# Enhancing material behavior discovery using embedding-oriented Physically-Guided Neural Networks with Internal Variables

arXiv: https://arxiv.org/abs/2508.00959 · HTML: https://mlsanity.com/paper/2508.00959/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 58 references that are in the map.
- First posted on arXiv 1 Aug 2025.

## Builds on (most cited first, 14 of 14)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Thermodynamics-informed graph neural networks](https://mlsanity.com/paper/2203.01874/) (2022)
- [Automated discovery of generalized standard material models with EUCLID](https://mlsanity.com/paper/2211.04453/) (2022)
- [NN-EUCLID: deep-learning hyperelasticity without stress data](https://mlsanity.com/paper/2205.06664/) (2022)
- [Discovering plasticity models without stress data](https://mlsanity.com/paper/2202.04916/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Recent Advances and Applications of Machine Learning in Experimental Solid Mechanics: A Review](https://mlsanity.com/paper/2303.07647/) (2023)
- [Physics-informed machine learning for Structural Health Monitoring](https://mlsanity.com/paper/2206.15303/) (2022)
- [On interpretability and proper latent decomposition of autoencoders](https://mlsanity.com/paper/2211.08345/) (2022)
- [Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variables](https://mlsanity.com/paper/2308.03915/) (2023)

## Abstract (arXiv)

Physically Guided Neural Networks with Internal Variables are SciML tools that use only observable data for training and and have the capacity to unravel internal state relations. They incorporate physical knowledge both by prescribing the model architecture and using loss regularization, thus endowing certain specific neurons with a physical meaning as internal state variables. Despite their potential, these models face challenges in scalability when applied to high-dimensional data such as fine-grid spatial fields or time-evolving systems. In this work, we propose some enhancements to the PGNNIV framework that address these scalability limitations through reduced-order modeling techniques. Specifically, we introduce alternatives to the original decoder structure using spectral decomposition, POD, and pretrained autoencoder-based mappings. These surrogate decoders offer varying trade-offs between computational efficiency, accuracy, noise tolerance, and generalization, while improving drastically the scalability. Additionally, we integrate model reuse via transfer learning and fine-tuning strategies to exploit previously acquired knowledge, supporting efficient adaptation to novel materials or configurations, and significantly reducing training time while maintaining or improving model performance. To illustrate these various techniques, we use a representative case governed by the nonlinear diffusion equation, using only observable data. Results demonstrate that the enhanced PGNNIV framework successfully identifies the underlying constitutive state equations while maintaining high predictive accuracy. It also improves robustness to noise, mitigates overfitting, and reduces computational demands. The proposed techniques can be tailored to various scenarios depending on data availability, resources, and specific modeling objectives, overcoming scalability challenges in all the scenarios.