Physics-Informed Radial Basis Function Method for Transport Processes
Abstract
We propose a new method, which uses radial basis functions (RBFs) to assimilate data in the solution of the advection-dispersion equation (ADE). In our method, the weighted sum of RBFs is used to represent the concentration field. Then, the weights in the RBF expansion are estimated by solving a least square problem that minimizes the square of residuals, which may include the residuals of ADEs (we may have a system of equations), the boundary and initial conditions, and measurements.
Comparably to the physics-informed machine learning methods such as PINN, the spatial and time derivatives of the ADE residuals are calculated analytically in the RBF method. The main advantage of the RBF method over PINN (and other methods involving deep neural networks) is that for linear ADEs, the RBF method leads to a linear least square problem, which has a unique solution that can be easily obtained. On the other hand, the PINN method results in a non-linear least square problem, which can be non-convex and might not have a unique solution. We demonstrate that the proposed method accurately and efficiently solves forward, backward, and inverse transport problems. We apply it to different transport equations, different boundary and initial conditions, and analyze the errors and ways to optimize the solutions.- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2022
- Bibcode:
- 2022AGUFM.H35H1207R