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

The gist The study presents a data-efficient workflow that recovers a high-load reference hardness directly from shallow, size-affected indentation data using physicsguided feature engineering, physically motivated data augmentation, and a physics-constrained neural network.

Introduction to the Problem

Instrumented indentation testing is the established standard for evaluating the mechanical properties of materials at the micro- and nanoscale By continuously recording the applied load (P) and resulting displacement (h), IIT enables the simultaneous extraction of hardness (H) and reduced elastic modulus (Er) from a single load-displacement curve, relying on the Oliver-Pharr method to estimate the projected contact area (Ac) from the elastic unloading stiffness However, this spatial confinement introduces fundamental physical tension The very condition that makes IIT indispensable for characterizing constrained micro-volumes, testing at low loads and shallow contact depths, is the condition that activates a depth-dependent hardening phenomenon known as the Indentation Size Effect (ISE) Consequently, the apparent hardness extracted from shallow micro-volume measurements is systematically elevated above the true macroscopic bulk hardness (H0) and cannot be directly compared to other reference values

Physics-Guided Feature Engineering and Data Augmentation

The study utilizes a data-efficient workflow that begins by augmenting conventional training data Over 700 indentations on three certified steel reference blocks were augmented with instrumental noise, session-level drift, and phase-boundary mixing Physics-guided feature engineering was implemented to create the machine learning input vector, which was populated with three categories of descriptors extracted from each P–h curve These primary load-geometric parameters included Pmax, hmax, hc, and S Energetic parameters obtained by numerical integration included Wtot, Wel, and Wp = Wtot − Wel The descriptors also included the dimensionless plastic-work ratio Wp/Wtot (energy partitioning between plastic flow and elastic recovery [41]), the dimensionless hardness-to-modulus ratio H/Er (scales with pile-up propensity [42]), and the area-function-free compliance quantity Pmax/S2 For the physics-constrained NN model, a second feature frame was constructed from the same raw indentation quantities This frame contained dimensionless contact-mechanics descriptors while keeping the apparent hardness Happ separate for the final physical reconstruction The bounded depth coordinate was defined as t = hs/hc + hs, where hs is a characteristic depth scale estimated as the median positive contact depth of the training split

Physics-Constrained Neural Network (PCNN) Architecture

A physics-constrained neural network (PCNN) was implemented to constrain the hardness prediction through an Nix-Gao-like algebraic form [7] Instead of predicting Href directly, the model predicts a signed correction factor q and reconstructs the corrected hardness as Hbref = Happ/√1 + q This corresponds to the hardness-ratio relation Happ/Href 2 = 1 + q The signed corrections were bounded according to qmin ≤ q ≤ qmax, with the bounds set to qmin = −0.5 and qmax = 3.0 The model was trained by minimizing a composite loss L = LH + λqLq + λloadLload The hardness loss LH was a Smooth L1 loss between Hbref and the high-load reference target Href The correction-factor loss used the physically implied target qtarget = Happ/Href 2 − 1 and penalized deviations of the predicted q from this value The load-independence loss penalized the within-material variance of Hbref, encoding the expectation that the corrected reference hardness should be approximately independent of indentation load within a given specimen

Conventional Model Benchmarking and Performance

Five conventional regression architectures were implemented and benchmarked against increasing representational capacity These included Ridge Regression, Random Forest, XGBoost, a standard feed-forward neural network, and a bottlenecked neural network The conventional models directly regress Href from the input feature vector The bottlenecked architecture (64×8×64) achieved the strongest internal accuracy, with test RMSE = 0.184 GPa and R2 = 0.990 Both PCNN variants produced slightly higher internal errors than the unconstrained neural networks The PCNN-DI, restricted to dimensionless contact-mechanics descriptors, yielded the largest internal error among the non-linear models

Generalization and Boundary Testing Results

The external validation on the quarantined specimen demonstrated that PCNN-DI achieved the best blind performance (RMSE = 0.284 GPa, MAPE = 3.6%) The PCNN-DI produced a stable, depth-independent prediction band across the full contact depth range, whereas Random Forest failed to track the specimen outside its training load range The external ranking of the models inverted compared to internal testing, confirming that the bounded physical reconstruction sacrifices a fraction of in-distribution fit and returns it as out-of-distribution stability

Conclusion and Implications

The framework’s practical relevance lies in material systems where an equivalent analytical baseline does not exist The PCNN-DI attained RMSE = 0.343 GPa (MAPE = 4.42%) on the quarantined specimen and did not destabilize beyond the training loads, whereas tree-based models failed structurally there The framework’s practical relevance lies in material systems where an equivalent analytical baseline does not exist The PCNN-DI attained RMSE = 0.343 GPa (MAPE = 4.

1. Bold header: Improved model generalization via physics-constrained reconstruction. The PCNN-DI achieves the best blind performance (RMSE = 0.28 GPa, MAPE = 3.6%) by reconstructing hardness through the bounded Nix–Gao-like form Hbref = Happ/√1 + q, which is shown to be structurally insensitive to this boundary [absolute load coordinate can push the model outside its trained input domain].

2. Bold header: Enhanced uncertainty quantification via measurement noise propagation. The PCNN-DI's prediction scatter propagates transparently into the corrected values, meaning the spread of corrected values across a grid of shallow indentations directly reflects the reliability of the underlying measurements, providing a rudimentary per-grid confidence estimate at no additional cost.

3. Bold header: Mechanistic interpretability through SHAP feature attribution. The SHAP analysis for PCNN-DI shows that the correction is controlled primarily by compliance, dimensionless load–depth state, and energy partitioning rather than by unnormalized instrument-scale quantities, linking the model's learned correction to established contact mechanics principles.