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

TL;DR

The paper introduces NIM, a hybrid meshfree differentiable programming approach combining physics-based discretization with deep learning, improving efficiency and accuracy in computational mechanics problems.

Contribution

It proposes a novel neural-integrated meshfree framework with two solvers, S-NIM and V-NIM, that enhance solution accuracy and training efficiency in computational mechanics.

Findings

NIM outperforms existing methods in accuracy and efficiency.

V-NIM demonstrates superior scalability and generalizability.

The hybrid approach reduces model size and computational cost.

Abstract

We present the neural-integrated meshfree (NIM) method, a differentiable programming-based hybrid meshfree approach within the field of computational mechanics. NIM seamlessly integrates traditional physics-based meshfree discretization techniques with deep learning architectures. It employs a hybrid approximation scheme, NeuroPU, to effectively represent the solution by combining 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 for spatial gradient computations based on automatic differentiation, 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 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.