Chen Zhao, Liu Yang, Sun Hao
Department of Civil and Environmental Engineering, Northeastern University, Boston, MA, 02115, USA.
Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA, 02115, USA.
Nat Commun. 2021 Oct 21;12(1):6136. doi: 10.1038/s41467-021-26434-1.
Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel approach called physics-informed neural network with sparse regression to discover governing partial differential equations from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this discovery approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to approximate the solution of system variables, compute essential derivatives, as well as identify the key derivative terms and parameters that form the structure and explicit expression of the equations. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of partial differential equation systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture.
利用数据来发现描述复杂物理系统行为的潜在支配定律或方程，可显著推动我们在各种科学和工程学科中对此类系统的建模、模拟及理解。这项工作引入了一种名为带稀疏回归的物理信息神经网络的新方法，用于从稀缺且有噪声的数据中发现非线性时空系统的支配偏微分方程。特别地，这种发现方法无缝集成了深度神经网络在丰富表示学习、物理嵌入、自动微分和稀疏回归方面的优势，以近似系统变量的解、计算基本导数，以及识别构成方程结构和显式表达式的关键导数项和参数。通过数值和实验，在发现具有不同数据稀缺程度和噪声水平且考虑不同初始/边界条件的各种偏微分方程系统时，证明了该方法的有效性和鲁棒性。所得的计算框架显示了在难以获取大型准确数据集的实际应用中进行闭式模型发现的潜力。
