We inspect the analogy between machine-learning (ML) applications based on the transformer architecture without self-attention, {\it neural chains} hereafter, and discrete dynamical systems associated with discretised versions of neural integral and partial differential equations (NIE, PDE). A comparative analysis of the numerical solution of the (viscid and inviscid) Burgers and Eikonal equations via standard numerical discretization (also cast in terms of neural chains) and via PINN's learning is presented and commented on. It is found that standard numerical discretization and PINN learning provide two different paths to acquire essentially the same knowledge about the dynamics of the system. PINN learning proceeds through random matrices which bear no direct relation to the highly structured matrices associated with finite-difference (FD) procedures. Random matrices leading to acceptable solutions are far more numerous than the unique tridiagonal form in matrix space, which explains why the PINN search typically lands on the random ensemble. The price is a much larger number of parameters, causing lack of physical transparency (explainability) as well as large training costs with no counterpart in the FD procedure. However, our results refer to one-dimensional dynamic problems, hence they don't rule out the possibility that PINNs and ML in general, may offer better strategies for high-dimensional problems.

翻译：本文考察了基于无自注意力Transformer架构的机器学习应用（以下简称神经链）与离散化神经积分方程和偏微分方程（NIE、PDE）所关联的离散动力系统之间的类比关系。我们展示并评述了通过标准数值离散化方法（亦表述为神经链形式）与物理信息神经网络（PINN）学习对（粘性与非粘性）Burgers方程和Eikonal方程进行数值求解的对比分析。研究发现，标准数值离散化与PINN学习提供了两条获取系统动力学本质相同知识的不同路径。PINN学习通过随机矩阵进行，这些矩阵与有限差分（FD）方法中高度结构化的矩阵无直接关联。能够产生可接受解的随机矩阵在矩阵空间中的数量远多于唯一的三对角形式，这解释了PINN搜索通常落在随机集合上的原因。代价是参数数量大幅增加，导致物理透明度（可解释性）缺失以及训练成本高昂，而这些在FD方法中并不存在。然而，我们的研究结果针对一维动力学问题，因此并不排除PINN及广义机器学习在高维问题上可能提供更优策略的可能性。