Physics-informed machine learning (PIML) as a means of solving partial
differential equations (PDE) has garnered much attention in the Computational
Science and Engineering (CS&E) world. This topic encompasses a broad array of
methods and models aimed at solving a single or a collection of PDE problems,
called multitask learning. PIML is characterized by the incorporation of
physical laws into the training process of machine learning models in lieu of
large data when solving PDE problems. Despite the overall success of this
collection of methods, it remains incredibly difficult to analyze, benchmark,
and generally compare one approach to another. Using Kolmogorov n-widths as a
measure of effectiveness of approximating functions, we judiciously apply this
metric in the comparison of various multitask PIML architectures. We compute
lower accuracy bounds and analyze the model's learned basis functions on
various PDE problems. This is the first objective metric for comparing
multitask PIML architectures and helps remove uncertainty in model validation
from selective sampling and overfitting. We also identify avenues of
improvement for model architectures, such as the choice of activation function,
which can drastically affect model generalization to "worst-case" scenarios,
which is not observed when reporting task-specific errors. We also incorporate
this metric into the optimization process through regularization, which
improves the models' generalizability over the multitask PDE problem.

将物理学定律纳入机器学习模型的训练过程中，用于解决偏微分方程问题的基于物理学的机器学习 (PIML) 在计算科学与工程中引起了广泛关注。通过应用 Kolmogorov n - 宽度这种度量方法，我们比较了各种多任务 PIML 架构，为其提供第一种客观指标，帮助消除模型验证中的不确定性，并发现了模型架构的改进途径，如激活函数的选择，可显著影响模型对最坏情况的泛化性能。