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Nicolás Valenzuela

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math.AP math.NA Machine Learning math.PR Numerical Analysis

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preprint2026arXiv

Error bounds for Physics Informed Neural Networks in Generalized KdV Equations placed on unbounded domains

In this paper we study a rigorous setting for the numerical approximation via deep neural networks of the generalized Korteweg-de Vries (gKdV) model in one dimension, for subcritical and critical nonlinearities, and assuming that the domain is the unbounded real line. The fact that the model is posed on the real line makes the problem difficult from the point of view of learning techniques, since the setting required to model gKdV is structured on intricate oscillatory estimates dating from Kato, Bourgain and Kenig, Ponce and Vega, among others. Therefore, a first task is to adapt the setting of these techniques to the deep learning setting. We shall use a battery of Kenig-Ponce-Vega suitable norms and Physics Informed Neural Networks (PINNs) to describe this approximative scheme, proving rigorous bounds on the approximation for each critical and subcritical gKdV model. We shall use this results to provide clear approximation results in the case of several gKdV nonlinear patterns such as solitons, multi-solitons, breathers, among other solutions.

preprint2026arXiv

Neural Discovery of Strichartz Extremizers

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 Schrödinger 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 $γ=1/q$, where existence is open, the optimization does not converge to any $L^2$ profile: instead, the iterates organize themselves as mKdV breathers $B(0,\cdot;α,1,0,0)$ with growing internal frequency $α$, and the discovered ratio approaches the Frank--Sabin universal lower bound $\widetilde A_{q,r}$ from below with a power-law gap $\simα^{-0.9}$. 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.

preprint2022arXiv

A new approach for the fractional Laplacian via deep neural networks

The fractional Laplacian has been strongly studied during past decades. In this paper we present a different approach for the associated Dirichlet problem, using recent deep learning techniques. In fact, intensively PDEs with a stochastic representation have been understood via neural networks, overcoming the so-called curse of dimensionality. Among these equations one can find parabolic ones in $\mathbb{R}^d$ and elliptic in a bounded domain $D \subset \mathbb{R}^d$. In this paper we consider the Dirichlet problem for the fractional Laplacian with exponent $α\in (1,2)$. We show that its solution, represented in a stochastic fashion can be approximated using deep neural networks. We also check that this approximation does not suffer from the curse of dimensionality.

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