FMS PINN: Flow-matching sampling for efficient solution of partial differential equations with source singularities
Keywords: physics informed neural networks, Adaptive sampling
Abstract: Singularities in the source functions of partial differential equations (PDEs) can pose significant challenges for physics-informed neural networks (PINNs), often leading to numerical instability and necessitating a large number of sampling points thereby increasing the computational time. In this paper, we introduce a novel sampling point selection method to address these challenges. Our approach is based on diffusion models capable of generative sampling from the distribution of PDE residuals. Specifically, we apply the optimal transport coupling flow-matching technique to generate more sampling points in regions where the PDE residuals are higher, enhancing the accuracy and efficiency of the solution. In contrast to existing approaches in the literature, our method avoids explicit modeling of the probability density proportional to residuals, instead using the benefits of flow matching to generate novel and probable samples from more complex distributions, thereby enhancing PINN solutions for problems with singularities.
We demonstrate that this method, in certain scenarios, outperforms existing techniques such as normalizing flow-based sampling PINN. Especially, our approach demonstrates effectiveness in improving the solution quality for the linear elasticity equation in the case of material with complex geometry of inclusion. A detailed comparison of the flow matching sampling method with other approaches is also provided.
Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.
Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.
Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.
No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.
Submission Number: 11074
Loading