Machine learning-accelerated peridynamics model for mechanical and failure behaviors of materials

In the realm of modern scientific research, computational methodologies have become indispensable tools, driving advancements in various fields such as material science, structural engineering, and geophysics. These methodologies enable the detailed simulation and analysis of complex physical phenomena, allowing researchers to predict the behavior of materials and structures under a wide range of conditions [[1], [2], [3], [4], [5], [6]] and from multiple scales [[7], [8], [9], [10]]. However, as the complexity of these simulations increases [[11], [12], [13], [14]], so do the computational demands, often leading to significant time and resource constraints. This is particularly true in the field of computational mechanics, where the need to accurately model material behaviors such as fracture, fatigue, and failure is of paramount importance [[15], [16], [17]]. Traditional continuum mechanics has long served as the foundation for modeling the mechanical behavior of materials. However, when it comes to handling discontinuities, such as cracks and voids, these classical approaches face significant limitations [18,19]. The mathematical treatment of discontinuities often requires complex regularizations or artificial corrections, which can introduce errors and limit the models’ applicability, especially in heterogeneous materials. These challenges are particularly evident in simulations that demand high precision, such as those used in the design and safety assessment of critical engineering structures.

Peridynamics (PD), a non-local continuum theory, has emerged as a promising alternative to classical continuum mechanics, offering a more robust mesh-free framework for modeling phenomena involving discontinuities [[20], [21], [22], [23], [24], [25], [26], [27]]. Unlike traditional models, which rely on local interactions, PD accounts for interactions over finite distances between material points, allowing it to naturally accommodate cracks, voids, and other discontinuities [[28], [29], [30], [31]]. This capability makes PD particularly well-suited for applications in fields such as aerospace engineering [32,33], geophysics [34,35], soft materials [22,36], and the study of composite materials [[37], [38], [39], [40]]. Moreover, PD is increasingly utilized in multi-physics simulations encompassing fluid-structure interactions and coupled phenomena [[41], [42], [43]]. Despite its proficiency in these areas, PD faces significant challenges, particularly in terms of computational costs [20]. However, the practical application of PD is often hindered by its high computational cost. The need to evaluate numerous pairwise interactions between material points results in substantial computational overhead, particularly in large-scale simulations involving complex geometries and loading conditions [20,44,45].

In recent years, machine learning (ML) has revolutionized various aspects of computational science by providing powerful tools for data-driven modeling and simulation [[46], [47], [48], [49]]. ML can adapt computational meshes dynamically to allocate resources effectively, thereby improving accuracy and reducing computational overhead in simulations involving dynamic boundaries or evolving phenomena [50,51]. Furthermore, ML facilitates the creation of reduced-order models that approximate complex computational mechanics with fewer degrees of freedom [52,53]. It optimizes the distribution of computational workloads across multiple processors in high-performance computing environments, thereby accelerating the analysis of large-scale problems [54,55]. Surrogate models-a machine learning model trained to approximate system responses, such as displacements and stresses-replace the need for computationally expensive simulations by mimicking the behavior of numerical simulations with lower computational demands [[56], [57], [58]]. Recent advances in integrating machine learning (ML) with computational mechanics have demonstrated the potential of ML in addressing complex physical and engineering problems. For example, Dong et al. [59] explored the application of coupling strategies to enhance numerical methods for fracture mechanics, providing a robust framework for modeling crack propagation. Similarly, the study by Liu et al. [60] showcased how machine learning could be tailored for large-scale molecular dynamics simulations, improving computational efficiency without sacrificing accuracy. Recent work by Zhu et al. [61] proposed innovative transfer learning strategies to identify material parameters of soft materials, significantly reducing computational costs while maintaining accuracy. In the context of PD, integrating ML aims to overcome its inherent computational challenges—typically involving intensive and time-consuming iterative processes. While previous studies have demonstrated the potential of ML-based surrogate models for accelerating simulations [62], many of these approaches rely on relatively simple surrogate models, such as linear regression, which are often not robust enough to handle the complexity of multidimensional, nonlocal systems like PD. For instance, prior work has utilized modal analysis data to train surrogate models for displacement prediction in PD simulations, focusing primarily on one-dimensional (1D) and two-dimensional (2D) cases. While these models provide some success in these simplified cases, they often struggle to generalize effectively to higher-dimensional scenarios or to offer valuable insights into the underlying physical relationships that govern the material behavior.

To address these limitations, this study proposes a novel ML-accelerated PD framework that incorporates several key innovations to enhance both accuracy and computational efficiency. By constructing a surrogate model that predicts the displacements of material points based on the displacements of their neighbors, we effectively bypass the need for repetitive, time-consuming computations inherent in traditional PD approaches. The primary distinction of our work lies in the use of artificial neural networks (ANNs) as surrogate models, enabling more accurate predictions of particle displacements in 1D, 2D, and 3D structures. This multi-dimensional capability is one of the key advantages of our approach, as it ensures our framework can handle more complex real-world problems that previous models, often limited to 1D or 2D, could not address. In addition to improving predictive power, we introduce Shapley Additive Explanations (SHAP) values to bridge the gap between ML predictions and the underlying physical principles of PD. While previous studies have emphasized accuracy as the main criterion, we integrate interpretability into our framework, providing insights into how different input features influence the model’s predictions. This results in a more transparent, physics-informed approach, which is essential for engineering applications where understanding the model’s decision-making process is as crucial as achieving accurate predictions. Moreover, unlike prior methods that primarily focus on enhancing prediction accuracy, our framework simultaneously improves computational efficiency. By replacing the traditional iterative computation steps in PD with a fast, pre-trained surrogate model, we achieve significant reductions in runtime, enabling more efficient simulations for larger and more complex problems. This innovation positions our approach as not only accurate but also highly scalable, allowing for simulations that would otherwise be computationally prohibitive.

A detailed overview of the ML-accelerated PD framework is provided in Section 2, covering the fundamentals of PD and the integration of ML techniques to enhance computational efficiency. This section also introduces the development of the ML surrogate model, including its training on modal analysis datasets and its implementation for 1D, 2D, and 3D structures. In Section 3, the performance of the proposed framework is validated through benchmark tests under various loading conditions, showcasing its capability to predict displacement fields and damage evolution accurately. In this section, the computational efficiency of the surrogate model is further explored, highlighting its substantial speedups compared to traditional PD methods. Finally, concluding remarks are presented in Section 4, summarizing the key findings and discussing future directions for enhancing the framework’s applicability to complex material behaviors and interdisciplinary challenges.