Nowadays, in the Scientific Machine Learning (SML) research field, the traditional machine learning (ML) tools and scientific computing approaches are fruitfully intersected for solving problems modelled by Partial Differential Equations (PDEs) in science and engineering applications. Challenging SML methodologies are the new computational paradigms named Physics-Informed Neural Networks (PINNs). PINN has revolutionized the classical adoption of ML in scientific computing, representing a novel class of promising algorithms where the learning process is constrained to satisfy known physical laws described by differential equations. In this paper, we propose a PINN-based computational study to deal with a non-linear partial differential equations system. In particular, using this approach, we solve the Gray-Scott model, a reaction–diffusion system that involves an irreversible chemical reaction between two reactants. In the unstable region of the model, we consider some a priori information related to dynamical behaviors, i. e. a supervised approach that relies on a finite difference method (FDM). Finally, simulation results show that PINNs can successfully provide an approximated Grey-Scott system solution, reproducing the characteristic Turing patterns for different parameter configurations.

中文翻译：

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用于 1D 和 2D Gray-Scott 系统的基于物理的神经网络方法

如今，在科学机器学习 (SML) 研究领域，传统的机器学习 (ML) 工具和科学计算方法在解决科学和工程应用中偏微分方程 (PDE) 建模的问题方面取得了丰硕的成果。具有挑战性的 SML 方法是名为物理信息神经网络 (PINN) 的新计算范式。PINN 彻底改变了 ML 在科学计算中的经典应用，代表了一类新的有前途的算法，其中学习过程被限制为满足由微分方程描述的已知物理定律。在本文中，我们提出了一种基于 PINN 的计算研究来处理非线性偏微分方程系统。特别是，使用这种方法，我们解决了 Gray-Scott 模型，一种反应-扩散系统，涉及两种反应物之间的不可逆化学反应。在模型的不稳定区域，我们考虑一些与动力学行为相关的先验信息，即依赖于有限差分法（FDM）的监督方法。最后，仿真结果表明，PINNs 可以成功地提供一个近似的 Grey-Scott 系统解决方案，再现不同参数配置的特征图灵模式。