# Non-linear mechanical field reconstruction coupling recurrent neural networks with physics-informed graph neural networks

arXiv: https://arxiv.org/abs/2606.10909 · HTML: https://mlsanity.com/paper/2606.10909/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 72 references that are in the map.
- First posted on arXiv 9 Jun 2026.

## Builds on (most cited first, 23 of 23)

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- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [Thermodynamics-informed graph neural networks](https://mlsanity.com/paper/2203.01874/) (2022)
- [Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions](https://mlsanity.com/paper/2205.11912/) (2022)
- [Physically recurrent neural networks for path-dependent heterogeneous materials: embedding constitutive models in a data-driven surrogate](https://mlsanity.com/paper/2209.07320/) (2022)
- [A Microstructure-based Graph Neural Network for Accelerating Multiscale Simulations](https://mlsanity.com/paper/2402.13101/) (2024)
- [Probabilistic modeling of discrete structural response with application to composite plate penetration models](https://mlsanity.com/paper/2011.11780/) (2020)
- [An efficient monolithic solution scheme for FE2 problems](https://mlsanity.com/paper/2101.01802/) (2021)
- [Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity](https://mlsanity.com/paper/2507.05291/) (2025)

## Abstract (arXiv)

Reconstructing local stress fields in heterogeneous microstructures under non-linear, history-dependent loading remains a major computational bottleneck in multi-scale simulations. We propose a coupled LSTM-GNN framework that links the temporal and spatial aspects of local stress field reconstruction. A Long Short-Term Memory network encodes macroscopic stress-strain sequences into a compact hidden state that captures the path-dependent constitutive response, while a physics-informed Graph Neural Network reconstructs the spatially-resolved stress field at each time step. We introduce a relative weighting strategy with linear warm-up to balance the data-driven reconstruction loss and a discrete divergence-based equilibrium penalty. This resolves the scale mismatch that prevents fixed-weight formulations from converging in the elasto-plastic regime. The model is trained on 10,000 non-proportional loading paths applied to a periodic plate-with-a-hole microstructure and von Mises elasto-plasticity. The model achieves three orders of magnitude speedup over finite element simulations and generalizes to loading sequences twice the training length, with 1.9% cumulative error. Because the graph relies on mesh connectivity instead of the specific element type, one trained surrogate can be applied directly without retraining to meshes with different element types and to both coarser and finer resolutions, while in all cases reproducing the high-fidelity quad-element FE field used during training. Indeed, the message passing characteristics inherent to GNN and MeshGraphNet architecture render the model mesh-agnostic. Analysis of the LSTM hidden states suggests a low-dimensional structure related to the internal state variables of the constitutive model.