Data-driven inverse design of multifunctional bicontinuous multiscale structures

Domain: chemistry
Published: 2026-01-08
Editorial outcome: major_revision

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.

Introduction
Natural materials such as femur bones1–4, wood5–8, and sponges9–11 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 biomedicine13–18, mechanical engineering19–21, sports equipment22,23, and energy and environmental applications24,25, where they fulfill critical needs for high stiffness26,27, extensive surface area28, superior breathability29, and varying levels of permeability30–32. This study explores bicontinuous multiscale structures and their potential to enhance material performance, starting with their definition.

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 efforts33,34. 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 methods16,38,39 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 microstructures40–42 (Fig. 1c left). Commonly employed techniques to enhance this connectivity include optimization43,44, 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 structured 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).

To expand the dataset, we implement a self-conditioning diffusion model47,48, 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 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.

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 structured design
The developed L-BOM datasets offer extensive options for bicontinuous multiscale structured 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 shielding50–53. 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 infiltration54–57. 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 accommodate 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 seek 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.

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).

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.

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).

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 optimized properties and established open-cell permeability. Finally, a multiscale design framework is implemented to enable multiscale 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 2.30GHz), 512GB of RAM, and 8 Nvidia GeForce RTX 3090 GPUs. The multiscale design process is carried out on a computer with 1 CPU (Intel i7-6700K 4.00GHz) and 16GB 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 structured 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 f(C)
C
s:t: K(C)U = F,
SDF(Ce) ≤ 0, 8e ∈ {1, ..., N}:

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, ..., CN} represent 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(Ce) ≤ 0 is used to ensure that Ce 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 se ∈ D where D is the proposed dataset that minimizes

se = argmin s∈D ||C(s) - Ce||2 : (1)

Here se 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).

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 (https://doi.org/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 (https://doi.org/10.5281/zenodo.17475876). The repository link for the LIVE3D framework is https://github.com/lavenklau/homo3d, and the Git commit version is 608f473.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript “Data-driven Inverse Design of Bicontinuous Multiscale Structures” presents an approach for the inverse design of bicontinuous, elastic multiscale structures. This method focuses on constructing datasets of cubic symmetric, bicontinuous microstructures with identical boundaries to ensure seamless connectivity across scales. Using topology optimization, denoising diffusion models, and active learning, four extensive datasets are generated. These datasets enable the selection of microstructures based on matching elastic coefficients, facilitating the design of multiscale structures with tailored properties.\n\nThe concept of creating extensive datasets of bicontinuous microstructures with identical boundaries, along with the reported applications, is intriguing. However, the performance, impact, and novelty of the proposed approach compared to existing methods remain unclear.\n\nComments.\n1. In the introduction, the authors state that commonly employed techniques to enhance connectivity between microstructures lead to “increased computational costs and limited properties”, struggling to “effectively address the above two key challenges”. Can the authors discuss and quantify these challenges? How does the proposed approach solve them? Does the proposed approach outperform them?\n2. Related to 1., benchmarks with existing methods, including graded spinodoid microstructures ( https://doi.org/10.1038/s41524-020-0341-6 ; https://doi.org/10.1016/j.cma.2021.113894) or growth-based approaches https://doi.org/10.1038/s41467-024-47831-2 , are missing. Can the proposed approach outperform these methods? What does this method achieve that other approaches cannot?\n3. Related to 1. and 2., are interpolations between microstructures more computationally expensive and have limited property range compared to the proposed method? If yes, can the authors show a benchmark?\n4. Why is the initial dataset optimized by maximizing the elastic moduli (bulk, shear, Young’s modulus, etc)? Do most of the proposed applications require stiffness values not achievable by using non-optimized microstructures with compatible 3D-printed materials (e.g., biocompatible)?\n5. Have any of the designs presented in the case studies been compared to existing commercial designs?\n6. The selection of the four masks is not entirely clear: “We evaluated the generated structures based on quality, Poisson’s ratio coverage, and bicontinuous.” How did you exactly select these masks? How do you define “quality”?\n7. Regarding the multiscale design, how is the distribution of elastic coefficients in the design domain determined? Looking at the bone implant as an example, to the best of my knowledge, the implant’s stiffness (entire stiffness matrix) should match that of the bone to promote cell growth and tissue regeneration. In the reported example as well as in all other case studies, stiffness is maximized, rather than matched with that of the bone. Furthermore, should the stress distribution within the implant be as uniform as possible?\n8. Following up on point 7, does the assumption of cubic symmetry pose limitations for certain applications, such as bone implants?\n9. About the filtering device example, the datasets were constructed accounting only for the stiffness. However, the main performance metric for such devices seems to be permeability. Would using other conditioning properties, such as permeability, in the diffusion model help finding better designs?", "reviewer_2": "1. The paper does not discuss how errors in the generative model (e.g., boundary misalignment or failure to meet elastic tensor constraints) propagate through the multiscale optimization process. Quantifying this could identify potential pitfalls in applications.\n2. The reliance on predefined elastic tensors and masks could limit the model's ability to generalize. Techniques like unsupervised learning or meta-learning could enhance adaptability for unseen property spaces.\n3. Include automated metrics for validating bicontinuity and boundary alignment during dataset generation, such as a) Fourier transform-based spatial analysis for structural periodicity, and b) Stress-testing using finite element analysis on randomly sampled outputs.\n4. Reducing the degrees of freedom from 21 to 3 could oversimplify certain material behaviors. The paper could explore whether this reduction impacts the fidelity of the generated microstructures for applications requiring anisotropic properties.\n5. Embedding physical constraints (e.g., Navier-Stokes equations for permeability or heat transfer equations for thermally conductive structures) directly into the model could improve performance on application-specific tasks.\n6. Use latent space analysis to identify how different latent dimensions correlate with specific structural properties, aiding in targeted property tuning.\n7. Instead of static boundary masks, incorporate adaptive masks that evolve during training to explore broader structural designs.", "reviewer_3": "The study investigates the machine learning (ML)-enabled design process of large-scale, boundary-identical, bicontinuous structures. While the study addresses an important research area, the current results fail to clearly demonstrate the novelty of the work. Several critical issues should be addressed to enhance its scientific contribution and overall quality.\n\nMajor Concerns:\n1. The manuscript shows no significant advantages in active learning or microstructure generation compared to their prior studies (reference 56 and https://doi.org/10.1002/advs.202410428). If the novelty lies in adding boundary conditions to achieve large-scale structure assembly, the authors must clarify how their approach improves computational efficiency or structure performance. Such as directly large-scale topology optimization methods (https://doi.org/10.1038/nature23911) or other ML-based methods (https://www.nature.com/articles/s41467-024-47831-2, https://www.nature.com/articles/s41467-023-42415-y)\n2. The two examples of bone implants (segmental implant and fusion cage) differ from actual requirements, and the performance description is insufficient.\n(a) The base material for constructing the structures is unclear. As load-bearing bone implants, materials such as PEEK or Ti are typically used, and optimization objectives differ significantly depending on the base material.\n(b) For example, metal implants often require reduced stiffness to avoid stress shielding, while polymer implants face strength limitations, making strength more critical than stiffness. Additionally, base material selection affects porosity, curvature, and overall structural properties.\n(c) What is the difference between assembling the final implant by selecting different structures within a defined range and repeatedly constructing it using a single structure optimized for the target performance?\n(d) The authors optimized the microstructures, but do the assembled large-scale structures still meet the target performance? In other words, the final porosity, mechanical properties, and other performance metrics of the large-scale structures should be clarified and compared. A good example would be the permeability test in this paper.\n3. How do the principles of microstructure arrangement enhance or control properties such as permeability? Such summaries or discussions should be provided to support the advancement of the field.\n\nMinor Concerns:\n4. The transitions between units with the preset boundary conditions are good. And the internal structures appear to have rounded transitions near the boundaries. How were these transitions considered and achieved?\n5. On Page 7, the authors claim that smaller pores contribute to faster cell seeding and that high curvature improves nutrient flow, cell attachment, and tissue regeneration. These statements are inaccurate. Accurate discussions of these can be referred to https://doi.org/10.1016/j.matt.2022.06.003 and https://doi.org/10.1038/s42254-024-00700-9 for accurate discussions of these phenomena."}, "author_response": "We are grateful to all three reviewers for their constructive and insightful comments. Based on the feedback from the reviewers, we have thoroughly revised the main manuscript and the Supplementary Information according to all comments, primarily including\n• This study considers two types of bone: compact bone and cancellous bone. To highlight the advantages of our optimization results in terms of computational efficiency and performance metrics, we compare them with parametric microstructure models, including the body-centered cubicsymmetry(BCC)truss,Schoen’sI-graph-wrappedpackage(IWP),andspinodalstructures.\nIn both cases, our optimized designs demonstrate clear advantages.\n• We develop and implement a novel experimental protocol that integrates numerical analysis, simulation modeling, and experimental testing to comprehensively validate the broad range of permeability represented in our dataset. Both simulation and experimental results demonstrate the outstanding performance of the constructed dataset.\n• We further enrich the dataset by incorporating a comprehensive set of properties, including mechanicalcharacteristics(Young’smodulus,Poisson’sratio,shearmodulus,andbulkmodulus),\naswellaspermeability,porosity(volumefraction),poresize, andsurface-to-volumeratio(S/V).\nPlease see below for a point-by-point response to the reviewers’ comments and concerns. All page/figure/table/line numbers refer to the revised manuscript file with tracked changes. All paragraphs that appear in the revised manuscript or supplementary are marked in red."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have thoroughly addressed all of my previous comments and concerns. The revised manuscript is significantly improved and is now suitable for publication in Nature Communications.", "reviewer_3": "The authors have made substantial revisions that address most major concerns. I think this paper can be accepted after addressing some issues remained blow:\nThe reply to their novelty is reasonably thorough and includes performance comparisons, e.g., in mechanical cloak design (5 minutes vs. 10 hours). Still, A more explicitly contrast the new boundary-aware generation strategy with their previous “boundary-free” designs should be clarified.\nAbout the discussion part,the author experimentally compared the permeability of two LBBM structures and four TPMS structures, achieving a wide range of permeability results. And the addition of application such as thermal conductivity to the permeability discussion is good. However, the author still have not addressed my previous question: Does this contribute to some theoretical understanding of the relationship between structural design and permeability, rather than merely comparing different results? For example, among parameters such as porosity, pore size, pore shape, and orientation, how do they rank in terms of influence on permeability? The authors do not need to experimentally or numerically verify each factor, but should at least provide a theoretical discussion.", "reviewer_4": "Comment 1: In general this is a quality piece of work. The manuscript and figures are well constructed. Most, but not all prior reviewer concerns appear to have been addressed.\nComment 2: I am confused by the need to resolve connectivity in conventional periodic lattices. This must stem from the discretization of the design domain, but why this is necessary for conventional lattice design is not explained. Continuous fields can be used to efficiently drive periodic lattice design variable such as cell size/shape, density, and tiling.\nComment 3: There are still some concerns over novelty. If the term “open cell” is substituted (quite appropriately) for bi-continuous, much of the implications of this work can be more readily understood.\nComment 4: There are too many complex multi-panel figures. To be fair these figures are executed extremely well, and are visually pleasing, but many features become too small to see, making interpretation of results difficult at times. A more straightforward quantitative presentation of results may help make direct comparisons, and remove any obfuscation.\nComment 5: I am also curious about the choice of boundary masks. Can the authors provide more detail on how they were selected and enforced. What order connectivity is enforced at the boundary, geometric, tangent, continuous (C0, C1, C2)? Can the authors comment more about the impact of boundary mask on properties? Is it useful to construct 100,000 internally different iterations of 4 boundary configurations, or would more performance gains be made by constructing 1000 different boundary masks with 100 internal designs each? The re-entrant boundary mask showing such wide spread in poissans ratio is telling that this initial constraint is significantly blocking/unlocking property spaces. Until the boundary conditions can be fully understood and modified where possible by machine design, human designs will often remain superior."}, "author_response": "We thank all three reviewers for their constructive and insightful comments. In response, we have thoroughly revised the main manuscript and the Supplementary Information. The key changes are as follows:\n• Importance of connectivity. We expanded the discussion of connectivity, detailing its implications for both mechanics and manufacturing. We clarify how connectivity governs effective properties and directly affects printability and practical feasibility.\n• Fabrication. We report additive manufacturing of both single-cell and multiscale structures and now include the fabricated samples.\n• Simplified analytical framework. We analyze permeability within the Kozeny–Carman framework—highlighting the roles of porosity, pore size, shape/tortuosity, and orientation—and complement it with a maximum-flow surrogate that captures connectivity and bottlenecks.\n• Definitions of bicontinuity and open-cell. We sharpen these definitions and revise the title and main text to ensure terminological precision and consistency throughout.\n• Language and figures. We streamline the prose and simplify figures to improve clarity, readability, and the intuitive presentation of key results.\nWe believe these revisions substantially strengthen the clarity, rigor, and impact of the work."}, {"round": 3, "reviewer_comments": {"reviewer_3": "The authors have well addressed my concerns and significantly improved the manuscript. I think it is now suitable for publication in Nature Communications.", "reviewer_4": "The modifications to this manuscript make it much better and address all the reviewer comments. It is clear the authors have done quite a bit of technical work, and work in preparation of the manuscript."}, "author_response": ""}]

Decision letter