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Bounds on the approximation error for deep neural networks applied to dispersive models: Nonlinear waves

Claudio Muñoz, Nicolás Valenzuela

Abstract

We present a comprehensive framework for deriving rigorous and efficient bounds on the approximation error of deep neural networks in PDE models characterized by branching mechanisms, such as waves, Schrödinger equations, and other dispersive models. This framework utilizes the probabilistic setting established by Henry-Labordère and Touzi. We illustrate this approach by providing rigorous bounds on the approximation error for both linear and nonlinear waves in physical dimensions $d=1,2,3$, and analyze their respective computational costs starting from time zero. We investigate two key scenarios: one involving a linear perturbative source term, and another focusing on pure nonlinear internal interactions.

Bounds on the approximation error for deep neural networks applied to dispersive models: Nonlinear waves

Abstract

We present a comprehensive framework for deriving rigorous and efficient bounds on the approximation error of deep neural networks in PDE models characterized by branching mechanisms, such as waves, Schrödinger equations, and other dispersive models. This framework utilizes the probabilistic setting established by Henry-Labordère and Touzi. We illustrate this approach by providing rigorous bounds on the approximation error for both linear and nonlinear waves in physical dimensions , and analyze their respective computational costs starting from time zero. We investigate two key scenarios: one involving a linear perturbative source term, and another focusing on pure nonlinear internal interactions.
Paper Structure (21 sections, 27 theorems, 330 equations)

This paper contains 21 sections, 27 theorems, 330 equations.

Table of Contents

  1. Introduction and Main Results
  2. Approximation of PDEs via DNN
  3. Setting
  4. Representation of waves
  5. Duhamel's formula for general wave models
  6. Probabilistic representation of linear wave models
  7. Nonlinear waves: the branching mechanism
  8. Main results
  9. Deep Neural Networks: A quick review and new results
  10. Neural Network setting
  11. Review of DNN algebra
  12. Additional contributions in DNN algebra
  13. Universal approximation: the linear case
  14. Setting
  15. Linear perturbation of waves
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $d=1,2,3$. Let $\widetilde{F}$, $f_2$ be bounded continuous functions. Then the problem eq:linear_wave2 has a unique solution $U\in C_b^0([0, \infty)\times\mathbb{R}^d,\mathbb{R})$ of the Duhamel's representation

Theorems & Definitions (63)

  • Theorem 1.1: HLT21
  • Remark 2.1: On the existence of $Z_{t,2}$
  • Lemma 2.2
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7: On the meaning of \ref{['eq:sol-linear']}
  • Definition 3.2: Branching mechanism, see HLT21
  • Theorem 3.3: Theorem 3.2 in HLT21
  • ...and 53 more