Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity
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Abstract (arXiv)
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 ~ 10-5 in the relative L2 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.
Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 33 of its 78 references that are in the map. First posted on arXiv 15 Jul 2024.
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References · 33
- 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
- 2015Automatic differentiation in machine learning: a surveycited by 846
- 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
- 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
- 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
- 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 135
- 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
- 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
- 2021Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problemscited by 100
- 2019A Differentiable Programming System to Bridge Machine Learning and Scientific Computingcited by 81
- 2020Physics informed deep learning for computational elastodynamics without labeled datacited by 79
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- 2019Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transportcited by 60
- 2020A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applicationscited by 53
- 2020Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticitycited by 49
- 2022JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data sciencecited by 40
- 2022JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flowscited by 36
- 2019VarNet: Variational Neural Networks for the Solution of Partial Differential Equationscited by 35
- 2024A review on data-driven constitutive laws for solidscited by 35
- 2022A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element methodcited by 33
- 2018A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materialscited by 31
- 2021The mixed deep energy method for resolving concentration features in finite strain hyperelasticitycited by 28
- 2020Meshless physics-informed deep learning method for three-dimensional solid mechanicscited by 27
- 2022A deep learning energy method for hyperelasticity and viscoelasticitycited by 20
- 2019A Physics-Constrained Data-Driven Approach Based on Locally Convex Reconstruction for Noisy Databasecited by 18
- 2021Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN)cited by 18
- 2021Partition of unity networks: deep hp-approximationcited by 15
- 2022A deep learning energy-based method for classical elastoplasticitycited by 12
- 2021NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEscited by 10
- 2022A Neural Network-enhanced Reproducing Kernel Particle Method for Modeling Strain Localizationcited by 8
- 2023A Neural Network-Based Enrichment of Reproducing Kernel Approximation for Modeling Brittle Fracturecited by 4
- 2023Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanicscited by 4
- 2024N-Adaptive Ritz Method: A Neural Network Enriched Partition of Unity for Boundary Value Problemscited by 1
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