Automatically Differentiable Model Updating (ADiMU): conventional, hybrid, and neural network material model discovery including history-dependency

TL;DR

ADiMU provides the first fully differentiable framework to automatically discover history-dependent constitutive models from either local strain-stress data or global displacement-force data, accommodating conventional, neural network, and hybrid architectures with no hyperparameter tuning. By formulating a force-equilibrium loss and leveraging automatic differentiation, it updates model parameters across tens to millions of degrees of freedom in both local and global discovery settings, including convexity-constrained yield surfaces like the Lou-Zhang-Yoon model. The framework is demonstrated across a range of synthetic data, highlighting the importance of data diversity and the potential of hybrid architectures to improve data efficiency, while revealing the relative strengths of local versus global discovery for neural networks. HookeAI, the accompanying open-source tool, provides a platform to test new material-model architectures and to benchmark discovery performance, with implications for multi-scale modeling, design optimization, and uncertainty quantification in future work.

Abstract

We introduce the first Automatically Differentiable Model Updating (ADiMU) framework that finds any history-dependent material model from full-field displacement and global force data (global, indirect discovery) or from strain-stress data (local, direct discovery). We show that ADiMU can update conventional (physics-based), neural network (data-driven), and hybrid material models. Moreover, this framework requires no fine-tuning of hyperparameters or additional quantities beyond those inherent to the user-selected material model architecture and optimizer. The robustness and versatility of ADiMU is extensively exemplified by updating different models spanning tens to millions of parameters, in both local and global discovery settings. Relying on fully differentiable code, the algorithmic implementation leverages vectorizing maps that enable history-dependent automatic differentiation via efficient batched execution of shared computation graphs. This contribution also aims to facilitate the integration, evaluation and application of future material model architectures by openly supporting the research community. Therefore, ADiMU is released as an open-source computational tool, integrated into a carefully designed and documented software named HookeAI.

Figures (46)

• Figure 1: Automatically Differentiable Model Updating (ADiMU) framework, enabling the automatic, indirect discovery of any history-dependent parametric material model, $\mathcal{M}(\bm{\theta})$, parameterized by $\bm{\theta}$, from full-field displacement and global force data. The gradient-based solution of a force equilibrium optimization problem is unlocked with automatic differentiation, as ADiMU's workflow is fully automatically differentiable.
• Figure 2: Material model discovery workflow with Automatically Differentiable Model Updating (ADiMU). In the local discovery setting, a general material model, $\mathcal{M}$, parameterized by $\bm{\theta}$, is discovered directly from strain-stress data, $(\bm{\varepsilon}, \, \bm{\sigma})$. In contrast, in the global discovery setting, the material model is indirectly discovered from displacement-force data, $(\bm{u}, \, \bm{f})$. Note that ADiMU's local forward propagation is shared between both local and global discovery settings.
• Figure 3: Hybrid material model architecture. The material input data flows into each hybridization channel, each containing one or more hybridized models (one per hybridization layer). The outputs from the hybridization channels are then processed by the hybridization model, which yields the final material output data.
• Figure 4: Algorithmic design of the Automatically Differentiable Model Updating (ADiMU) framework. Vectorizing maps significantly reduce computational costs by leveraging batched execution with shared computation graphs.
• Figure 5: VM model local discovery history from VM strain-stress data ($E=110\,$GPa, $\nu=0.33$, $s_{0}=900\sqrt{3}\,$MPa, $s_{1}=700\sqrt{3}\,$MPa, $s_{2}=0.5$) with random model initialization: \ref{['subfig:local_vm_training_loss_history']} Training loss (MSE) history throughout the discovery process; \ref{['subfig:local_vm_model_parameter_history_E']} Young modulus $E$; \ref{['subfig:local_vm_model_parameter_history_v']} Poisson ratio $\nu$; \ref{['subfig:local_vm_model_parameter_history_s0']} Yield parameter $s_{0}$; \ref{['subfig:local_vm_model_parameter_history_a']} Yield parameter $s_{1}$; \ref{['subfig:local_vm_model_parameter_history_b']} Yield parameter $s_{2}$. Black dashed lines correspond to the optimization upper and lower bounds. Red dashed lines correspond to the parameters 'ground-truth'.

Theorems & Definitions (21)

• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Remark 9
• Remark 10
• Remark 11
• Remark 12