Recent studies have demonstrated the success of deep learning in solving forward and inverse problems in engineering and scientific computing domains, such as physics-informed neural networks (PINNs). Source inversion problems under sparse measurements for parabolic partial differential equations (PDEs) are particularly challenging to solve using PINNs, due to their severe ill-posedness and the multiple unknowns involved including the source function and the PDE parameters. Although the neural tangent kernel (NTK) of PINNs has been widely used in forward problems involving a single neural network, its extension to inverse problems involving multiple neural networks remains less explored. In this work, we propose a weighted adaptive approach based on the NTK of PINNS including multiple separate networks representing the solution, the unknown source, and the PDE parameters. The key idea behind our methodology is to simultaneously solve the joint recovery of the solution, the source function 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 apply our method on the advection-diffusion equation and we present various 2D and 3D numerical experiments using different types of measurements data that reflect practical engineering systems. Our proposed method is successful in estimating the unknown source function, the velocity and diffusion parameters as well as recovering the solution of the equation, while remaining 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的正反问题求解中取得了成功，例如物理信息神经网络（PINNs）。针对抛物型偏微分方程（PDE）的稀疏测量源项反演问题，由于其严重的不适定性以及涉及源函数和PDE参数等多个未知量，使用PINNs求解尤为困难。尽管PINNs的神经切线核（NTK）已广泛用于涉及单一神经网络的正问题，但其在涉及多个神经网络的逆问题中的拓展研究仍较少。本文提出一种基于NTK的加权自适应方法，该方法采用多个独立网络分别表征解函数、未知源项及PDE参数。方法论的核心思想在于，通过将偏微分方程作为耦合多个未知函数参数的约束条件，同步求解解函数、源函数及未知参数的联合恢复问题，从而更高效地利用测量数据中的有限信息。我们将该方法应用于对流-扩散方程，并利用反映实际工程系统的多类型测量数据进行了大量二维与三维数值实验。结果表明，本方法不仅能成功估计未知源函数、速度场及扩散参数，还能恢复方程的解，且对测量噪声具有稳健性。