A modular PINN recipe (sinh sampling, hard parity and nondegeneracy constraints, exact far-field asymptotics, self-scaled BFGS) yields much lower PDE residuals for self-similar blowup profiles of the 1D Burgers and 2D Boussinesq equations.
On the stability of blowup solutions to the complex Ginzburg-Landau equation in R^d
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abstract
Building upon the idea in \cite{HNWarXiv24}, we establish stability of the type-I blowup with log correction for the complex Ginzburg-Landau equation. In the amplitude-phase representation, a generalized dynamic rescaling formulation is introduced, with modulation parameters capturing the spatial translation and rotation symmetries of the equation and novel additional modulation parameters perturbing the scaling symmetry. This new formulation provides enough degrees of freedom to impose normalization conditions on the rescaled solution, completely eliminating the unstable and neutrally stable modes of the linearized operator around the blowup profile. It enables us to establish the full stability of the blowup by enforcing vanishing conditions via the choice of normalization and using weighted energy estimates, without relying on a topological argument or a spectrum analysis. The log correction for the blowup rate is captured by the energy estimates and refined estimates of the modulation parameters.
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High precision PINNs in unbounded domains: application to singularity formulation in PDEs
A modular PINN recipe (sinh sampling, hard parity and nondegeneracy constraints, exact far-field asymptotics, self-scaled BFGS) yields much lower PDE residuals for self-similar blowup profiles of the 1D Burgers and 2D Boussinesq equations.