Machine-Assisted Inverse Design of Patchy Particles for Self-Assembly of Archimedean Tilings

Abstract

Self-assembly holds great promise for the development of next-generation materials with highly ordered micro- or nanoscale structures. A primary challenge involves the efficient exploration of high-dimensional parameter spaces to achieve user-desired architectures. In this study, we developed an inverse design strategy for the self-assembly of various Archimedean tilings using one-component patchy particles. The cornerstone of our approach resides in the seamless integration of design space decomposition with machine-assisted optimization techniques and specialized simulation evolution pathways, culminating in a stepwise modular protocol for determining critical particle attributes. Specifically, we employed a genetic algorithm-based backward evolution learning protocol followed by a Bayesian-based forward optimization protocol to sequentially determine the patch position and binding strength of the patchy particle. Consequently, the self-assembly of various Archimedean tilings and even more intricate exotic superlattices was successfully realized at a reduced computational cost. Moreover, our strategy showcases a robust capability to explore the design space, offering simplified or enhanced design schemes for target structures. Overall, our work advances inverse design strategies for fabricating intricately structured and high-performance interfacial materials within the scope of patchy particle models.

1. Introduction

Self-assembly offers a powerful bottom-up strategy for constructing micro- and nanoscale materials with intricate hierarchical architectures. As nature’s fundamental mechanism for organizing living systems across different scales, this approach enables the engineering of advanced high-performance materials. − Self-assembly relies on colloidal or nanoscale building blocks, among which patchy particlescharacterized by distinct, localized surface regions (″patches″) exhibiting tailored chemical or physical propertieshave garnered significant interest. Recent progress in surface patterning, including ligand decoration and DNA functionalization, − has enabled precise control over patch characteristics, including their number, position, and binding strength. These particles act as ″chemical valences,″ critically influencing the enthalpy–entropy competition and directing self-assembly pathways, particularly for nonclose-packed architectures. − However, the target-driven design of building blocks that spontaneously assemble into user-desired architecturesthe so-called inverse design problem in self-assemblyremains a formidable challenge to date.

Recent advances in machine-assisted approaches have created new opportunities to tackle this problem. ,− For instance, Kim et al. employed a generative adversarial network trained on 31,713 known zeolite structures to design crystalline porous materials with desired methane adsorption heats. Coli et al. proposed a design framework integrating evolutionary parameter optimization with a convolutional neural network, enabling the engineering of crystals, quasicrystals, and liquid crystals. Other approachesincluding genetic algorithm (GA) and Bayesian optimization (BO)have been also widely employed to accelerate the design of diverse functional materials and nanostructures. , However, given the highly complex relationship between building blocks and potential assembled structures, further efforts are still urgently required.

In this work, a machine-assisted inverse design strategy was developed to efficiently engineering of patchy particles that spontaneously self-assemble into desired structures. Specifically, we focus on Archimedean tilingsa class of two-dimensional superlattices comprising in-plane tessellations with two or more distinct polygon types (Table and Figure S1). , These superlattices demonstrate distinctive structure–property relationships, offering potential applications in optoelectronics, catalytic systems, and low-friction coatings. ,− However, achieving the self-assembly of Archimedean-like structures remains challenging. ,,,,−

1. Target Structures (Archimedean Tilings), the Designed Patchy Particles, and Their Corresponding Self-Assembled Arrangements.

2. Results and Discussion

2.1. A Modular Two-step Inverse Design Protocol

To address this challenge, our inverse design strategy integrates machine-assisted optimization and tailored Monte Carlo simulations within a modular framework (Figure ). The cornerstone of our approach is the decomposition of the high-dimensional design space into two complementary subspaces: one for patch positioning and the other for binding strength. We acknowledge this separation is a nontrivial assumption, as it presumes that the optimal parameters for position and strength are not strongly coupled. However, as our comparative analysis later demonstrates, this strategy yields outcomes comparable to those from unified, higher–dimensional parameter explorations, validating its efficacy for the systems studied. This dimensional reduction enables our targeted, two-step protocol: a Genetic Algorithm (GA) is first employed to efficiently explore the complex combinatorial space of patch positions, followed by Bayesian Optimization (BO) to finely tune the continuous binding strengths.

1.

Schematic drawing illustrating the inverse design strategy of patchy particles for self-assembly. Top: Decomposition of the patchy particle design space into subspaces for patch position (θ) and strength (u­(θ)); Middle: Two-step modular protocol for patch position and strength design; Bottom: Archimedean tilings (Target structures).

This in silico strategy was implemented using a one-component minimal patchy particle model as the fundamental building block (here, ‘one-component’ signifies that all particles within a given system are identical). The only tunable parameters are the patches’ angular positions (θ∈[0,2π) and their binding affinities (u(θ)∈[0,1]) (see Materials and Methods for model specifications). A key feature of our framework is the ability to dynamically determine the optimal patch number, as the optimization can effectively turn patches ‘off’ by setting their binding strength to zero. Our design intentionally employs spherical particles with simple, nonselective short-range interactions. This minimal approach contrasts sharply with previous Archimedean tiling designs, which often required additional control parameters beyond basic patch characteristics. ,,,− , Ultimately, this combination of a modular strategy and a minimal model allows for the efficient design of particles that self-assemble into both Archimedean and non-Archimedean superlattices (Table ).

Specifically, in the initial phase, we developed and implemented a GA-based backward evolution protocol to optimize patch positions (Figure ). GA, as an evolutionary computational approach, possesses robust global search capabilities due to its inherent mechanisms such as crossover and mutation. This enables it to effectively explore complex solution landscapes and address optimization challenges within large parameter spaces. ,− This algorithm operates based on the concepts of genome and phenome, which correspond to patch positions and self-assembling structures, respectively. In Generation 1, the GA was initialized with a population of N = 1000 randomly generated genomes, and the corresponding phenomes were determined through Hard Particle Monte Carlo (HPMC) simulations (see Materials and Methods). Specifically, we designed 1000 distinct patchy particle configurations with randomized patch positions (genomes). For each configuration, HPMC simulations were conducted to characterize their respective self-assembly behaviors (phenomes). Consequently, a total of 1000 independent MC simulations were performed, with each simulation system comprising 100 identical patchy particles (see Materials and Methods for details of GA). It is worth noting that HPMC, executed using the HOOMD-blue package, has been widely applied in studies investigating self-assembly behaviors. ,,,,,,, Subsequently, utilizing a scoring module that evaluates structural similarity based on local particle arrangements (see Materials and Methods), we identified and selected the top 10 highest-scoring phenomes. These were then employed to generate a new population of 1000 genomes (i.e., a subsequent generation) through crossover and mutation operations. This iterative procedure continued until the system met predefined convergence criteria (Figure S2).

Especially, to enhance computational efficiency, we incorporated a backward-evolution path within the MC framework in this step (Figure ). In this pathway, simulations are initialized with particles prearranged in the target lattice, followed by MC sampling to assess structural stability. This approach effectively avoids kinetic traps that are prevalent in forward-evolution simulations, thereby significantly improving computational efficiency. As demonstrated below, our GA-based backward evolution protocol proves particularly effective for exploring the high-dimensional patch position design space. However, it is noted that the high-scoring configurations obtained in this step may not always demonstrate comparable performance in realistic self-assembly processeseven with correct patch positionsrequiring subsequent validation through forward-evolution simulations and additional optimization (Figure S3).

The optimized patch positions served as the foundation for the second-step patch strength design, which was conducted using a Bayesian-based forward optimization protocol (Figure ). The essence of this step involves leveraging Bayesian optimization to efficiently explore the parameter space and predict the ″optimal object″ by utilizing all available information from prior optimizations (see Figure and Materials and Methods for details of BO). − Specifically, a distribution of patch strengths (u) was sampled from the parameter space, and the objective function f(u)a black-box functionwas employed to evaluate the self-assembly outcome for a given u via the scoring function. Given the black-box nature of f(u), Gaussian process regression acted as the surrogate model to approximate its behavior. In general, our Bayesian optimization operates iteratively: it selects a new set of u values using the Upper Confidence Bound (UCB) acquisition function, evaluates the corresponding f(u), and updates the surrogate model until a stopping criterion is satisfied. Furthermore, throughout this process, both u and f(u) values were systematically recorded for evaluation and facilitation of the optimization toward an improved distribution of patch binding strengths (Figure S2). To assess the performance of Bayesian optimization, MC simulations were performed for each optimization trial. Unlike the GA-based step, these simulations followed a forward-evolution path starting from random particle configurations (Figure ), ensuring that the designs incorporate realistic assembly information. Additionally, an annealing technique was integrated into the MC simulations to assist particles in escaping kinetic traps during assembly (see Figure and Materials and Methods). These combined methodologies are critical for evaluating the particles’ capacity to overcome kinetic barriers and for optimizing patch interaction strengths.

While forward-evolution simulations are computationally expensive, Bayesian optimization tackles this challenge by employing ″informed″ sampling strategies and integrating prior knowledge. Consequently, this approach facilitates efficient exploration of the parameter space with rapid convergence, requiring substantially fewer evaluations compared to traditional methods. As a result, the Bayesian-based forward optimization protocol also enables the effective determination of patch strength.

2.2. Case Study: Self-Assembly of the {3, 12, 12} Archimedean Tiling

To illustrate our strategy, we selected the {3, 12, 12} Archimedean tiling (truncated hexagonal tiling) as a representative case study (Table ). This open network structure exhibits the largest pore size among all Archimedean tilings (12-sided polygons), which poses significant challenges for its self-assembly. , Initially, we applied the GA-based backward evolution learning protocol for designing patch positions (Figure A). This method demonstrated remarkable efficiency by converging to optimal patch positions within only 10 generations, and each generation’s evaluation required merely 10³ Monte Carlo steps (Figure B, approximately 0.033 core-hours per simulation on a single 2.50 GHz Intel Xeon E5–2678 v3 CPU). Consequently, the total computational cost for this step amounted to approximately 333 core-hours (10 generations × 1000 genomes × 0.033 core-hours). Such high efficiency in patch position design was consistently observed across all Archimedean tilings examined in this study (Figures S4–S10). Moreover, while early generations explored complex and unstable patch distributions, substantial improvements in scores were achieved through both the reduction of patch numbers and the optimization of their positions (Figure A–C). Eventually, the protocol maintained stable structures with properly designed patches: for the {3, 12, 12} tiling, only three patches were needed (Figure A), while other Archimedean tilings required no more than five patches (Table and Figures S4–S10).

2.

GA-based backward evolution learning for patch position design in the {3, 12, 12} Archimedean tiling (Step 1). (A) Highest score per generation and its corresponding self-assembled structure. (B) Score evolution with MC steps for representative generations. (C) Stability testing of preset target structures for designed particles in the backward-evolution simulations. In all these cases, u­(θ) = 1. (D–F) Bayesian-based forward optimization for patch strength design in the {3, 12, 12} Archimedean tiling (Step 2). (D) Score per optimization trial and its corresponding self-assembled structure. (E) Score evolution with MC steps for representative trials; the blue arrow marks the final self-assembly in Trial 31. (F) Self-assembly of designed particles for representative trials in the forward-evolution simulations.

The three-patch particle design served as the foundation for implementing our Bayesian-based forward optimization protocol to determine optimal patch strengths. This methodology demonstrated remarkable efficacy in achieving target structural designs (Figure D). In initial forward-evolution MC simulations, particles failed to self-assemble into the target structure from initial random conditions, and even no pore formation was observed (Figures E,F and S3). However, through iterative optimization, the scores improved rapidly, with an open pore emerging by the 10th trial (Figure D). After 30 optimization trials, the system successfully achieved the targeted network structure (Figure E,F). The forward-evolution simulations in this step were computationally expensive, requiring approximately 8 core-hours per run (using a single 2.50 GHz Intel Xeon E5–2678 v3 CPU core). Nonetheless, the optimization process typically converged within ∼30 trials, resulting in a total computational cost of ∼240 core-hours. Notably, for all examined Archimedean tilings, our protocol efficiently identified optimal patch interaction strengths in fewer than 50 trials, with particularly rapid convergence (<10 trials) for specific patterns such as the {3,6,3,6} tiling (see Figures S4–S10). These findings underscore the computational efficiency of our strategy. Furthermore, while our design intentionally employs a minimal patchy particle model to enhance experimental feasibility and practical applications, other factorssuch as particle core properties that govern softnessmay also influence both the design process and assembly outcomes (Figure S11).

2.3. Discovering Alternative Pathways and Simplified Designs

Efficient exploration of the vast parameter space is essential to successful inverse design. Our strategy integrates design space decomposition with machine-assisted optimization and tailored molecular simulations, providing attractive advantages in exploring or even exploiting the design space. Remarkably, our approach requires only minimal design information (patch positions and binding strengths of one-component particles) to achieve self-assembly of diverse Archimedean lattices. This represents a substantial simplification over conventional methods, which typically demand additional constraints including particle geometry and type, and selective patch–patch interactions. ,,,− , Furthermore, our strategy employs a simple scoring module that directly evaluates structural similarity between assembled and target structures, thereby obviating the necessity to comprehend the target’s complexity. This contrasts with many machine-learning-based strategies that require careful selection of scoring metrics.

Especially, our strategy is able to provide an alternative design scheme for complex target structures. For instance, in a pioneering study, Whitelam et al. employed a neuroevolutionary learning algorithm to develop a 3-patch particle model for the {3,12,12} Archimedean tiling - similar to one of our optimized designs (red arrow in Figure D). Although both particle models share the same number of patches, they differ in terms of patch strength distribution: their W1S2 model (one weak patch and two strong patches) contrasts with our W2S1 design (one strong patch and two weak patches). Notably, these design differences result in distinct assembly pathways. In the case of W1S2-type particles, trimer formation is initially driven by the interaction between the two strong patches (a-b patches; Figure A–C), followed by subsequent particle recruitment through weak c–c patch bonding with these trimers. This process progressively leads to the formation of a pore structure, achieving a score of approximately 0.7 at ∼90 × 10⁶ steps (blue arrow in Figure A). In contrast, W2S1-type particles first assemble into dimers via their single strong patch (c–c patch; Figure D–F); and these dimers then serve as the foundation for open pore structures that develop through weak a-b patch interactions. Ultimately, this results in an assembly structure achieving a higher score (∼1) by ∼60 × 10⁶ steps (red arrow in Figure D). Another example is the design of the {3,3,3,4,4} Archimedean tiling. Prior investigations presumed that a 5-patch particle model was required due to geometric symmetry constraints (each lattice particle having five neighbors; Figure A). Surprisingly, our approach successfully achieved this tiling using only a 3-patch particle model. As shown in Figures A and S5, the comprehensive utilization of patch interactionsespecially the nonpatch contact (red arrow, Figure A)played a pivotal role in enabling the self-assembly of this structure. Recall the simplifying assumption introduced earlier regarding the decomposition of the design space into separable subspaces for patch position and binding strength optimization, which inherently risks neglecting potential correlations between these parameters. However, it should be noted that, while design space decomposition might influence correlations between patch positions and binding strengths, the aforementioned comparative analysis indicates that our strategy yields outcomes comparable to those obtained through higher–dimensional parameter exploration. This robust agreement strongly suggests that, for the specific class of patchy particle systems and target properties studied here, the correlations between patch positions and binding strengths, while potentially present, were not the dominant factor governing the optimal designs identified. Consequently, the decomposition assumption proved to be a valid and highly effective strategy, enabling significant computational savings while achieving results on par with a more exhaustive search.

3.

Self-assembly of two types of designed patchy particle types for the {3, 12, 12} Archimedean tiling (W1S2-type Vs. W2S1-type). (A) Evolution of the score, patch contacts, and neighboring particle numbers during self-assembly. Red and blue arrows indicate the final formation of self-assembled structures. (B, C) Representative snapshots at different assembly stages and schematic illustration of the corresponding particle packing arrangements. Color code: gray (monomers), red (dimers/trimers), blue (intermediate states).

4.

Self-assembly of target structures using minimal patchy particle models. (A) {3, 3, 3, 4, 4} Archimedean tiling assembly with 3-patch particles. (B) {3, 12, 12; 3, 4, 3, 12} non-Archimedean tiling assembly using one-component patchy particles. Left and middle panels: schematic representations of designed particles and their packing states, respectively. Right panel: simulation snapshots of assembled structures with corresponding vertex schematics.

2.4. Generalization to Complex Non-Archimedean Tilings

Furthermore, it was argued self-assembling non-Archimedean tilings is particularly challenging due to their multiple vertex types, implying that multiple particle species would be necessary. Our strategy challenged this assumption. As demonstrated in Figure B for the {3,12,12; 3,4,3,12} tilingwhich contains two distinct vertex types (green and black)we achieved successful self-assembly of its fundamental pore structure using only one-component 3-patch particles (Figure S9). While some defects persist due to the structure’s inherent complexity, this result clearly demonstrates the capability of our design strategy.

Our choice of a two-stage strategy combining a GA and BO is deliberate, designed to synergistically address the distinct facets of the inverse design problem. The initial phase employs GA to efficiently navigate the high-dimensional (i.e., 36-dimensional), discrete search for patch positionsa combinatorial challenge well-suited to evolutionary exploration. This search is guided by a computationally inexpensive scoring function within our backward-evolution protocol, enabling the rapid identification of promising configurations. Subsequently, the second phase utilizes BO to fine-tune the few (typically 3–5) continuous parameters of binding strength. This task is governed by computationally expensive forward-evolution simulations where BO’s sample efficiency is critical for finding the optimum with minimal evaluations. A standalone GA would be prohibitive for this costly optimization, while a standalone BO would be inefficient for the initial high-dimensional discrete search. Thus, this hybrid approach is essential for effectively exploring the complex design space within practical computational limits.

While the precise control over patch strength required by our designs (Table ) remains experimentally challenging, ongoing advances in DNA functionalization are creating increasingly viable pathways to its realization. For instance, programmable interaction strengths are now achievable by modulating parameters like DNA grafting density and linker length. Beyond simple strength tuning, complex patch geometries such as multihandle DNA origami ‘nanoflowers’ enable explicit control over bonding symmetry and interaction valency. Furthermore, techniques like photopatterning facilitate the spatial encoding of DNA on particle surfaces, establishing a direct link between angular position and binding specificity. Collectively, these methodologies are bringing the experimental synthesis of particles with the precise angular and energetic specificity required for our designed tessellations within reach, making such in silico blueprints increasingly accessible for fabrication.

3. Conclusion

In summary, we developed a machine-assisted inverse design strategy for engineering patchy particles that spontaneously self-assemble into target high-order architectures. By integrating genetic algorithm-based backward evolution learning with Bayesian forward optimization, our modular approach enables efficient exploration of high-dimensional parameter space for Archimedean tilings while significantly reducing computational costs.

Our machine-assisted approach acts as a form of in silico ″directed evolution″, mirroring the optimization processes that guide self-assemblynature’s fundamental mechanism for creating hierarchical structures. While mathematically perfect Archimedean tilings are ideals, the widespread occurrence of tessellated and periodic patterns in nature, from diatom frustules to honeycomb arrays, suggests a convergence of design principles. Consequently, the rules uncovered by our strategy may not only guide the fabrication of advanced synthetic materials but also provide valuable hypotheses for understanding how natural evolution fashions complex functional architectures from simple components.

4. Methods

4.1. Patchy Particle Model

To construct the patchy particles, the particle surface is uniformly divided into 36 segments, each possessing an identical central angle and size. Subsequently, a specific binding strength u is assigned to each segment (with the initial value of u being randomly determined). Segments where u ≠ 0 are defined as patches. Each patch is characterized by its angular position θ and binding strength u(θ). Herein, θ is defined as the clockwise angular displacement between the center of the patch and the preset principal axis (as illustrated in Figure ). Notably, dynamic conversion between patched and nonpatched segments is permitted during evolutionary optimization, enabling automatic adjustment of the patch number in the GA-based step.

4.2. Simulation Models

In our study, we utilized Hard Particle Monte Carlo (HPMC) simulations implemented with the HOOMD-blue software in the NVT ensemble. A two-dimensional square substrate measuring 15a in size was employed to investigate self-assembly on a surface, with periodic boundary conditions applied in both x and y directions. The system consisted of N = 100 patchy particles, each with a hard-core diameter a. All simulations were performed using N = 100 particles, which is more than ten times the number of particles typically found within a target unit cell (as target unit cells generally contain fewer than 10 particles), within a square simulation box with a lateral dimension of 15a (where a represents the size of the target unit cell). This system size was chosen to balance the computational tractability required for our extensive optimization search with the necessity of capturing the target structures’ fundamental assembly motifs. The ensemble size is sufficient to mitigate significant finite-size effects for the structures studied, though we acknowledge that more complex targets may necessitate larger systems in future work. The large box dimension was deliberately selected to be incommensurate with and significantly larger than the characteristic unit cells of the tilings. This approach minimizes potential artifacts from periodic boundary conditions and ensures that the intrinsic, often nonsquare, geometries of the target structures can emerge without being artificially constrained by the simulation box. The Monte Carlo moves encompassed particle translations, rotations, and collective motions for all particles within the system, which were primarily governed by interparticle interactions. These interactions are characterized as short-range, patch-dependent, and nonselective

$$E_{ij}(r,{\theta }_{ij},{\theta }_{ji})=\{\begin{array}{l}\infty \\ -\epsilon ·\min (u({\theta }_{ij}),u({\theta }_{ji}))\\ 0\end{array}\begin{array}{l}r<a\\ a<r\le 1.1a\\ r>1.1a\end{array}$$ 1

Here, r denotes the distance between particles i and j, while θ _ij specifies the patch on particle i located at the contact region between particles i and j. The initial particle configuration can either be preset as the target structure (i.e., backward evolution path) or initialized randomly (i.e., forward evolution path), as detailed in the Supporting Information. The system temperature was set to k _B T = 3, allowing for the use of a reduced temperature $T^{*}=\frac{k_{\mathrm{B}}T}{\epsilon }$ in simulations, consistent with prior studies. , In forward-evolution simulations, an annealing technique was employed, wherein the system was gradually cooled by incrementing the parameter ε by approximately 0.1 every million Monte Carlo steps. ,, This approach facilitates the overcoming of kinetic traps during assembly, thereby minimizing defect formation (see Table ). Furthermore, it is important to note that while the simulation box size has minimal influence in the GA-based step (owing to the backward-evolution path), it becomes a crucial parameter governing particle assembly in forward-evolution simulations (Figure S12).

4.3. Genetic Algorithm Implementation

A standard genetic algorithm was employed for two distinct optimization tasks: patch position design and the determination of particle orientations within the target lattice. A detailed flowchart of the GA is provided in the Supporting Information. For both tasks, the GA operated with a population size of 1000 genomes, evolved over 10 generations. Parent selection was performed using tournament selection (size 3), and elitism was applied to preserve the top 5% of genomes in each generation. The specific implementation details for each task are outlined below.

4.3.1. Patch Position Design

For this task, a genome was represented as a 36 bit binary string (corresponding to 36 equally spaced 10-degree angular sectors around the particle), where each bit corresponded to the on/off state of a potential patch location on the particle surface. The fitness function was defined as F = S – λ·N _p , where S is the structural similarity score (see Scoring Module), N _p is the number of active patches (i.e., the number of ‘1’s in the genome), and λ is a penalty coefficient set to 0.1. This function balances the drive for structural stability with a preference for minimalist designs. New generations were created using single-point crossover (probability P _c = 0.7) and bit-flip mutation (per-bit probability P _m = 0.01).

4.3.2. Particle Orientation Determination

To find the optimal orientations that minimize the energy of the target lattice, a genome was represented as a vector of continuous orientation angles, θ _i ∈[0,2π), for each of the N particles in the minimal repeating unit. Here, θ _i defines the orientation of particle i as the angle (in radians) between the y-axis and the vector connecting the particle center to its designated first patch position. The fitness function was simply the negative of the total system energy, F = −E _total . Offspring were generated using simulated binary crossover (SBX) with a probability P _c = 0.7 (distribution index η _c = 15). Each orientation angle was subject to polynomial mutation with a per-component probability P _m = 0.05 (distribution index η _m = 20), ensuring that angles remained within the valid [0, 2π) range.

4.4. Bayesian Optimization

This is a statistical technique based on Bayes’ Theorem, which enables the updating of the probability distribution of a model’s parameters using observed data. In this study, it was employed to design the patch strength. The Bayesian optimization methodology utilized in our work consists of two steps. In the first step, a Gaussian process (GP) surrogate model is used to evaluate the computationally expensive black-box function f­(X). For the GP prior, we employed the widely used squared exponential (Radial Basis Function, RBF) kernel. This includes incorporating a prior distribution that represents our beliefs about the behavior of f­(X), as well as an observation model that explicitly defines rules for data generation. The second step involves employing an acquisition function to determine criteria for selecting subsequent query design points within the parameter space. We adopted the Upper Confidence Bound (UCB) method, which strikes a balance between exploration and exploitation by adjusting weights assigned to predicted means and uncertainties from the surrogate model. Specifically, we define the UCB acquisition function as α­(X) = μ­(X) + κ σ­(X), where μ­(X)­represents the predicted mean of the surrogate model at pointX, σ­(X) denotes its standard deviation (representing uncertainty), and κ controls trade-offs between exploration and exploitation. The κ parameter was chosen as a constant value of 2.0 throughout the optimization process. While tuning κ could potentially improve performance, the computational budget required for extensive parameter tuning was not available in this study. Therefore, we adopted the widely used default value of κ = 2.0, as implemented in the GPyOpt package’s UCB acquisition function. Initial empirical testing indicated that this value provided a suitable exploration-exploitation balance for our problem. These two steps are iterated until a stopping criterion is satisfied (see Supporting Information). After each iteration, both parameter X and objective function value f­(X)­are recorded for evaluating subsequent iterations. GPyOpt package is used to implement Bayesian optimization in our work. ,

4.5. Scoring Analysis

The scoring module is based on a neighbor-matching approach to assess the potential deviation of the self-assembled structure generated by HPMC simulations from the target structure. This is achieved by comparing the neighbor configuration of each particle with predefined target structures, encompassing both structural alignment comparison (to determine if the particle’s structural configuration matches the features of the target structure) and orientation comparison (to evaluate the relative position of particles, including patches within the overall structure). In details, $\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(t)=\frac{1}{N}\sum _i\sum _{j\in N_g^t(i)}\frac{\delta ({\theta }_{ij}^t,{\theta }_{ji}^t)}{|N_g^t(i)|}$ , where N _g (i) represents neighboring particles of particle i and |N _g (i)| denotes the number of these neighboring particles; δ is the Delta function: δ = 1 if the patch contacts match the target structure, otherwise δ = 0; θ denotes patch position; t refers to the time point.

All the Supporting Information are provided in the main text and Supporting Information.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsnano.5c10787.

• Schematics of the target Archimedean tilings (Figure S1); detailed flowcharts of the GA, BO, and scoring algorithms (Figure S2); a comprehensive, step-by-step application of our inverse design protocol to a wide range of additional Archimedean and non-Archimedean targets, detailing the optimization convergence and final self-assembly dynamics (Figure S3–S10); and further analyses on the influence of key physical and simulation parameters, such as particle core type (Figure S11) and simulation box size (Figure S12) (PDF)

K.Y. and B.Y. conceived, designed and supervised the project. Y.X., X.Y. and X.D. performed simulations. Y.X. and T.X. conducted machine learning and data analysis. K.Y. and B.Y. wrote and finalized the manuscript. All authors approved the final version of the manuscript.

The authors declare no competing financial interest.