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arXiv · 2610.30492

Discovery of Singularities in Partial Differential Equations Using Adaptive Numerical Discretizations

Allen Yang·Lucas Amoudruz·Petros Koumoutsakos
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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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Allen Yang, Lucas Amoudruz, Petros Koumoutsakos. 2026-10-21. Discovery of Singularities in Partial Differential Equations Using Adaptive Numerical Discretizations. https://arxiv.org/abs/2610.30492

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