Teacher-Student Guided Inverse Modeling for Steel Final Hardness Estimation

TL;DR

The paper tackles inverse steel heat-treatment planning by addressing the many-to-one mapping that links input conditions to final hardness. It introduces a Teacher-Student framework where a fixed forward Teacher predicts hardness from $13$ inputs, while an inverse Student learns to generate plausible input configurations from a target hardness through iterative supervision. On a public tempered steel dataset, the approach achieves high $R^2$ (≈0.98) and rapid convergence, outperforming baseline regressors and a TD3 reinforcement learning agent in both accuracy and training time. This method offers a data-efficient, interpretable route for inverse process modeling with potential for real-time control and extensions to uncertainty and diversity in the predicted inputs.

Abstract

Predicting the final hardness of steel after heat treatment is a challenging regression task due to the many-to-one nature of the process -- different combinations of input parameters (such as temperature, duration, and chemical composition) can result in the same hardness value. This ambiguity makes the inverse problem, estimating input parameters from a desired hardness, particularly difficult. In this work, we propose a novel solution using a Teacher-Student learning framework. First, a forward model (Teacher) is trained to predict final hardness from 13 metallurgical input features. Then, a backward model (Student) is trained to infer plausible input configurations from a target hardness value. The Student is optimized by leveraging feedback from the Teacher in an iterative, supervised loop. We evaluate our method on a publicly available tempered steel dataset and compare it against baseline regression and reinforcement learning models. Results show that our Teacher-Student framework not only achieves higher inverse prediction accuracy but also requires significantly less computational time, demonstrating its effectiveness and efficiency for inverse process modeling in materials science.

Figures (6)

• Figure 1: Overview of the Teacher-Student framework for inverse hardness prediction. (a) The Teacher model is trained to map process inputs ($X$) to final hardness ($Y$). The Student model learns to predict valid input configurations ($X_{\text{pred}}$) from target hardness values. Predictions are evaluated by the fixed Teacher model, and the loss is used to iteratively update the Student. (b) A schematic view highlighting the knowledge distillation process between Teacher and Student models.
• Figure 2: Exploratory data analysis. (a) Relationship between tempering temperature, time, and final hardness (HRC). Higher temperatures and longer durations result in lower hardness, illustrating a clear inverse trend. (b) Multiple input configurations leading to the same hardness value, highlighting the many-to-one nature of the inverse problem.
• Figure 3: Baseline model performance. (a) The Random Forest inverse model shows high test error and low generalization. (b) The MLP baseline failed to converge, with persistent high loss across training and validation. These results highlight the difficulty of solving many-to-one inverse prediction problems with standard models.
• Figure 4: Training and validation loss curves. (a) The forward (Teacher) MLP model converges quickly within the first 50 epochs, with both losses approaching zero, demonstrating strong generalization and suitability as a reference for inverse training. (b) The inverse (Student) model effectively learns the mapping over 500 epochs, with both training and validation losses sharply decreasing and stabilizing.
• Figure 5: Evaluation metrics for the Student model on training and test data. The model achieved high $R^2$ values (0.98) and low mean squared and absolute errors, demonstrating accurate and generalizable inverse predictions when supervised by the Teacher model.