Discovery of Singularities in Partial Differential Equations Using Adaptive Numerical Discretizations
Abstract
We compute self-similar singularity profiles of partial differential equations (PDEs) using adaptive numerical discretizations based on optimizing a discrete loss (ODIL). The fields are represented in overlapping local Chebyshev patches joined with a mapped asymptotic patch. After factoring out the leading power laws of solution ansatzes, their limiting amplitudes at infinity are determined by the optimization, eliminating domain truncation and prescribed far-field conditions. Adaptive refinement is guided by Chebyshev-coefficient decay, and the nonlinear least-squares problem is solved using automatically differentiated Gauss--Newton iterations with Levenberg--Marquardt damping. The method achieves O(10^{-14})-level accuracy for Burgers profiles, residuals of O(10^{-14}) to O(10^{-10}) for the generalized De Gregorio family, and of O(10^{-8}) in a local region for the two-dimensional Boussinesq equations. These results demonstrate that ODIL can identify singularity profiles for PDEs with high accuracy using adaptive numerical methods, complementing ongoing efforts that rely on artificial intelligence.
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BibTeX
@misc{yang2026discovery,
title = {{Discovery of Singularities in Partial Differential Equations Using Adaptive Numerical Discretizations}},
author = {Allen Yang and Lucas Amoudruz and Petros Koumoutsakos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30492},
eprint = {2610.30492},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30492},
}
RIS
TY - UNPB
TI - Discovery of Singularities in Partial Differential Equations Using Adaptive Numerical Discretizations
AU - Allen Yang
AU - Lucas Amoudruz
AU - Petros Koumoutsakos
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30492
UR - https://arxiv.org/abs/2610.30492
AB - We compute self-similar singularity profiles of partial differential equations (PDEs) using adaptive numerical discretizations based on optimizing a discrete loss (ODIL). The fields are represented in overlapping local Chebyshev patches joined with a mapped asymptotic patch. After factoring out the leading power laws of solution ansatzes, their limiting amplitudes at infinity are determined by the optimization, eliminating domain truncation and prescribed far-field conditions. Adaptive refinement is guided by Chebyshev-coefficient decay, and the nonlinear least-squares problem is solved using automatically differentiated Gauss--Newton iterations with Levenberg--Marquardt damping. The method achieves O(10^{-14})-level accuracy for Burgers profiles, residuals of O(10^{-14}) to O(10^{-10}) for the generalized De Gregorio family, and of O(10^{-8}) in a local region for the two-dimensional Boussinesq equations. These results demonstrate that ODIL can identify singularity profiles for PDEs with high accuracy using adaptive numerical methods, complementing ongoing efforts that rely on artificial intelligence.
ER -