Neural Discovery of Strichartz Extremizers

TL;DR

This paper introduces a neural network approach to identify extremizers of Strichartz inequalities in dispersive PDEs, successfully recovering known extremizers and supporting conjectures in uncharted cases.

Contribution

The authors develop a neural-network-based pipeline that systematically searches for extremizers of Strichartz inequalities, including conjectural and open cases, providing new insights and validation.

Findings

Recovered Gaussian extremizers in dimensions 1 and 2 with high accuracy.

Supported the conjecture that Gaussians are universal extremizers in certain ranges.

Discovered breather solutions approaching universal bounds in open cases.

Abstract

Strichartz inequalities are a cornerstone of the modern theory of dispersive PDEs, but their extremizers are known explicitly only in a handful of sharp cases. The non-convexity of the underlying functional makes the problem hard, and to our knowledge no systematic numerical attack has been attempted. We propose a simple neural-network-based pipeline that searches for extremizers as critical points of the Strichartz ratio, and apply it in three settings. First, on the Schr\"odinger group we recover the Gaussian extremizers of Foschi and Hundertmark--Zharnitsky in dimensions $d=1,2$ to within $10^{-3}$ relative error, with no analytical prior. Second, on $59$ further admissible pairs in $d=1$ where the answer is conjectural, the method consistently finds Gaussians, supporting the conjecture that Gaussians are the universal extremizers in the admissible range. Third, on the critical Airy--Strichartz inequality at $\gamma=1/q$, where existence is open, the optimization does not converge to any $L^2$ profile: instead, the iterates organize themselves as mKdV breathers $B(0,\cdot;\alpha,1,0,0)$ with growing internal frequency $\alpha$, and the discovered ratio approaches the Frank--Sabin universal lower bound $\widetilde A_{q,r}$ from below with a power-law gap $\sim\alpha^{-0.9}$. We confirm the same picture with an independent Hermite-basis ansatz. We propose a precise conjecture: the supremum equals $\widetilde A_{q,r}$ and is approached, but not attained, along the breather family. The pipeline thus serves both as a validator on known cases and as a discovery tool when no extremizer exists.