Reaction-diffusion systems are ubiquitous in nature and in engineering applications, and are often modeled using a non-linear system of governing equations. While robust numerical methods exist to solve them, deep learning-based reduced ordermodels (ROMs) are gaining traction as they use linearized dynamical models to advance the solution in time. One such family of algorithms is based on Koopman theory, and this paper applies this numerical simulation strategy to reaction-diffusion systems. Adversarial and gradient losses are introduced, and are found to robustify the predictions. The proposed model is extended to handle missing training data as well as recasting the problem from a control perspective. The efficacy of these developments are demonstrated for two different reaction-diffusion problems: (1) the Kuramoto-Sivashinsky equation of chaos and (2) the Turing instability using the Gray-Scott model.

中文翻译：

---

反应扩散系统的深度对抗 Koopman 模型

反应扩散系统在自然界和工程应用中无处不在，并且通常使用非线性控制方程系统进行建模。虽然存在强大的数值方法来解决这些问题，但基于深度学习的降阶模型 (ROM) 正在获得吸引力，因为它们使用线性化动力学模型及时推进解决方案。其中一个算法系列基于 Koopman 理论，本文将这种数值模拟策略应用于反应扩散系统。引入了对抗性和梯度损失，并发现它们可以加强预测。所提出的模型被扩展为处理丢失的训练数据以及从控制的角度重新解决问题。这些发展的功效在两个不同的反应扩散问题上得到了证明：