# Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics

arXiv: https://arxiv.org/abs/2207.09908 · HTML: https://mlsanity.com/paper/2207.09908/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2022.

## Built on by (most cited first, 13 of 13)

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- [Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance](https://mlsanity.com/paper/2303.03918/) (2023)
- [Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems](https://mlsanity.com/paper/2409.04067/) (2024)
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- [Physics-informed DeepONet with stiffness-based loss functions for structural response prediction](https://mlsanity.com/paper/2409.00994/) (2024)
- [Physics-Audited Agentic Discovery in Scientific Machine Learning](https://mlsanity.com/paper/2607.07379/) (2026)
- [A thermodynamically consistent physics-informed deep learning material model for short fiber/polymer nanocomposites](https://mlsanity.com/paper/2403.18310/) (2024)
- [A physics-informed U-Net-LSTM network for nonlinear structural response under seismic excitation](https://mlsanity.com/paper/2511.21276/) (2025)

## Abstract (arXiv)

We present a new Integrated Finite Element Neural Network framework (I-FENN), with the objective to accelerate the numerical solution of nonlinear computational mechanics problems. We leverage the swift predictive capability of neural networks (NNs) and we embed them inside the finite element stiffness function, to compute element-level state variables and their derivatives within a nonlinear, iterative numerical solution. This process is conducted jointly with conventional finite element methods that involve shape functions: the NN receives input data that resembles the material point deformation and its output is used to construct element-level field variables such as the element Jacobian matrix and residual vector. Here we introduce I-FENN to the continuum damage analysis of quasi-brittle materials, and we establish a new non-local gradient-based damage framework which operates at the cost of a local damage approach. First, we develop a physics informed neural network (PINN) to resemble the non-local gradient model and then we train the neural network offline. The network learns to predict the non-local equivalent strain at each material point, as well as its derivative with respect to the local strain. Then, the PINN is integrated in the element stiffness definition and conducts the local to non-local strain transformation, whereas the two PINN outputs are used to construct the element Jacobian matrix and residual vector. This process is carried out within the nonlinear solver, until numerical convergence is achieved. The resulting method bears the computational cost of the conventional local damage approach, but ensures mesh-independent results and a diffused non-local strain and damage profile. As a result, the proposed method tackles the vital drawbacks of both the local and non-local gradient method, respectively being the mesh-dependence and additional computational cost.