Error bounds for Physics Informed Neural Networks in Generalized KdV Equations placed on unbounded domains
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918269553737728 |
|---|---|
| author | Freire, Ricardo Muñoz, Claudio Valenzuela, Nicolás |
| author_facet | Freire, Ricardo Muñoz, Claudio Valenzuela, Nicolás |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00779 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Error bounds for Physics Informed Neural Networks in Generalized KdV Equations placed on unbounded domains Freire, Ricardo Muñoz, Claudio Valenzuela, Nicolás Analysis of PDEs Primary: 65K10, 65M99, Secondary: 68T07 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. |
| title | Error bounds for Physics Informed Neural Networks in Generalized KdV Equations placed on unbounded domains |
| topic | Analysis of PDEs Primary: 65K10, 65M99, Secondary: 68T07 |
| url | https://arxiv.org/abs/2601.00779 |