Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks
Alexandre M. Tartakovsky Carlos Ortiz Marrero Paris Perdikaris Guzel D. Tartakovsky David Barajas-Solano Pacific Northwest National Laboratory University of Pennsylvania
Abstract
We present a physics informed deep neural network (DNN) method for estimating parameters and unknown physics (constitutive relationships) in partial differential equation (PDE) models. We use PDEs in addition to measurements to train DNNs to approximate unknown parameters and constitutive relationships as well as states. The proposed approach increases the accuracy of DNN approximations of partially known functions when a limited number of measurements is available and allows for training DNNs when no direct measurements of the functions of interest are available. We employ physics informed DNNs to estimate the unknown space-dependent diffusion coefficient in a linear diffusion equation and an unknown constitutive relationship in a non-linear diffusion equation. For the parameter estimation problem, we assume that partial measurements of the coefficient and states are available and demonstrate that under these conditions, the proposed method is more accurate than state-of-the-art methods. For the non-linear diffusion PDE model with a fully unknown constitutive relationship (i.e., no measurements of constitutive relationship are available), the physics informed DNN method can accura
中文速览
用深度神经网络（DNN）来反推偏微分方程（PDE）中未知的参数和本构关系，是个数据少、问题病态的难题。作者提出了一种"物理知情"深度神经网络方法：同时用有限的测量数据和偏微分方程本身作为约束来联合训练两个神经网络，一个拟合方程状态量，另一个拟合未知的扩散系数或非线性本构关系，通过自动微分将PDE残差直接嵌入损失函数。在线性扩散方程的参数估计任务中，该方法在数据稀缺时比现有主流方法更精确；在非线性扩散方程中，即便完全没有本构关系的直接观测，仅凭状态量测量就能准确重建非线性本构关系；方法在含噪声数据下同样表现稳健。这项工作表明，将物理方程显式融入神经网络训练可以大幅降低对观测数据量的依赖，为复杂物理系统中未知物理规律的数据驱动发现提供了一条切实可行的路径。
原文 arXiv:1808.03398;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1808.03398v2