The free energy plays a fundamental role in descriptions of many systems in continuum physics. Notably, in multiphysics applications, it encodes thermodynamic coupling between different fields. It thereby gives rise to driving forces on the dynamics of interaction between the constituent phenomena. In mechano-chemically interacting materials systems, even consideration of only compositions, order parameters and strains can render the free energy to be reasonably high-dimensional. In proposing the free energy as a paradigm for scale bridging, we have previously exploited neural networks for their representation of such high-dimensional functions. Specifically, we have developed an integrable deep neural network (IDNN) that can be trained to free energy derivative data obtained from atomic scale models and statistical mechanics, then analytically integrated to recover a free energy density function. The motivation comes from the statistical mechanics formalism, in which certain free energy derivatives are accessible for control of the system, rather than the free energy itself. Our current work combines the IDNN with an active learning workflow to improve sampling of the free energy derivative data in a high-dimensional input space. Treated as input-output maps, machine learning accommodates role reversals between independent and dependent quantities as the mathematical descriptions change with scale bridging. As a prototypical system we focus on Ni-Al. Phase field simulations using the resulting IDNN representation for the free energy density of Ni-Al demonstrate that the appropriate physics of the material have been learned. To the best of our knowledge, this represents the most complete treatment of scale bridging, using the free energy for a practical materials system, that starts with electronic structure calculations and proceeds through statistical mechanics to continuum physics.

中文翻译：

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尺度桥接材料物理学：用于合金中自由能函数表示的主动学习工作流程和可集成的深度神经网络

自由能在连续介质物理学中的许多系统的描述中起着重要作用。值得注意的是，在多物理场应用中，它编码了不同场之间的热力学耦合。因此，它产生了构成现象之间相互作用的动力的驱动力。在机械化学相互作用的材料系统中，即使只考虑成分、有序参数和应变，也可以使自由能达到合理的高维。在提出自由能作为尺度桥接的范式时，我们之前已经利用神经网络来表示这种高维函数。具体来说，我们开发了一个可集成的深度神经网络 (IDNN)，可以训练从原子尺度模型和统计力学中获得的自由能导数数据，然后进行分析积分以恢复自由能密度函数。动机来自统计力学形式主义，其中某些自由能衍生物可用于控制系统，而不是自由能本身。我们目前的工作将 IDNN 与主动学习工作流相结合，以改进高维输入空间中自由能导数数据的采样。作为输入-输出图，机器学习适应独立量和相关量之间的角色反转，因为数学描述随着尺度桥接而变化。作为原型系统，我们专注于 Ni-Al。使用由此产生的 IDNN 表示的 Ni-Al 自由能密度的相场模拟表明，已经了解了材料的适当物理特性。据我们所知，