---
title: "Rigorously Certified Neural-Network Computation of Nonlinear Water Waves at Near-Machine Precision"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2611.07395v1"
permalink: "https://feed.astralor.com/items/itm_468e46a2637e9d356461c447"
published: "2026-11-05T03:44:55.000Z"
score: 61
channel: "tech"
tags: ["LLM","Rigorously Certified Neural-Network","Computation","Nonlinear Water Waves","Near-Machine Precision","Physics-informed","Through","Stokes"]
---

# Rigorously Certified Neural-Network Computation of Nonlinear Water Waves at Near-Machine Precision

## At a glance

We present a computational framework that combines machine-learning-based discovery with rigorous computer-assisted numerical certification and formal verification for strongly nonlinear partial differential equations. …

## Key points

- We present a computational framework that combines machine-learning-based discovery with rigorous computer-assisted numerical certification and formal verification for strongly nonlinear partial differential equations.
- We demonstrate the approach for periodic gravity-capillary traveling water waves.
- Physics-informed neural networks are used as a search mechanism, with the wave profile, wave speed and, in continuation experiments, surface tension learned directly from the governing equations without training data.

## Full text

We present a computational framework that combines machine-learning-based discovery with rigorous computer-assisted numerical certification and formal verification for strongly nonlinear partial differential equations. We demonstrate the approach for periodic gravity-capillary traveling water waves. Physics-informed neural networks are used as a search mechanism, with the wave profile, wave speed and, in continuation experiments, surface tension learned directly from the governing equations without training data. Through a staged training and refinement procedure, the neural networks achieve near-machine-precision residuals of the governing equations, even in strongly nonlinear regimes. A single formulation recovers high-amplitude Stokes waves, Crapper waves and Wilton ripples. Continuation from a high-amplitude Wilton ripple in surface tension reveals new high-amplitude candidate solutions. We freeze one candidate as an explicit finite representation and rigorously certify a uniform bound on its governing-equation residual using interval arithmetic, accounting for floating-point roundoff, Fourier truncation, aliasing and grid-to-continuum errors. This independently validates the neural-network accuracy. We formalize the certification argument in Lean, which machine-checks the deductions from certified numerical bounds to the residual theorem. Both the interval certification and formalization were developed with substantial assistance from a large language model, while the validity of the result remains independent of the language model: the numerical bounds are rigorously enclosed by interval arithmetic and the subsequent mathematical deductions are checked by the Lean kernel. Together, these steps provide a pipeline from neural discovery to rigorous numerical certification and machine-checked verification with auditable mathematical guarantees.

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