This work is concerned with discovering the governing partial differential equation (PDE) of a physical system. Existing methods have demonstrated the PDE identification from finite observations but failed to maintain satisfying performance against noisy data, partly owing to suboptimal estimated derivatives and found PDE coefficients. We address the issues by introducing a noise-aware physics-informed machine learning (nPIML) framework to discover the governing PDE from data following arbitrary distributions. Our proposals are twofold. First, we propose a couple of neural networks, namely solver and preselector, which yield an interpretable neural representation of the hidden physical constraint. After they are jointly trained, the solver network approximates potential candidates, e.g., partial derivatives, which are then fed to the sparse regression algorithm that initially unveils the most likely parsimonious PDE, decided according to the information criterion. Second, we propose the denoising physics-informed neural networks (dPINNs), based on Discrete Fourier Transform (DFT), to deliver a set of the optimal finetuned PDE coefficients respecting the noise-reduced variables. The denoising PINNs' structures are compartmentalized into forefront projection networks and a PINN, by which the formerly learned solver initializes. Our extensive experiments on five canonical PDEs affirm that the proposed framework presents a robust and interpretable approach for PDE discovery, applicable to a wide range of systems, possibly complicated by noise.

翻译：这项工作涉及从物理系统中发现管理部分差异方程(PDE) 。 现有方法通过有限观察显示PDE的识别方法,但未能保持对噪音数据的满意性能,部分原因包括估计衍生物的不最佳估计值和发现PDE系数。 我们通过引入一个有噪音意识的物理知情机器学习(nPIML)框架,从任意分布后的数据中发现管理PDE(PDE) 。 我们的提议是双重的。 首先,我们建议建立几个神经网络,即求解器和预选器,以产生隐藏物理约束的可解释的神经代表。 在经过联合培训后, 求解器网络接近潜在候选人, 例如部分衍生物, 后被反馈到根据信息标准决定的稀释回归算算中。 其次, 我们提议取消有噪音的物理知情神经网络(dPINN) 。 我们提议基于Discrete Fourier变换(DFDT), 来提供一套最优化的PDE系数, 与噪音受控变异的系统有关, 初步的PIN 模型将一个我们所了解的系统变成的系统。