Discovery of Unstable Singularities
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The authors present a playbook for finding unstable finite-time singularities in fluid PDEs, uncovering new self-similar blow-up solutions in three canonical systems and training neural solvers to near machine precision, which enables downstream computer-assisted proofs.
What they found. New families of unstable self-similar singularities are discovered for the incompressible porous media equation and the 2D Boussinesq system (analogous to axisymmetric 3D Euler with a boundary), plus a higher-order unstable profile for the Córdoba-Córdoba-Fontelos model.
Key pattern. The inverse scaling rate grows roughly linearly with the instability order in IPM and Boussinesq, providing a simple empirical rule to seed higher-order searches.
How they did it. They reformulate each PDE in self-similar coordinates, embed symmetry and decay constraints directly in the network outputs, and train physics-informed neural networks with a full-matrix Gauss-Newton optimizer plus multi-stage refinement to drive residuals down to 10⁻¹³ for certain CCF solutions.
Validation. Accuracy is quantified via maximum residuals on dense grids and by linear stability analysis of the profiled solutions, matching n unstable modes for the n-th unstable solution. Funnel plots around admissible λ values confirm significant digits and admissibility.
Why it matters. Unstable singularities are expected in boundary-free Euler and Navier-Stokes settings. This work supplies high-precision candidates, scalable heuristics for λ, and numerics precise enough to support computer-assisted proofs, pushing toward the resolution of long-standing questions in fluid singularity formation.
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