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

Ricardo Freire, Claudio Muñoz, Nicolás Valenzuela

TL;DR

This work establishes a rigorous error-bound framework for Physics-Informed Neural Networks approximating the generalized Korteweg-de Vries equation on the unbounded real line. By leveraging Kenig-Ponce-Vega dispersive norms and a PINN residual-contraction approach, the authors prove a quantitative stability result (Theorem MT) that ties neural approximations to true gKdV solutions in the $Y_{k,s}$ framework, with errors controlled by a small residual parameter $\varepsilon$. The theory is complemented by extensive numerical validation across solitons, multi-solitons, breathers, and defocusing mKdV kinks, demonstrating high accuracy and practical viability of PINNs for dispersive PDEs on unbounded domains. The results pave the way for rigorous analysis of long-time dynamics and stability for neural-augmented PDE solvers, while also highlighting current limitations and avenues for future work in higher dimensions, supercritical regimes, and uncertainty quantification.

Abstract

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.

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

TL;DR

This work establishes a rigorous error-bound framework for Physics-Informed Neural Networks approximating the generalized Korteweg-de Vries equation on the unbounded real line. By leveraging Kenig-Ponce-Vega dispersive norms and a PINN residual-contraction approach, the authors prove a quantitative stability result (Theorem MT) that ties neural approximations to true gKdV solutions in the framework, with errors controlled by a small residual parameter . The theory is complemented by extensive numerical validation across solitons, multi-solitons, breathers, and defocusing mKdV kinks, demonstrating high accuracy and practical viability of PINNs for dispersive PDEs on unbounded domains. The results pave the way for rigorous analysis of long-time dynamics and stability for neural-augmented PDE solvers, while also highlighting current limitations and avenues for future work in higher dimensions, supercritical regimes, and uncertainty quantification.

Abstract

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.
Paper Structure (26 sections, 9 theorems, 152 equations, 9 figures, 8 tables)

This paper contains 26 sections, 9 theorems, 152 equations, 9 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. g-KdV models and deep learning techniques
  4. Main results
  5. Method of proof
  6. Numerical testing of Theorem \ref{['MT']}
  7. Preliminaries
  8. Loss function
  9. Activation function
  10. Methodology
  11. Results
  12. The soliton case
  13. The $N$-Soliton case in integrable models
  14. The breather solution
  15. The defocusing mKdV kink case
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $k \in \{2,3,4,5\}$ and let $s \geq s_k$ for $k \in\{3,4,5\}$ and $s>s_k$ for $k=2$. Assume now the following Then there exists a solution $u \in C(I;H^s(\mathbb{R}))\cap Y_{k,s}(I\times\mathbb{R})$ to eq:KdV on $I\times\mathbb{R}$ with initial datum $u_0 \in H^s(\mathbb{R})$ such that for all $R>0$ sufficiently large and constants $C$, there is smallness in the $Y_{k,s}$ norm: Additionally,

Figures (9)

  • Figure 1: Exact $u$ solution and predicted $u_{\text{DNN},\#}$ approximate solution in the 3-solitonic mKdV case at times $t=-3,0$ and $2$.
  • Figure 2: Numerical simulation of breathers. Exact solution $u$ and predicted $u_{\text{DNN},\#}$ in the mKdV breather case with parameters $\alpha=1.3, \beta = 0.2$, and at times $t=-2, 0$ and $1$.
  • Figure 3: Exact and predicted solution in the solitonic case, for different values of $k$.
  • Figure 4: Exact and predicted solution in the 2-solitonic case: left: $k=2$, right: $k=3$.
  • Figure 5: Exact and predicted solution in the 3 solitonic case.
  • ...and 4 more figures

Theorems & Definitions (21)

  • Definition 1.1: Quadratures
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 3.1
  • Definition 3.1
  • Lemma 3.2: Product rule
  • Lemma 3.3: Improved product rule
  • Lemma 3.4
  • Theorem 3.1: LWP gKdV
  • ...and 11 more