Neural-Integrated Meshfree (NIM) Method: A differentiable
  programming-based hybrid solver for computational mechanics
Honghui Du QiZhi He
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.