Recent studies have shown the success of deep learning in solving forward and inverse problems in engineering and scientific computing domains, such as physics-informed neural networks (PINNs). In the fields of atmospheric science and environmental monitoring, estimating emission source locations is a central task that further relies on multiple model parameters that dictate velocity profiles and diffusion parameters. Estimating these parameters at the same time as emission sources from scarce data is a difficult task. In this work, we achieve this by leveraging the flexibility and generality of PINNs. We use a weighted adaptive method based on the neural tangent kernels to solve a source inversion problem with parameter estimation on the 2D and 3D advection-diffusion equations with unknown velocity and diffusion coefficients that may vary in space and time. Our proposed weighted adaptive method is presented as an extension of PINNs for forward PDE problems to a highly ill-posed source inversion and parameter estimation problem. The key idea behind our methodology is to attempt the joint recovery of the solution, the sources along with the unknown parameters, thereby using the underlying partial differential equation as a constraint that couples multiple unknown functional parameters, leading to more efficient use of the limited information in the measurements. We present various numerical experiments, using different types of measurements that model practical engineering systems, to show that our proposed method is indeed successful and robust to additional noise in the measurements.

翻译：近期研究表明，深度学习在工程与科学计算领域（如物理信息神经网络PINNs）的正、反问题求解中取得了成功。在大气科学与环境监测领域，排放源位置估计是一项核心任务，其进一步依赖于决定速度廓线和扩散参数的多项模型参数。从稀缺数据中同时估计排放源与这些参数是一项困难任务。本研究通过利用PINNs的灵活性和通用性实现了这一目标。我们采用基于神经正切核的加权自适应方法，求解二维与三维对流-扩散方程中具有未知时空变化速度及扩散系数的源反演与参数估计问题。所提出的加权自适应方法将PINNs从适用于正PDE问题扩展至高度病态的源反演与参数估计问题。该方法的核心思想是联合恢复解、源项及未知参数，从而将底层偏微分方程作为约束条件，耦合多个未知函数参数，更高效地利用测量中的有限信息。通过模拟实际工程系统的多种测量类型实验表明，所提方法成功且对测量噪声具有鲁棒性。