# Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs

arXiv: https://arxiv.org/abs/2412.05719 · HTML: https://mlsanity.com/paper/2412.05719/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 39 references that are in the map.
- First posted on arXiv 7 Dec 2024.

## Builds on (most cited first, 8 of 8)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios](https://mlsanity.com/paper/2205.07731/) (2022)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Eighty Years of the Finite Element Method: Birth, Evolution, and Future](https://mlsanity.com/paper/2107.04960/) (2021)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)
- [NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs](https://mlsanity.com/paper/2110.01601/) (2021)
- [Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics](https://mlsanity.com/paper/2311.12915/) (2023)

## Built on by (most cited first, 3 of 3)

- [A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2504.02459/) (2025)
- [SplineNet: An Isogeometric Deep Learning Method for Complex Shells](https://mlsanity.com/paper/2607.06026/) (2026)
- [Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees](https://mlsanity.com/paper/2608.02036/) (2026)

## Abstract (arXiv)

We present the Finite Element Neural Network Interpolation (FENNI) framework, a sparse neural network architecture extending previous work on Embedded Finite Element Neural Networks (EFENN) introduced with the Hierarchical Deep-learning Neural Networks (HiDeNN). Due to their mesh-based structure, EFENN requires significantly fewer trainable parameters than fully connected neural networks, with individual weights and biases having a clear interpretation. Our FENNI framework, within the EFENN framework, brings improvements to the HiDeNN approach. First, we propose a reference element-based architecture where shape functions are defined on a reference element, enabling variability in interpolation functions and straightforward use of Gaussian quadrature rules for evaluating the loss function. Second, we propose a pragmatic multigrid training strategy based on the framework's interpretability. Third, HiDeNN's combined rh-adaptivity is extended from 1D to 2D, with a new Jacobian-based criterion for adding nodes combining h- and r-adaptivity. From a deep learning perspective, adaptive mesh behavior through rh-adaptivity and the multigrid approach correspond to transfer learning, enabling FENNI to optimize the network's architecture dynamically during training. The framework's capabilities are demonstrated on 1D and 2D test cases, where its accuracy and computational cost are compared against an analytical solution and a classical FEM solver. On these cases, the multigrid training strategy drastically improves the training stage's efficiency and robustness. Finally, we introduce a variational loss within the EFENN framework, showing that it performs as well as energy-based losses and outperforms residual-based losses. This framework is extended to surrogate modeling over the parametric space in Part II.