用材料力学知识增强深度学习，提升小样本下材料逆问题预测精度

Advancements in deep learning and machine learning have improved the ability to model complex, nonlinear relationships, such as those encountered in complex material inverse problems. However, the effectiveness of these methods often depends on large datasets, which are not always available. In this study, the incorporation of domain-specific knowledge of the mechanical behavior of material microstructures is investigated to evaluate the impact on the predictive performance of the models in data-scarce scenarios. To overcome data limitations, a two-step framework, Learning Latent Hardening (LLH), is proposed. In the first step of LLH, a Deep Neural Network is employed to reconstruct full stress-strain curves from randomly selected portions of the stress-strain curves to capture the latent mechanical response of a material based on key microstructural features. In the second step of LLH, the results of the reconstructed stress-strain curves are leveraged to predict key microstructural features of porous materials. The performance of six deep learning and/or machine learning models trained with and without domain knowledge are compared: Convolutional Neural Networks, Deep Neural Networks, Extreme Gradient Boosting, K-Nearest Neighbors, Long Short-Term Memory, and Random Forest. The results from the models with domain-specific information consistently achieved higher $R^2$ values compared to models without prior knowledge. Models without domain knowledge missed critical patterns linking stress-strain behavior to microstructural changes, whereas domain-informed models better identified essential stress-strain features predictive of microstructure. These findings highlight the importance of integrating domain-specific knowledge with deep learning to achieve accurate outcomes in materials science.