The authors propose a new general (quasi) Monte Carlo PINN method for solving fractional partial differential equations (fPDEs) on irregular domains. The key aspects are:
The method extends the Monte Carlo approximation to right-sided fractional derivatives, enabling it to handle different types of fractional derivatives beyond the Caputo definition used previously.
The generated sample nodes exhibit a block-like dense distribution, which provides computational efficiency advantages over the original fPINN method. This distribution is similar to finite difference methods on non-equidistant or nested grids, allowing better adaptation to irregular boundaries.
Numerical examples demonstrate the effectiveness of the proposed method in solving 2D space-fractional Poisson equations and 2D time-space fractional diffusion equations on irregular domains like unit disks and heart-shaped regions. The method achieves the same or better accuracy compared to the original fPINN, with faster computation times.
The method is also applied to solve a 3D coupled time-space fractional Bloch-Torrey equation on the ventricular domain of the human brain, with results compared to classical numerical methods.
An interesting "fuzzy boundary location" problem is tested, where the method shows advantages over fPINN in handling uncertain boundary information.
Overall, the new general Monte Carlo PINN method provides an efficient and accurate approach for solving fPDEs on irregular domains, with potential applications in various scientific and engineering fields.
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by Shupeng Wang... um arxiv.org 05-02-2024
https://arxiv.org/pdf/2405.00217.pdfTiefere Fragen