In recent years, there has been growing interest in nanotechnology. Nanomaterials and nanostructures are used to manufacture micro-electromechanical and nanoelectromechanical systems, which are widely used in the fields of machinery, aerospace, electronics, and biomedical engineering. Micro-nano scale systems, including biosensors, electrochemical actuators, supercapacitors, nanotubes, and nanoprobes, predominantly use nanorods, nanobeams, nanoplates, and nanoshells as fundamental structural units [1–4]. Subsequently, advanced composites, such as functionally graded materials and graphene-based reinforced composites, have been developed, which have attracted scientists to establish mechanical models for their analysis. Among them, as a typical representative of the gradient structure, the mechanical behavior of mechanical metamaterial composite beams with a negative Poisson’s ratio effect has attracted increasing attention from scholars.

Numerous experiments and theoretical studies have been conducted to effectively analyze and design metamaterial composite structures to characterize the mechanical behavior of the nanostructures. Zhao et al. unveiled the mechanism of a highly tunable negative Poisson’s ratio and developed micromechanical models based on genetic programming for the accelerated prediction of mechanical properties in GOri-based metallic metamaterials [5,6]. They then developed a model for the first time to study the bending [7], buckling, and postbuckling properties [8]. Their findings indicated that graphene origami can significantly influence the buckling and postbuckling behavior of metamaterial composite beams. Karami and Ghayesh [9] investigated the dynamics of auxetic metamaterials using various shear deformation theories. Ezzati et al. [10] and Ebrahimi and Ahari [11] investigated the free vibration behavior and vibration control of a graphene origami auxetic beam. Ebrahimi and Parsi [12] revealed the elastic wave characteristics of functionally graded graphene origami-reinforced metamaterial beams. Recently, several papers have been published on the thermal-mechanical coupling behavior of functionally graded graphene origami-enabled metamaterials [13]. On the other hand, certain experimental studies have proven the significance of the material microstructure and long-range interaction force on the mechanical response of micro-and nanostructured components [14–16]. Classical continuum theory cannot describe these small-scale effects well, but is widely used in macro-level structural modeling. Among the different modeling possibilities, one feasible method is to provide the so-called nonlocal continuum theory by directly modifying the classical theory. Over the years, researchers have proposed several non-local theories. The theories of stress gradient, strain gradient, and even integral type [17–20] provide some classic examples in which the small-scale effect is considered by introducing one or more internal length scale parameters. When structural dimensions are at the micrometer/nanometer scale, microscopic effects become particularly significant, especially for the study of size effects in graphene origami-enhanced microstructures.

In addition, advanced computational methods have significantly expanded the analytical capabilities of metamaterial composite systems. Neural networks, particularly deep-learning frameworks, have emerged as powerful tools in engineering disciplines such as materials science [21], solid mechanics [22], and fluid mechanics [23]. In the above research, the data-driven method was mainly used to minimize the error norm between network prediction and real data (training data) to train the model parameters [24]. However, their reliance on empirical correlations often limits their physical interpretability, and large-scale datasets are required for reliable predictions. To address these limitations, PINNs have gained traction by integrating governing differential equations into the learning process [25]. This framework introduces loss functions that enforce boundary conditions and conservation laws, thereby enabling the extraction of physics-compliant solutions without exhaustive data requirements. PINNs have been successfully applied to solve complex partial differential equations in fluid dynamics (e.g., Navier-Stokes equation [26]), phase transition phenomena (Cahn-Hilliard and Allen-Cahn equation [27]), and heat transfer processes [28]. In the field of solid mechanics, PINNs have demonstrated potential for modeling mechanical behaviors. For instance, Xue et al. [29] incorporated two loss functions, mass transfer and stress field, and used them to numerically obtain the solution of diffusion stress in an elastic sphere. Fallah and Aghdam [30] applied PINNs to study the bending and free vibration characteristics of three-dimensional functionally graded porous beams. Kianian et al. [31] extended the PINN approach to model the static bending response of micro/nanoelectromechanical systems. Moreover, Bazmara et al. [32] proposed an NN framework based on physical information to predict the nonlinear buckling behavior of 3D functionally graded (FG) porous beams on elastic foundations. These studies highlight PINN’s ability to handle higher-order differential equations and achieve close agreement with the numerical solutions. However, PINN is rarely used to solve the mechanical behavior of graphene origami composite beams, considering the size effect.

The key contributions of this work are as follows: On the one hand, a large body of scholarly work has employed PINN to investigate the bending behavior of functionally graded material (FGM) nanobeams and nanoplates. These studies typically rely on small-scale theories, such as the strain gradient theory and nonlocal elasticity theory, yet their research focus remains confined to traditional functionally graded materials rather than mechanical metamaterials. However, some scholars have applied PINN to analyze the bending or buckling responses [33] of origami metamaterial composite beams/plates; the critical factor of the size effect has been overlooked in their investigations. Against this backdrop, this study addresses the bending problem of beams subjected to the combined influence of the size effect and metamaterial characteristics. Since the differential equations incorporate physical information, only the loss associated with boundary conditions needs to be considered during model training, eliminating the need for mesh partitioning and effectively improving computational efficiency. By employing both numerical methods and mesh-free PINN methods for parallel solution, this study aims to contribute to the advancement of research in this field.

In this study, a physics-informed neural network was used to analyze the bending behavior of graphene origami-reinforced metamaterial beams. The governing equation, a fourth-order differential equation with variable coefficients, was combined with the boundary conditions formulated using Hamilton’s principle. A deep feedforward neural network is constructed to approximate the beam bending deflection. Its parameters are optimized by minimizing a composite loss function that forces both the governing equations and boundary constraints. To validate the proposed approach, PINN predictions were compared against available analytical solutions. The convergence and accuracy of the PINN in predicting deflections under different boundary conditions were studied. Then, the influence of the graphene origami content, distribution, and size parameter on the deflection of graphene origami-reinforced metamaterial beams is investigated.

This section employs a general PINN model to solve the bending Eq. (18). PINNs, a specialized class of artificial neural networks, are used to obtain an approximate solution of the governing differential equations. To solve Eq. (18), a neural network is constructed with the spatial variable x as its input to approximate the solution. Within the PINN framework, a feedforward neural network (FNN) [39] is commonly adopted as the neural network architecture [40,41] due to its strong function approximation capability and structural simplicity. An illustration of the FNN is shown in Fig. 1. Thus, the approximate solution can be written as (19)w^(x)=F(x)=N L(x;W,b)where w^(x) is the approximate solution, F(x) denotes the FNN. The FNN is structured as an L-layer deep neural network. It accepts an input vector x, produces an output vector w^, and is parameterized by a weight matrix W and a bias vector b. The network was trained using a backpropagation algorithm [42], which iteratively reduces a composite loss function. This loss function quantifies the discrepancy between the network’s predictions and the target solution [43]. In the context of PINNs, the loss function incorporates differential equations and boundary conditions (20)Loss=LPDE+LBCwhere LPDE and LBC represent the losses associated with partial differential equations and boundary conditions, respectively. It is noteworthy that the partial differential term within the loss function can be analytically derived using automatic differentiation [44].

The loss function endowed with physical constraints is a crucial element in PINNs. It facilitates the training of neural networks in accordance with the physical laws. Within the current PINN-based computational mechanics framework, two main types of loss functions are commonly employed: the allocation loss function and the energy-based loss function. In the allocation loss function, the residuals of the equilibrium equation and boundary conditions were compelled to be zero at each sample point. In this study, an allocation loss function was adopted to formulate the loss function. In this approach, the mean-square error is used to define the loss function: (21)Loss=1n∑i=1nRPDE2(xi)+1m∑i=1mRBC2(xi)=1n∑i=1n[De∂w^(xi)∂x4−q(xi)]2+(w^(0))2+(w^(L))2+(∂2w^(0)∂x2)2+(∂2w^(L)∂x2)2where RPDE and RBC are the residuals of the equilibrium equation and the boundary conditions, respectively. Here, n represents the total number of sample points and m is the number of sample points on the boundary. Subsequently, the training algorithm is implemented to search for the neural network mapping that satisfies the governing equations at sample points. A schematic of the PINN algorithm applied to the bending equation is illustrated in Fig. 1b.

The developed physics-informed neural network was used for numerical calculations. In addition, to verify the reliability of the PINN approximate solution, the numerical results obtained based on finite element and analytical expressions are also presented and used as reference numerical solutions, in which the focus of the analysis is on the bending behavior of GOri/Sn metamaterial microbeams. In this study, all tasks were trained using an NVIDIA Titan RTX GPU. As depicted in Fig. S1 (Supporting Information), we trained three different neural network architectures with varying numbers of hidden layers. Based on the training speed, convergence curve volatility, and forecasting performance of each neural network, we selected a three-layer network with 40 nodes as the architecture for this study. As shown by the error curve in Fig. S2 (Supporting Information), compared to other adaptive stochastic gradient descent methods such as stochastic gradient descent (SGD) and root mean square propagation (RMSPro), the Adam algorithm demonstrates superior performance in model optimization. It enables faster computation and convergence, making it an ideal optimization algorithm for neural network training. Additionally, the Adam optimization scheme [45] is prominent, extensively applied, and highly adaptable within the PINN methodology. In this study, an Adam optimizer with an initial learning rate of 10⁻⁴ is used to train the deep neural network (DNN). Fig. S3 (Supporting Information) shows the variation in loss function values during model training under different activation functions. When using rectified linear unit (ReLU) as the activation function, the loss decreases more slowly, ultimately leading to underfitting in the final training results. When training models with tanh and Sigmoid as activation functions, the loss function converges more rapidly. Furthermore, upon convergence, the loss value of the tanh model is lower than that of the Sigmoid model. Thus, the hyperbolic tangent (tanh) function was used as the activation function within the FNNs [46]. In the workflow described in this paper, 200 sample points within the domain and 100 boundary sample points were generated using the Latin hypercube sampling (LHS) method. The maximum training iterations were set to 50,000, and parameters were initialized using the Glorot normal distribution initializer. Geometric parameters of pristine graphene: lGr=83.76A∘ and tGr=3.4A∘ [5]. The material parameters at room temperature are presented in Table 1.

Inspired by deep learning technology, we propose a physically constrained deep learning method to solve the bending problem of graphene origami-reinforced composite microbeams. Bending control equations and boundary conditions are embedded into the loss function of the network as physical information constraints, which can reflect the complex mechanical behavior and ensure the physical rationality of the identification results.

Through an example of composite microbeam bending under a uniformly distributed load, the accuracy of PINNs and traditional analytical solution methods in the calculation results is compared. In addition, a numerical example of the influence of the graphene origami content and the folding degree of the composite beam on the bending behavior under the size effect verifies the possibility of using a small amount of training data to accurately predict the mechanical behavior in different calculation domains in PINNs. It is worth noting that PINNs are still a promising research field in calculation methods and are excellent candidates for solving inverse and other mechanical problems. For the inverse problem, the traditional forward finite element method is difficult to deal with and usually requires additional optimization algorithms, such as the genetic algorithm and the gradient descent method. In the future, we plan to conduct related research on inverse problems.