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.

翻译：我们提出神经集成无网格（NIM）方法，这是一种基于可微编程的计算力学领域混合无网格方法。NIM将基于传统物理的无网格离散化技术与深度学习架构无缝融合，采用混合近似方案NeuroPU，通过结合连续深度神经网络（DNN）表示与底层空间离散化的单位分解（PU）基函数来有效表征解。这种神经-数值混合方法不仅通过函数空间分解增强了解的表达能力，还减少了DNN模型的规模以及基于自动微分的空间梯度计算需求，从而显著提高了训练效率。在NIM框架下，我们提出两种真正无网格求解器：基于强形式的NIM（S-NIM）和基于局部变分形式的NIM（V-NIM）。S-NIM求解器直接在损失函数中考虑强形式控制方程，而V-NIM求解器采用局部Petrov-Galerkin方法，允许基于任意重叠子域构建变分残差，从而既保证底层物理规律的满足，又保留无网格特性。我们针对稳态和瞬态基准问题进行大量数值实验，从精度、可扩展性、泛化能力和收敛性等方面评估所提NIM方法的有效性。此外，与其他物理信息机器学习方法的对比分析表明，NIM（尤其是V-NIM）在端到端预测能力上显著提升了精度和效率。