Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer very high approximation capabilities. However, when the PDE solutions are low regular, an expert insight is required to build deep learning formulations that do not incur in variational crimes. Optimization solvers are also significantly challenged, and can potentially spoil the final quality of the approximated solution due to the convergence to bad local minima, and bad generalization capabilities. In this paper, we present an exhaustive numerical study of the merits and limitations of these schemes when solutions exhibit low-regularity, and compare performance with respect to more benign cases when solutions are very smooth. As a support for our study, we consider singularly perturbed convection-diffusion problems where the regularity of solutions typically degrades as certain multiscale parameters go to zero.

翻译：以深层次学习为基础的数字计划,如物质知情神经网络(PINNs)最近成为解决局部差异的典型数字计划的一种替代方案。它们最初非常吸引人,因为实施基于强效残余形式的香草版PINNs很容易,神经网络提供非常高的近似能力。然而,当PDE解决方案比较低时,需要专家的深入了解才能形成在变异犯罪中不会发生的深层次学习配方。优化解决方案的解决者也面临重大挑战,并有可能由于与不良的当地微型和不良的概括能力趋同而破坏近似解决方案的最终质量。在本文件中,我们详尽地用数字研究这些解决方案的优点和局限性,并在解决方案非常顺利的情况下,将绩效与更温和的案例进行比较。作为对我们研究的支持,我们认为,在解决方案的常规性通常会降低为某些多尺度参数的异常偏向零的情况下,我们可考虑奇地受困的同化-融合问题。