[论文解读] Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing

Salah A Faroughi, Nikhil Pawar+5 more|arXiv (Cornell University)|Nov 14, 2022

Model Reduction and Neural Networks被引用 47

一句话总结

本论文回顾四种神经网络框架——PgNNs、PiNNs、PeNNs，以及神经算子（NOs）——在科学计算中强制物理规律的做法，讨论流体与固体力学领域的架构、应用、局限性及未来机会。

ABSTRACT

Recent breakthroughs in computing power have made it feasible to use machine learning and deep learning to advance scientific computing in many fields, including fluid mechanics, solid mechanics, materials science, etc. Neural networks, in particular, play a central role in this hybridization. Due to their intrinsic architecture, conventional neural networks cannot be successfully trained and scoped when data is sparse, which is the case in many scientific and engineering domains. Nonetheless, neural networks provide a solid foundation to respect physics-driven or knowledge-based constraints during training. Generally speaking, there are three distinct neural network frameworks to enforce the underlying physics: (i) physics-guided neural networks (PgNNs), (ii) physics-informed neural networks (PiNNs), and (iii) physics-encoded neural networks (PeNNs). These methods provide distinct advantages for accelerating the numerical modeling of complex multiscale multi-physics phenomena. In addition, the recent developments in neural operators (NOs) add another dimension to these new simulation paradigms, especially when the real-time prediction of complex multi-physics systems is required. All these models also come with their own unique drawbacks and limitations that call for further fundamental research. This study aims to present a review of the four neural network frameworks (i.e., PgNNs, PiNNs, PeNNs, and NOs) used in scientific computing research. The state-of-the-art architectures and their applications are reviewed, limitations are discussed, and future research opportunities in terms of improving algorithms, considering causalities, expanding applications, and coupling scientific and deep learning solvers are presented. This critical review provides researchers and engineers with a solid starting point to comprehend how to integrate different layers of physics into neural networks.

研究动机与目标

• 总结物理如何被整合到神经网络中，以应对科学计算中的数据稀疏性。
• 比较PgNNs、PiNNs、PeNNs和神经算子在架构、优势和局限性方面的差异。
• 分析这些框架在流体力学和固体力学中的应用。
• 强调当前局限并概述将物理学与深度学习结合的未来研究方向。

提出的方法

• 通过将PgNNs描述为数据驱动的替代模型，并通过对基于物理的数据集进行监督学习来整合已知物理规律。
• 通过在损失函数中嵌入支配方程的残差，并利用自动微分来实现物理规律的约束，解释PiNNs为模型。
• 介绍PeNNs为直接将物理规律编码到网络结构中的架构，以在数据稀疏情况下实现更强的泛化能力。
• 讨论神经算子（NOs）为学习连续算子的模型，能够实现实时预测，并可与PiNNs/PeNNs框架耦合。

实验结果

研究问题

• RQ1 在科学计算中强制物理约束的不同神经网络框架有哪些（PgNNs、PiNNs、PeNNs、NOs），它们在理论和实践上有何差异？
• RQ2 在流体力学和固体力学中，这些物理信息模型的主要应用、优点和局限性是什么？
• RQ3 这些方法如何解决数据稀疏和泛化问题，实时或多物理场仿真的前景如何？
• RQ4 有哪些未来研究机会，以改进算法、因果性考量和求解器耦合？

主要发现

• PgNNs 通过利用基于物理的数据来加速科学计算中的预处理、建模和后处理步骤。
• PiNNs 通过方程残差来强制物理规律，使得在数据稀疏时也能学习，但面临收敛性、稳定性和边界条件处理等问题。
• PeNNs 将物理规律编码进网络结构，在数据稀缺下以及相较于 PgNNs 和 PiNNs 的泛化性能有所提升。
• 神经算子（NOs）学习连续算子，提供实时推断的鲁棒性，并可与PiNNs和PeNNs结合用于复杂非线性多物理场学习。
• 将神经网络架构与传统求解器耦合的混合方法可以实现显著的加速（例如以更粗的网格达到相近精度），并提升CFD和多物理问题的效率。
• 综述强调了前沿架构、跨领域应用，以及在因果性、收敛性和更广泛应用方面需要进一步的基础研究。