Honghui Du

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.

Honghui Du, Binyao Guo, QiZhi He

The present study aims to extend the novel physics-informed machine learning approach, specifically the neural-integrated meshfree (NIM) method, to model finite-strain problems characterized by nonlinear elasticity and large deformations. To this end, the hyperelastic material models are integrated into the loss function of the NIM method by employing a consistent local variational formulation. Thanks to the inherent differentiable programming capabilities, NIM can circumvent the need for derivation of Newton-Raphson linearization of the variational form and the resulting tangent stiffness matrix, typically required in traditional numerical methods. Additionally, NIM utilizes a hybrid neural-numerical approximation encoded with partition-of-unity basis functions, coined NeuroPU, to effectively represent the displacement and streamline the training process. NeuroPU can also be used for approximating the unknown material fields, enabling NIM a unified framework for both forward and inverse modeling. For the imposition of displacement boundary conditions, this study introduces a new approach based on singular kernel functions into the NeuroPU approximation, leveraging its unique feature that allows for customized basis functions. Numerical experiments demonstrate the NIM method's capability in forward hyperelasticity modeling, achieving desirable accuracy, with errors among $10^{-3} \sim 10^{-5}$ in the relative $L_2$ norm, comparable to the well-established finite element solvers. Furthermore, NIM is applied to address the complex task of identifying heterogeneous mechanical properties of hyperelastic materials from strain data, validating its effectiveness in the inverse modeling of nonlinear materials. To leverage GPU acceleration, NIM is fully implemented on the JAX deep learning framework in this study, utilizing the accelerator-oriented array computation capabilities offered by JAX.

Honghui Du, QiZhi He

Differentiable programming has emerged as a powerful paradigm in scientific computing, enabling automatic differentiation through simulation pipelines and naturally supporting both forward and inverse modeling. We present JAX-MPM, a general-purpose differentiable meshfree solver based on the material point method (MPM) and implemented in the modern JAX architecture. The solver adopts a hybrid Eulerian-Lagrangian framework to capture large deformations, frictional contact, and inelastic material behavior, with emphasis on geomechanics and geophysical hazard applications. Leveraging GPU acceleration and automatic differentiation, JAX-MPM enables efficient gradient-based optimization directly through its time-stepping solvers and supports joint training of physical models with deep learning to infer unknown system conditions and uncover hidden constitutive parameters. We validate JAX-MPM through a series of 2D and 3D benchmark simulations, including dam-break and granular collapse problems, demonstrating both numerical accuracy and GPU-accelerated performance. Results show that a high-resolution 3D granular cylinder collapse with 2.7 million particles completes 1000 time steps in approximately 22 seconds (single precision) and 98 seconds (double precision) on a single GPU. Beyond high-fidelity forward modeling, we demonstrate the framework's inverse modeling capabilities through tasks such as velocity field reconstruction and the estimation of spatially varying friction from sparse data. In particular, JAX-MPM accommodates data assimilation from both Lagrangian (particle-based) and Eulerian (region-based) observations, and can be seamlessly coupled with neural network representations. These results establish JAX-MPM as a unified and scalable differentiable meshfree platform that advances fast physical simulation and data assimilation for complex solid and geophysical systems.