Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics

Abstract

While deep learning and data-driven modeling approaches based on deep neural networks (DNNs) have recently attracted increasing attention for solving partial differential equations, their practical applications to real-world scientific and engineering problems remain limited due to the relatively low accuracy and high computational costs. In this study, we present a differentiable programming-based hybrid meshfree approach within the field of computational mechanics. This approach seamlessly integrates neural network-based deep learning architectures with physics-based meshfree discretization techniques, thus referred to as the neural-integrated meshfree (NIM) method. To effectively approximate the solution, NIM employs a hybrid scheme called NeuroPU approximation, which combines continuous DNN representations with partition of unity (PU) basis functions associated with the underlying spatial discretization. This neural-numerical hybridization not only enhances the solution representation through functional space decomposition but also reduces both the size of DNN model and the need of automatic differentiation for spatial gradient computations, leading to a significant improvement in training efficiency. Under the NIM framework, we propose two truly meshfree solvers: the strong form-based NIM (S-NIM) and the local variational form-based NIM (V-NIM). In the S-NIM solver, the strong-form governing equation is directly considered in the loss function, while the V-NIM solver employs a local Petrov–Galerkin approach that allows the construction of variational residuals based on arbitrary overlapping subdomains. This ensures both the satisfaction of underlying physics and the preservation of the meshfree property. We perform extensive numerical experiments on both stationary and transient benchmark problems to assess the effectiveness of the proposed NIM methods in terms of accuracy, scalability, generalizability, and convergence properties. Moreover, comparative analysis with other physics-informed machine learning methods demonstrates that NIM, especially V-NIM, significantly enhances both accuracy and efficiency in end-to-end predictive capabilities.