditions affect isotropic and anisotropic lattices differently. While bulk moduli also vary, the differences are minor. Therefore, mask 3 is preferable for designing isotropic lattices with superior stiffness, while mask 4 is better suited for anisotropic lattices to enhance stiffness.

Multi-scene bicontinuous multiscale structure design

The developed L-BOM datasets offer extensive options for bicontinuous multiscale structure design. With a tailored application configuration, the top-down approach³⁸ using L-BOM datasets achieves improved results and high computational efficiency. Topology optimization focuses on determining the elastic tensor with application-dependent objectives.

Femoral implant design

The design of femoral implants entails balancing multiple objectives such as Young’s modulus, pore size, S/V, and weight constituting a prototypical multifunctional optimization problem. Young’s modulus is pivotal for replicating the mechanical behavior of native bone, enabling effective load transmission while minimizing stress shielding^50–53. Compact and cancellous bone exhibit Young’s modulus of approximately 5000 MPa and 2500 MPa, respectively⁵³. Pore sizes in the range of 200–400 μm have been shown to support cellular proliferation, nutrient diffusion, and vascular infiltration^54–57. A higher S/V promotes bone ingrowth⁵⁸, while sufficient permeability facilitates nutrient transport and metabolic waste clearance⁵⁹. Finally, lightweight architectures reduce physiological loading, thereby improving postoperative comfort and implant longevity⁵¹.

Taking the rabbit femur as a representative case, the initial configuration is depicted in Figure 4a left. The design domain, measuring 1 × 1 × 3 cm, is discretized into a 3 × 3 × 9 macroscale elements. For implant design, we select the third dataset owing to its broader S/V range, which better accommodates the structural variability inherent to femoral applications. A two-stage strategy is employed to address the implant design problem. In the first stage, candidate microstructures are screened from the dataset based on physiological constraints characteristic of compact bone: porosity must lie within the range of 0.2–0.85, and the average pore size (APS) must exceed 26 μm. The filtered microstructures, highlighted in blue, are illustrated in Fig. 4b (right). In the second stage, direct computation of the Young’s modulus is intractable; hence, the equivalent stiffness is utilized as a surrogate objective. The optimization seeks to minimize the relative deviation between the implant’s effective stiffness and the target value representative of host bone properties (Fig. 4a middle). Titanium serves as the base material, and the effective stiffness of a fully dense implant is evaluated. For each macroscale element, the elastic tensor is calculated (Fig. 4a right) and matched to a corresponding microstructure within the dataset (Fig. 4b left). The final implant is then constructed by assembling the selected microstructures across the global domain using solid intersection operations (Fig. 4b middle). Additional implementation details are provided in Supplementary Section 6.1.

Fig. 4. Multiscale design of compact and cancellous bone implants.

a Initial configuration defining the fixed boundaries (yellow triangles) and applied external forces (red arrows) within the design domain (3 × 3 × 9 units, corresponding to a physical size of 1 × 1 × 3 cm). Topology optimization is performed with the objective function ∣f − f₀∣/f₀, where f is the compliance and f₀ is the target compliance. b Generation of multiscale structures through microstructure matching, assembly, and intersection handling. The corresponding selected microstructures are shown alongside. c Comparisons among our result (I), Schoen’s I-graph-wrapped package (IWP) (II), BCC truss (III), and spinodal structure (IV). For structures (II–IV), the interpolation operation is performed. d Computation time (T, in seconds), Young’s modulus (E, in MPa), and average pore size (APS, in μm) of the IWP, BCC truss, and spinodal structures compared with our design, which is also compared with real cancellous bone. Source data are provided as a Source Data file.

The performance and efficiency of the proposed femoral implant design are benchmarked against three representative microstructures: Schoen’s I-graph-wrapped package (IWP), BCC truss, and spinodal structures, with comparisons covering computational time and key structural metrics including Young’s modulus, porosity, S/V, and APS (Fig. 4c, Supplementary Section 6.2, Fig. S6 and Tables S6 and S7). Computational time consists of five stages: optimization, matching, assembly, boundary connectivity, and intersection handling. In conventional methods, resolving connectivity that adopted an interpolation algorithm⁶⁰ accounts for over two-thirds of the total time due to post processing. The proposed method eliminates this step by ensuring connectivity at the design stage, substantially improving efficiency. Mechanically, the optimized structure achieves the closest match to compact bone properties, with a Young’s modulus approaching 5000 MPa, followed by BCC truss and IWP. Spinodal structures, with volume fractions exceeding 0.3⁶¹, yield the poorest mechanical performance. The APS of the proposed compact bone design is 286.64 μm, well within the optimal 200–400 μm range, unlike the reference structures. Similar trends are observed in cancellous bone (Fig. 4d), where the optimized APS (299.31 μm) remains ideal. Structurally, the method generates plate-like forms for compact bone and truss-like architectures for cancellous bone, closely mimicking natural morphology. Owing to its data-driven formulation, the design framework exhibits reduced sensitivity to initial conditions compared to conventional microstructures.

Lumbar interbody fusion cage design

Low back pain from degenerative disc disease, spinal deformities, and injuries are significant concerns for middle-aged and elderly people⁶², affecting approximately 80% of the individuals and potentially leading to disability or paralysis⁶³. Typically, 3D-printed interbody fusion cages are popular for mimicking the stability and porosity of natural bones⁶⁴, delivering osteoinductive factors that promote bone growth. To prevent compression of surrounding nerve tissues, we adopt the first dataset, which offers a broad range of negative Poisson’s ratio (v). The multiscale structure is designed using a top-down inverse engineering approach, aiming to achieve predefined stiffness. The elastic tensors at different macro units are optimized, and the corresponding microstructures are subsequently identified from the dataset to populate the design domain. The final optimized designs are shown in Fig. 5a, b (see Supplementary Section 6.1 and Table S8).

Fig. 5. Lumbar interbody fusion cage and chair design.

a Initial configuration of the lumbar interbody fusion cage (11 × 8 × 3 units) showing the applied boundary conditions and external loads. Topology optimization is performed with the objective function ∣f − f₀∣/f₀, where f is the compliance and f₀ is the target compliance. Red arrows indicate applied loads. b Final optimized structure, iteration curves for achieving the predefined stiffness, and the location of the corresponding microstructures from the first dataset. c Initial configurations of the chair seat board (10 × 12 × 1 units) and backrest (14 × 6 × 2 units). d Final optimized results, iteration curves, and the distribution of effective properties. e Microstructure matching, assembly, intersection handling, and data distribution used in the fourth dataset. Source data are provided as a Source Data file.

Chair component design

The seat board and backrest are key components of a chair. A breathable design with bicontinuous open-cell materials can reduce heat and sweat buildup, promote air circulation, and prevent bacterial growth, enhancing comfort and overall health during prolonged sitting. The initial configurations are shown in Fig. 5c. The upper surface of the seat board serves as the primary load-bearing area, with the surrounding edges and diagonal lines as fixed boundaries. The backrest is designed to follow the natural curvature of the human spine. The performance distribution of the seat board and backrest is determined through topology optimization (Fig. 5d). To better withstand the pressure of the human body, the fourth dataset is characterized by more exceptional stiffness, which is selected for design. The final designs are presented in Fig. 5e. Detailed optimization is shown in Supplementary Section 6.3.

Filtering device design

To enable tunable permeability, a 4 × 4 × 1 bicontinuous multiscale filter is designed (Fig. 6a, Supplementary Section 6.6, and Figs. S9–S12). A three-stage validation framework comprising numerical analysis, simulation, and experimental testing is implemented to assess the wide permeability range supported by the dataset. Permeability and S/V are computed for four datasets and benchmarked against TPMS structures. For instance, in the second dataset, the results exhibit a broader distribution in both permeability and S/V compared to TPMS (Fig. 6b, left). These trends are further confirmed by fluid simulations at porosity levels of 0.5, 0.6, 0.7, and 0.8. To benchmark against Gyroid, Diamond, and IWP structures, 20 representative microstructures (M₁–M₂₀) are selected across porosity levels, capturing high, medium, and low permeability cases. Their porosity and S/V distributions are summarized in box plots (Fig. 6b, right). COMSOL simulations (Fig. 6c) confirm that the proposed designs span a wider range of geometric and transport properties than conventional TPMS structures (Fig. 6d). Fluid simulation experiments are conducted using COMSOL Multiphysics. For each group, the steady-state response of four structures is tested under unidirectional creeping flow conditions. Water is used as the testing fluid, with an inlet positioned at the top of a 4 × 4 × 1 structure and an applied pressure of 500 Pa. The outlet is located at the outer surface of four microstructures at the bottom center, where the pressure is set to 0 Pa. A custom experimental setup (Fig. 6e, left; and Supplementary Fig. S13) is built to quantify permeability via water flow velocity. To demonstrate tunability, three samples with porosity 0.8 (M₁₆, M₁₉, and M₂₀) are tested and compared to Gyroid, Diamond, and IWP analogs. Experimental measurements align well with numerical and simulated predictions, validating the design’s performance across a broad permeability spectrum (Fig. 6e, right and Supplementary Movie 5). A simplified analytical framework links channel connectivity and tortuosity to permeability, offering theoretical insight into governing structural parameters shown in Supplementary Section 7 and Fig. S14.

Fig. 6. Multiscale design of filtering devices.

a Schematic illustration (top) of the multiscale structure used in the filtering device and the corresponding initial configuration (bottom) showing the design domain (4 × 4 × 1 units) and boundary conditions. Yellow triangles represent fixed boundaries. b Comparison between microstructures of mask 2 and triply periodic minimal surface (TPMS) structures (Gyroid, Diamond, and IWP) in terms of porosity (1 − V), surface-to-volume ratio (S/V), and permeability (P). c Solid (gray) and void (semi-transparent) phases of the bicontinuous open-cell structure, along with fluid simulation results obtained from COMSOL Multiphysics. d Fluid velocity distributions of Gyroid, Diamond, and IWP structures, together with three optimized multiscale designs (M16, M19, and M20). e Experimental validation of permeability for the six structures. Blue arrows indicate water injection, and yellow triangles represent boundary supports. The snapshots correspond to t = 1.90s, and the total infiltration time T_total is given below each sample. Source data are provided as a Source Data file.

More cases

We design an invisibility cloak (see Supplementary Section 6.4 and Fig. S7) that reduces displacement distortion from 33.13% to 4.85%, thereby enhancing the invisibility effect. Compared to the method in work³⁹, our approach reduces the total design time from 617 min to just 5 min. Additionally, we customize surf shoes based on the stress distribution of the sole by embedding microscale structures with varying elastic moduli across different pressure zones. This design effectively reduces stress in the metatarsal region and improves overall comfort (see Supplementary Section 6.5 and Fig. S8).

Discussion

The main contribution of this work lies in efficiently generating bicontinuous structures. It is based on a key principle: a multiscale structure retains its bicontinuous property if its building blocks are bicontinuous open-cell and share identical boundaries. The identical boundaries can also simplify the assembly process and accelerate the multiscale optimization process. Moreover, the datasets also feature a larger design space, thus enabling a large range of tailorable properties for the designed structures. We introduce L-BOM datasets under four boundary conditions (masks 1–4), each offering a wide range of structural performance. These datasets offer several advantages. First, the extensive range of datasets provides a broader feasible design space, enabling more optimized solutions. Second, the identical boundary condition ensures seamless connectivity between adjacent units, significantly enhancing the efficiency of multiscale optimization (see Supplementary Table S6). Both simulations and experiments confirm their excellent performance.

In addition to a large design space and improved performance, the bicontinuous nature of the proposed structures offers key practical advantages, particularly in biocompatibility and 3D printing. Their open architecture supports nutrient flow, cell migration, and tissue integration while also preventing trapped powder or resin during fabrication, especially in powder bed fusion and stereolithography, ensuring high manufacturability despite the design complexity.

While the L-BOM datasets demonstrate advantageous properties, there remain unexplored opportunities in expanding toward more multifunctional capabilities and asymmetric microstructure datasets. To explore multifunctional properties, integrating physical constraints such as the Navier–Stokes equations for permeability or heat transfer equations for thermally conductive structures into the model can enhance performance in application-specific tasks. For asymmetric microstructure datasets, the theory of 3D space groups offers a pathway to relax cubic symmetry constraints, thereby enabling the generation of comprehensive, full-space microstructure datasets. This study marks a meaningful step toward realizing the broader vision of multiphysical, multifunctional, and bicontinuous multiscale structural design. These designs hold significant promise for applications in bone implants, environmental protection, health and wellness, and next-generation engineering solutions.

Methods

This work first employs a high-performance optimization framework, LIVE3D⁴⁶, to reformulate multiple objective functions and generate the initial dataset. Subsequently, an AL strategy is introduced to iteratively augment the data, producing a bicontinuous microstructure dataset with consistent boundaries and open porosity. Finally, a multiscale optimization scheme is established to enable unified modeling and optimization across a broad range of application scenarios. The training and inference of our network are conducted on a server equipped with 2 CPUs (Intel Xeon Silver 4316 2.30 GHz), 512 GB of RAM, and 8 Nvidia GeForce RTX 3090 GPUs. The multiscale design process is carried out on a computer with 1 CPU (Intel i7-6700 K 4.00 GHz) and 16 GB of RAM.

Active learning strategy

We develop a conditional diffusion model integrated with an AL strategy for microstructure generation to facilitate rapid dataset expansion. First, the diffusion model serves as the primary generative framework, progressively denoising Gaussian noise into structured patterns. We utilize elastic tensors as conditional inputs to guide the generation process. Critically, these target elastic properties are strategically selected from the boundary surface of the training set’s property space, enabling systematic exploration of the structural design boundaries. Second, the sampling strategy incorporates classifier-free guidance—a technique that enhances the model’s adherence to specified elastic properties by creating an implicit classifier through unconditional diffusion. This approach ensures that generated structures closely match the desired elastic properties while maintaining physical feasibility.

The AL pipeline proceeds as follows. A baseline diffusion model is trained using an initial dataset. We then sampled a batch of target elasticity tensors from the boundary of the property space of the initial dataset for generation (see Supplementary Sections 4.7.1 and 4.7.2). These boundary-sampled tensors condition the model to produce a new dataset. After rigorous quality filtering focused on structural connectivity (see Supplementary Section 4.7.3), we merge the qualified samples with the dataset to create an expanded dataset, which systematically pushes the boundaries of achievable elastic properties. Through multiple iterations of this process, we successfully constructed a comprehensive L-BOM dataset, where the properties of each structure are precisely controlled through strategically selected conditional elastic tensor inputs and classifier-free guided sampling.

Bicontinuous multiscale structure design

Multiscale topology optimization is a design method that optimizes both the overall shape and internal material layout to meet specific performance goals under certain constraints. It consists of two stages: first, computing the elastic tensor for each macro unit; second, finding a matching microstructure that achieves this tensor. The optimization model for the first stage is formulated as follows,

$${\min }_{\mathit{C}}f\left(\mathit{C}\right)\mathrm{s}.\mathrm{t}.\mathbf{K}\left(\mathit{C}\right)\mathit{U}=\mathit{F},\mathrm{SDF}\left({\mathit{C}}_e\right)\le 0,\forall e\in \left\{1,\dots ,N\right\}.$$

Here, f is the application-dependent objective function. In designing bone implants and spinal fusions, the goal is to meet specific stiffness requirements while staying within set porosity limits (see Supplementary Section 6.1). For the chair design, we aim to maximize stiffness at a given porosity level (see Supplementary Section 6.3). C = {C₁, ⋯  , C_N} represents the elastic tensor and is taken as the design variable. N is the number of macroscale elements. K(C) is macroscale stiffness matrix which is derived from C. U denotes the displacement field, and F represents the applied external force. The inequality constraint SDF(C_e) ≤ 0 is used to ensure that C_e is within the feasible range of the dataset, where SDF stands for the Signed Distance Function (see Supplementary Section 4.7.1). Then, we find ${\mathit{s}}_e\in \mathcal{D}$ where $\mathcal{D}$ is the proposed dataset that minimizes

$${\mathit{s}}_e=\arg {\min }_{\mathit{s}\in \mathcal{D}}{∥\mathit{C}\left(\mathit{s}\right)-{\mathit{C}}_e∥}_2.$$ 1

Here s_e is the material distribution of the e-th optimized microstructure. C(s) is calculated by homogenization theory⁶⁵.

Printability verification

The bicontinuous open-cell microstructures demonstrate excellent printability due to their open-hole characteristics. We select five microstructures from each dataset. Then, we fabricate 3 cm full-size specimens using stereolithography (SLA) on the BN600 3D printer and 3 mm half-unit cell structures using projection micro-stereolithography (P μSL) technology on the BMF S130 3D printer. Figure 7a–d illustrates these microstructures and their corresponding numerical indicators. In each group, the microstructures in orange represent isotropic lattices that approach the HS upper limit of stiffness, while the microstructures in blue represent anisotropic lattices that approach the Voigt upper limit of stiffness. For full-scale printing results and validation of unit-cell fidelity, see Supplementary Section 8 (Figs. S15–S17 and Table S9).

Fig. 7. Partial structures of four datasets shown in different colors.

a–d Radar charts summarize key mechanical and geometric properties, including bulk modulus (K), shear modulus (G), Young’s modulus (E), elastic tensor components (C₁₁₁₁, C₁₁₂₂, C₁₂₁₂), porosity (1–V), surface-to-volume ratio (S/V), and Zener ratio (Z). Colored regions indicate different structural categories: orange for isotropy, blue for anisotropy, and the remaining colors for random selections. 3D-printed examples of both complete microstructures and halved views are displayed beneath each chart. Each color in the radar chart corresponds to the representative microstructure shown below. Source data are provided as a Source Data file.

Author contributions

L.W., J.F., X.Z., L.L., and X.F. conceived and designed the research; L.W. constructed the 3D L-BOM dataset, implemented the numerical simulations, and performed the data analysis; L.W. and X.Z. printed the metamaterial samples; J.F. and K.C. designed the machine learning framework; L.W., J.F., and J.H. wrote the relevant code and conducted the experiments; L.W., X.Z., and X.F. drafted the manuscript; L.W., J.F., X.Z., J.H., W.W.S.M., and X.F. revised the manuscript; X.Z., L.L., and X.F. supervised the project; X.Z., L.L., X.F., and L.W. provided funding support and resources. All authors participated in the discussion of the results.

Peer review

Peer review information

Nature Communications thanks Marco Maurizi, Akshansh Mishra, Yu Qin, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The tables of physical properties for the microstructures in the four datasets, as well as those for the microstructures shown in each figure of the main text and supplementary materials, are available on Zenodo⁶⁶ (10.5281/zenodo.17662421). Additionally, the aforementioned Zenodo repository also includes selected example data and the corresponding visualization scripts. Source data are provided with this paper.

Code availability

The code for the microstructure generation network and the corresponding training dataset are available at Zenodo⁶⁷ (10.5281/zenodo.17475876). The repository link for the LIVE3D framework is https://github.com/lavenklau/homo3d, and the Git commit version is 608f473.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-68089-2.