Recent advances in scientific machine learning have shed light on the modeling of pattern-forming systems. However, simulations of real patterns still incur significant computational costs, which could be alleviated by leveraging large image datasets. Physics-informed machine learning and operator learning are two new emerging and promising concepts for this application. Here, we propose "Phase-Field DeepONet", a physics-informed operator neural network framework that predicts the dynamic responses of systems governed by gradient flows of free-energy functionals. Examples used to validate the feasibility and accuracy of the method include the Allen-Cahn and Cahn-Hilliard equations, as special cases of reactive phase-field models for nonequilibrium thermodynamics of chemical mixtures. This is achieved by incorporating the minimizing movement scheme into the framework, which optimizes and controls how the total free energy of a system evolves, instead of solving the governing equations directly. The trained operator neural networks can work as explicit time-steppers that take the current state as the input and output the next state. This could potentially facilitate fast real-time predictions of pattern-forming dynamical systems, such as phase-separating Li-ion batteries, emulsions, colloidal displays, or biological patterns.

翻译：科学机器学习的最新进展为图案形成系统的建模提供了新思路。然而，真实图案的模拟仍需要高昂的计算成本，而利用大规模图像数据集可有效缓解这一问题。物理信息机器学习与算子学习是两种新兴且具有前景的应用概念。本文提出"Phase-Field DeepONet"——一种基于物理信息的算子神经网络框架，能够预测自由能泛函梯度流支配系统的动态响应。我们通过Allen-Cahn方程和Cahn-Hilliard方程（作为化学反应混合物非平衡热力学反应型相场模型的典型特例）验证了该方法的可行性与准确性。该框架的核心在于引入最小移动方案，通过直接优化并控制系统总自由能的演化路径（而非直接求解控制方程）来实现预测。训练完成的算子神经网络可作为显式时间步进器，以当前状态作为输入并输出下一状态。这一方法有望实现图案形成动力系统的快速实时预测，例如相分离锂离子电池、乳液、胶体显示器或生物图案等场景。