[论文解读] State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

Sourav Das, Solomon Tesfamariam|arXiv (Cornell University)|Feb 13, 2022

Model Reduction and Neural Networks被引用 22

一句话总结

本文综述了设计实验设计 DoE 方法在 physics-informed 深度学习 PINNs 中的应用，聚焦取样策略、边界/初始条件管理以及损失函数的构造。

ABSTRACT

This paper presents a comprehensive review of the design of experiments used in the surrogate models. In particular, this study demonstrates the necessity of the design of experiment schemes for the Physics-Informed Neural Network (PINN), which belongs to the supervised learning class. Many complex partial differential equations (PDEs) do not have any analytical solution; only numerical methods are used to solve the equations, which is computationally expensive. In recent decades, PINN has gained popularity as a replacement for numerical methods to reduce the computational budget. PINN uses physical information in the form of differential equations to enhance the performance of the neural networks. Though it works efficiently, the choice of the design of experiment scheme is important as the accuracy of the predicted responses using PINN depends on the training data. In this study, five different PDEs are used for numerical purposes, i.e., viscous Burger's equation, Shrödinger equation, heat equation, Allen-Cahn equation, and Korteweg-de Vries equation. A comparative study is performed to establish the necessity of the selection of a DoE scheme. It is seen that the Hammersley sampling-based PINN performs better than other DoE sample strategies.

研究动机与目标

• Motivate the integration of design-of-experiments techniques with physics-informed neural networks.
• Survey sampling strategies for initial, boundary, and residual terms in PINN loss functions.
• Analyze how DoE choices impact training efficacy and accuracy in PINN frameworks.

提出的方法

• 综述并综合文献中在 PINNs 中使用的 DoE 方法。
• Present core loss components for PINNs: L0 for initial data, Lb for boundary conditions, and Lf for residual physics terms.
• 讨论损失项的构造与组合方式，例如 L = L0 + Lb + Lf。

实验结果

研究问题

• RQ1 在 PINNs 中，哪些 DoE 策略最有效地对初始数据、边界和残差点进行取样？
• RQ2 不同的 DoE 驱动的损失函数形式如何影响物理信息深度学习模型的准确性与收敛？
• RQ3 PINNs 中用于初始、边界和控制方程残余的损失项的典型形式有哪些？
• RQ4 将 DoE 应用于 PINNs 的开放挑战有哪些以及未来方向？

主要发现

• 本文汇总了 PINNs 常用的损失分量，包括初始数据损失 (L0)、边界条件损失 (Lb) 和控制方程残差损失 (Lf)。
• 边界项损失 Lb 通常涉及在边界处匹配函数值和导数值。
• PINN 损失常写成 L = L0 + Lb + Lf，说明数据保真、边界条件和物理残差如何被整合。
• 讨论强调了取样策略在塑造 PINN 的训练数据以及对模型性能的影响中的作用。