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arxiv: 2605.04918 · v2 · submitted 2026-05-06 · 🧮 math.AP · cs.LG· cs.NA· math.NA

Neural Discovery of Strichartz Extremizers

Nicol\'as Valenzuela , Ricardo Freire , Claudio Mu\~noz This is my paper

Pith reviewed 2026-05-08 16:42 UTC · model grok-4.3

classification 🧮 math.AP cs.LGcs.NAmath.NA
keywords Strichartz inequalitiesextremizersneural optimizationAiry equationmKdV breatherssharp constantsdispersive PDEsnumerical discovery
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The pith

Neural optimization shows mKdV breathers approach the sharp Airy-Strichartz bound without attaining it.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a neural-network pipeline that treats Strichartz extremizers as critical points of the ratio functional and applies it to several dispersive inequalities. In known cases it recovers the expected Gaussian profiles to high accuracy. In the critical Airy-Strichartz setting, where existence of an extremizer is open, the iterates instead organize into mKdV breathers whose internal frequency grows without bound; the attained ratio approaches the Frank-Sabin lower bound from below according to a power law. The authors conjecture that this bound is the true supremum but is never attained by any fixed profile.

Core claim

In the critical Airy-Strichartz inequality the optimization does not converge to any L2 profile. Instead the iterates self-organize as mKdV breathers B(0,·;α,1,0,0) with growing frequency α, and the discovered ratio approaches the universal lower bound Ã_{q,r} from below with a gap that scales like α^{-0.9}. The same behavior is recovered with an independent Hermite-basis ansatz. The authors conjecture that the supremum equals Ã_{q,r} and is approached but not attained along this breather family.

What carries the argument

A neural-network pipeline that searches for critical points of the Strichartz ratio functional without analytical priors.

If this is right

  • Gaussians are the extremizers for all 59 further admissible Strichartz pairs tested in one dimension.
  • The method can serve as a validator for conjectural sharp constants when analytical proofs are unavailable.
  • In settings where an extremizer does not exist, the supremum may still be characterized by an asymptotic family of soliton-like objects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar numerical pipelines could be used to conjecture the structure of extremizers or their absence in other nonlinear dispersive inequalities.
  • Analytical work on the Airy-Strichartz problem might usefully focus on the asymptotic behavior of high-frequency breathers rather than on existence of a fixed profile.
  • If the conjecture holds, the Airy-Strichartz inequality would furnish a concrete example where the sharp constant is known but never achieved.

Load-bearing premise

The network iterates reliably locate the relevant minimizing sequences and the observed power-law improvement continues for arbitrarily large breather frequencies.

What would settle it

A direct computation or proof showing that the ratio for breathers eventually stops improving or that some other profile exceeds the Frank-Sabin bound.

Figures

Figures reproduced from arXiv: 2605.04918 by Claudio Mu\~noz, Nicol\'as Valenzuela, Ricardo Freire.

Figure 1
Figure 1. Figure 1: Neural optimization pipeline. The network view at source ↗
Figure 2
Figure 2. Figure 2: Schrödinger d = 1, p = 6: the neural network recovers Foschi’s Gaussian extremizer with no analytical input. 5.1 Baseline: soliton and breather ratios For any w ∈ L 2 (R) we write R[w] := ∥ |Dx| γ e −t∂3 x w∥L q t Lr x ∥w∥L2 x . By the Airy scaling u(t, x) 7→ λu(λ 3 t, λx), R[Qp,c] is independent of c and R[B(0, ·; α, β, ·, ·)] depends only on α/β. The effective parameter spaces are therefore the discrete … view at source ↗
Figure 3
Figure 3. Figure 3: Schrödinger d = 1: the neural optimization finds Gaussians as extrem￾izers for any admissible pair view at source ↗
Figure 4
Figure 4. Figure 4: Airy–Strichartz ratios for solitons and breathers at view at source ↗
Figure 5
Figure 5. Figure 5: AI-based discovery for the Airy–Strichartz inequality, pair view at source ↗
Figure 6
Figure 6. Figure 6: Approximate extremizer ϕθ ⋆ (x) for the pair (8, 8). 5.5 Main conjecture Combining the soliton/breather baseline, the independent Hermite check, and the NN results, we arrive at the following. 11 view at source ↗
Figure 7
Figure 7. Figure 7: Ratio comparison between three solitons Qp, with p ∈ {2, 3, 4}, and the gaussian e − π 2 |x| 2 . The comparison is made in function of 1 r ∈ (0, 1 2 ), taking 59 different values. In the case r = 2 we have the classical L 2 conservation and every L 2 solution has the same ratio. d = 1, 2. In both dimensions, a neural network with 4 hidden layers and 20 neurons per hidden layer will be considered. The domai… view at source ↗
Figure 8
Figure 8. Figure 8: compares the absolute value of the approximate extremizer with the modulus of the theoretical one (Remark 2); in all cases, ϕθ ∗ coincides in modulus with the gaussian e − π 2 |x| 2 , so the approximate extremizers are gaussians view at source ↗
Figure 9
Figure 9. Figure 9: Schrödinger d = 1: NN results for four different admissible pairs (q, r). (a) Neral extremizer ϕθ⋆ ratio versus analytical gaussian ratio. (b) Relative error of Sˆ1,q,r, with respect to the gaussian ratio view at source ↗
Figure 10
Figure 10. Figure 10: Schrödinger d = 1: the neural optimization finds Gaussians as extremizers for any admissible pair. of a Gaussian, Se1,q,r. Error bars are obtained from 5 independent runs of the algorithm for each pair and report the standard deviation. As can be seen in the figure, the neural-approximated profile corresponds to a Gaussian as well, with relative error ≲ 10−3 for all the admissible pairs. A.2.2 Dimension 1… view at source ↗
Figure 11
Figure 11. Figure 11: Absolute value of uneven extremizers |ϕθ ∗ (x)|, for (p, q) = (6, 6) in four different realizations. For these simulations, in average Sˆ 1,6,6 = 0.8128 and errorrel,1,6,6 = 1.590 × 10−3 . A.2.3 Dimension 2 For this dimension, the functionals JL2 x (Rd) and JL q t Lr x (Rd+1) are computed with R = 10, T = 1, M = 32 and N = 128. Due to the higher computational cost in dimension 2, the number of grid points… view at source ↗
Figure 12
Figure 12. Figure 12: Absolute value |ϕθ ∗ (x)|, for three different admissible pairs (q, r). From the findings regarding the Strichartz estimates in the case of the linear Schrödinger model, we find supporting evidence for the validity of the well-known conjecture on the extremizers of (1): In dimension d = 1, 2, Strichartz estimates with admissible pair (p, q) are extremized by Gaussians. Moreover, the optimal constant is Sd… view at source ↗
Figure 13
Figure 13. Figure 13: Schrödinger d = 2: NN results for three different admissible pairs (q, r). B Airy models The Airy initial-value problem ∂tv + ∂ 3 x v = 0, v(0, x) = v0(x), (19) is the linear part of the (focusing) generalized Korteweg–de Vries family ∂tu + ∂x(∂ 2 xu + u p ) = 0, p ≥ 2 [50, 45, 13, 23, 34, 49, 47, 48], whose soliton solutions are Qc,p,x0 (t, x) := Qp,c(x − ct − x0), Qp,c(s) = c 1/(p−1)Qp( √ cs), (20) with… view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of Airy-Strichartz (Airy) and Strichartz (Schrödinger) view at source ↗
Figure 15
Figure 15. Figure 15: In red: Airy-Strichartz ratio of initial conditions view at source ↗
Figure 16
Figure 16. Figure 16: In red: Airy-Strichartz ratio of initial conditions view at source ↗
Figure 17
Figure 17. Figure 17: In black, difference between Ae6,6 and R[B(0, x; α, 1, 0, 0)], for increas￾ing α. In blue, power-law decay ∼ 0.0139 α −0.9041 . where Hen(·) is the probabilistic Hermite polynomial Hen(x) = (−1)n e x 2 2 d n dxn e − x 2 2 . We consider the first 5 Hermite functions and we compute the ratios for those functions as initial conditions, by changing the value of the time-domain bound￾29 view at source ↗
Figure 18
Figure 18. Figure 18: In black, the lower bound Ae6,6. In “plus" dots, the ratios for Hermite functions with n = 0, 1, 2, 3, 4, when increasing time boundary [−T, T]x view at source ↗
Figure 19
Figure 19. Figure 19: In black, the lower bound Ae6,6. In red, the ratios for Hermite functions with n ∈ {0, 1, . . . , 20}. Before attacking the problem with neural networks, we use the L 2 x Hermite 30 view at source ↗
Figure 20
Figure 20. Figure 20: In blue, the function u0(x) defined in (28) with coefficients b ∗ n given in view at source ↗
Figure 21
Figure 21. Figure 21: Left: Neural extremizer obtained by means of view at source ↗
Figure 22
Figure 22. Figure 22: AI-based discovery for the Airy–Strichartz inequality, pair view at source ↗
Figure 23
Figure 23. Figure 23: Best function fitting the approximated profile, for the pair view at source ↗
Figure 24
Figure 24. Figure 24: In blue, Estimation of the sharp constant with its standard deviation. view at source ↗
Figure 25
Figure 25. Figure 25: Relative error of Aˆ 6,6 with respect to Ae6,6. 35 view at source ↗
Figure 26
Figure 26. Figure 26: AI-based discovery for the Airy–Strichartz inequality, pair view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a neural-network optimization pipeline to numerically locate critical points of the Strichartz ratio functional. It recovers the known Gaussian extremizers for the Schrödinger equation in d=1 and d=2 to relative error ~10^{-3}, obtains consistent Gaussian profiles across 59 additional admissible (q,r) pairs in d=1, and for the critical Airy-Strichartz inequality finds that the iterates organize as mKdV breathers B(0,·;α,1,0,0) whose ratios approach the Frank-Sabin bound Ã_{q,r} from below with an observed gap scaling as ~α^{-0.9}. An independent Hermite-basis ansatz yields the same picture, leading to the conjecture that the supremum equals Ã_{q,r} but is not attained along the breather family.

Significance. If the numerical evidence and extrapolation hold, the work supplies both a practical discovery tool for non-convex extremal problems in dispersive PDEs and a concrete conjecture resolving the open existence question for Airy-Strichartz extremizers. The validation on known cases, cross-check with an independent ansatz, and parameter-free nature of the optimization (no fitted parameters or self-referential definitions) are genuine strengths that could guide future analytical work.

major comments (1)
  1. [Airy-Strichartz section (breather family B(0,·;α,1,0,0) and associated ratio plots)] The central conjecture (stated in the abstract and elaborated in the Airy-Strichartz discussion) that the supremum equals Ã_{q,r} and is not attained rests on the observed power-law gap ~α^{-0.9} for finite-α breathers. No asymptotic analysis, a-priori estimate, or rigorous justification is supplied showing that this scaling persists as α→∞ or that no other L² sequence can exceed the breather ratios; the functional landscape is non-convex and all computations use finite discretization. This extrapolation is load-bearing for both the identification of the supremum and the non-attainment statement.
minor comments (2)
  1. [Abstract] The abstract introduces the notation Ã_{q,r} without a brief definition or reference; a parenthetical reminder of the Frank-Sabin universal lower bound would improve readability.
  2. [Numerical method section] The description of the neural-network pipeline should include explicit statements on the number of independent random initializations, convergence tolerances, and how local minima are distinguished from the reported breathers.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the thoughtful review and for identifying the central role of the Airy-Strichartz conjecture. We address the major comment below, maintaining that the manuscript accurately presents a numerically supported conjecture rather than a theorem.

read point-by-point responses

  1. Referee: [Airy-Strichartz section (breather family B(0,·;α,1,0,0) and associated ratio plots)] The central conjecture (stated in the abstract and elaborated in the Airy-Strichartz discussion) that the supremum equals Ã_{q,r} and is not attained rests on the observed power-law gap ~α^{-0.9} for finite-α breathers. No asymptotic analysis, a-priori estimate, or rigorous justification is supplied showing that this scaling persists as α→∞ or that no other L² sequence can exceed the breather ratios; the functional landscape is non-convex and all computations use finite discretization. This extrapolation is load-bearing for both the identification of the supremum and the non-attainment statement.

    Authors: We agree that the conjecture is based on numerical extrapolation and that the manuscript supplies no rigorous asymptotic analysis or a-priori estimate. The paper is a computational study whose primary contribution is the discovery, via two independent methods (neural optimization and Hermite-basis ansatz), that the iterates consistently organize as the mKdV breather family B(0,·;α,1,0,0) and that the ratio approaches the Frank-Sabin bound from below with an observed power-law gap. The non-convexity of the functional and the finite discretization are implicit limitations of any numerical approach and are consistent with the framing of the result as a conjecture. We do not claim that the observed scaling persists for all α or that the breather family is the unique maximizing sequence; we only report the empirical behavior and propose the precise statement for future analytical investigation. Because the manuscript already presents the claim as a conjecture rather than a theorem, we see no need to alter the statement itself. revision: no

standing simulated objections not resolved
  • Rigorous asymptotic analysis of the breather ratios as α→∞ or a proof that no other L² sequence can exceed the observed ratios.

Circularity Check

0 steps flagged

No circularity: numerical optimization on the ratio functional yields an observation-based conjecture

full rationale

The paper optimizes the Strichartz ratio directly via neural network (and independent Hermite ansatz) without fitting parameters that are then renamed as predictions. Known cases recover Gaussians to 10^{-3} error with no analytical prior. For the Airy case the iterates organize as mKdV breathers whose ratios approach the externally cited Frank-Sabin bound Ã_{q,r} with observed power-law gap; the conjecture that the supremum equals this bound and is approached but not attained is stated as an inference from the finite-α numerics rather than a self-definitional or fitted-input reduction. No self-citation chain, uniqueness theorem, or ansatz smuggling is load-bearing; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that gradient-based neural optimization finds the relevant critical points of the Strichartz ratio and that the observed breather behavior extrapolates to the infinite-frequency limit; no explicit free parameters or invented entities are introduced beyond standard neural-network training choices.

axioms (2)
  • domain assumption The Strichartz ratio functional admits critical points that can be located by gradient-based optimization of a neural-network parametrization.
    Invoked to justify the pipeline; appears in the description of the method applied to both Schrödinger and Airy cases.
  • standard math Admissible pairs (q,r) for Strichartz inequalities satisfy the usual scaling and Sobolev embedding conditions.
    Background assumption from dispersive PDE theory used to select the 59 test cases and the critical Airy regime.

pith-pipeline@v0.9.0 · 5636 in / 1542 out tokens · 69882 ms · 2026-05-08T16:42:59.346397+00:00 · methodology

discussion (0)

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