Traditional entropy-based methods - such as cross-entropy loss in classification problems - have long been essential tools for representing the information uncertainty and physical disorder in data and for developing artificial intelligence algorithms. However, the rapid growth of data across various domains has introduced new challenges, particularly the integration of heterogeneous datasets with intrinsic disparities. To address this, we introduce a zentropy-enhanced neural network (ZENN), extending zentropy theory into the data science domain via intrinsic entropy, enabling more effective learning from heterogeneous data sources. ZENN simultaneously learns both energy and intrinsic entropy components, capturing the underlying structure of multi-source data. To support this, we redesign the neural network architecture to better reflect the intrinsic properties and variability inherent in diverse datasets. We demonstrate the effectiveness of ZENN on classification tasks and energy landscape reconstructions, showing its superior generalization capabilities and robustness-particularly in predicting high-order derivatives. ZENN demonstrates superior generalization by introducing a learnable temperature variable that models latent multi-source heterogeneity, allowing it to surpass state-of-the-art models on CIFAR-10/100, BBCNews, and AGNews. As a practical application in materials science, we employ ZENN to reconstruct the Helmholtz energy landscape of Fe$_3$Pt using data generated from density functional theory (DFT) and capture key material behaviors, including negative thermal expansion and the critical point in the temperature-pressure space. Overall, this work presents a zentropy-grounded framework for data-driven machine learning, positioning ZENN as a versatile and robust approach for scientific problems involving complex, heterogeneous datasets.

翻译：传统的基于熵的方法——例如分类问题中的交叉熵损失——长期以来一直是表征数据信息不确定性与物理无序性以及开发人工智能算法的重要工具。然而，各领域数据的快速增长带来了新的挑战，尤其是如何整合具有内在差异性的异构数据集。为此，我们提出了一种增熵增强神经网络（ZENN），通过引入本征熵将增熵理论拓展至数据科学领域，从而实现对异构数据源更有效的学习。ZENN能够同时学习能量与本征熵分量，以捕捉多源数据的底层结构。为支持这一目标，我们重新设计了神经网络架构，以更好地反映多样化数据集中固有的本征特性与变异性。我们在分类任务和能量景观重构任务上验证了ZENN的有效性，展示了其卓越的泛化能力与鲁棒性——尤其是在预测高阶导数方面。ZENN通过引入可学习的温度变量来建模潜在的多源异质性，从而实现了优异的泛化性能，在CIFAR-10/100、BBCNews和AGNews数据集上超越了当前最先进的模型。作为在材料科学中的实际应用，我们利用ZENN重构了Fe$_3$Pt的亥姆霍兹能量景观，所用数据基于密度泛函理论（DFT）生成，并成功捕捉了包括负热膨胀和温压空间临界点在内的关键材料行为。总体而言，本研究提出了一个基于增熵理论的数据驱动机器学习框架，将ZENN定位为一种适用于涉及复杂异构数据集的科学问题的通用且鲁棒的方法。