# Data-Efficient Indentation Size Effect Correction in Steels Using Machine Learning and Physics-Constrained Neural Network

arXiv: https://arxiv.org/abs/2604.27775 · HTML: https://mlsanity.com/paper/2604.27775/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 60 references that are in the map.
- First posted on arXiv 30 Apr 2026.

## Builds on (most cited first, 5 of 5)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Machine Learning for Material Characterization with an Application for Predicting Mechanical Properties](https://mlsanity.com/paper/2010.06010/) (2020)

## Abstract (arXiv)

Shallow nanoindentation enables mechanical characterization of thin films, individual phases, and other volume-constrained materials, but the measured hardness is inflated by the indentation size effect (ISE). Classical corrections such as Nix-Gao require a deep linear regime and fail when only shallow measurements are accessible. We present a data-efficient workflow that recovers a high-load reference hardness directly from shallow, size-affected indentation data. Over 700 indentations on three certified steel reference blocks (2-6.5 GPa) were expanded by physics-guided augmentation and used to train conventional regressors (Ridge, Random Forest, XGBoost, neural networks) and a physics-constrained neural network (PCNN) that reconstructs hardness through the bounded form H_ref = H_app/sqrt(1+q), with the signed correction q learned from dimensionless contact-mechanics descriptors (E_r,ref*P_max/S^2, W_p/W_tot, H/E_r). On a quarantined fourth steel tested at loads offset from the training schedule, only the constrained formulation generalized: the dimensionless-input PCNN achieved RMSE = 0.28 GPa and MAPE = 3.6%, and remained stable beyond the training load range, where tree-based models failed structurally. Ablation attributed this robustness to the bounded reconstruction and a load-independence regularizer. Deliberate application to fused silica produced systematic failure, delimiting the method to dislocation-mediated crystalline plasticity. A few hundred laboratory indentations thus suffice to train a physically constrained ISE correction operating on single shallow indentations, offering a template for materials lacking analytical size-effect models.