Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients

Seungwan Han, Kwanghyuk Park, Jiaxi Gu, Jae-Hun Jung

Abstract

We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of $n$-dimensional reaction-diffusion equations with various reaction and diffusion coefficients. By applying a scaling transformation with the traveling wave form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation with unit reaction and diffusion coefficients. This reduction leads to the proposed framework, termed scaled TW-PINN, in which a single PINN solver trained on the scaled equation is reused for different coefficient choices and spatial dimensions. We also prove a universal approximation property of the proposed PINN solver for traveling wave solutions. Numerical experiments in one and two dimensions, together with a comparison to the existing wave-PINN method, demonstrate the accuracy, flexibility, and superior performance of scaled TW-PINN. Finally, we explore an extension of the framework to the Fisher's equation with general initial conditions.

A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients

Abstract

We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of -dimensional reaction-diffusion equations with various reaction and diffusion coefficients. By applying a scaling transformation with the traveling wave form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation with unit reaction and diffusion coefficients. This reduction leads to the proposed framework, termed scaled TW-PINN, in which a single PINN solver trained on the scaled equation is reused for different coefficient choices and spatial dimensions. We also prove a universal approximation property of the proposed PINN solver for traveling wave solutions. Numerical experiments in one and two dimensions, together with a comparison to the existing wave-PINN method, demonstrate the accuracy, flexibility, and superior performance of scaled TW-PINN. Finally, we explore an extension of the framework to the Fisher's equation with general initial conditions.
Paper Structure (18 sections, 3 theorems, 44 equations, 11 figures, 11 tables)

This paper contains 18 sections, 3 theorems, 44 equations, 11 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Scaling and traveling wave form of reaction-diffusion equations
  3. Reaction-diffusion equations
  4. Scaling
  5. Dimension-independent traveling wave reduction
  6. Reaction terms and corresponding exact traveling wave solutions
  7. Scaling PINN framework for traveling wave solutions
  8. Architecture
  9. Loss functions
  10. Training configuration and convergence analysis
  11. Solution pipeline
  12. Universal approximation property for traveling wave solutions
  13. Numerical results for scaled TW-PINN
  14. One-dimensional numerical results
  15. Two-dimensional numerical results
  16. ...and 3 more sections

Key Result

Theorem 4.1

Let $\sigma$ be any continuous discriminatory function. Then finite sums of the form are dense in $C(I_n)$. In other words, given any $F \in C(I_n)$ and $\varepsilon>0$, there is a sum, $G(\zeta)$, of above form, for which

Figures (11)

  • Figure 1: Schematic of the PINN architecture.
  • Figure 2: Training behavior on the physical (blue) and spurious (orange) convergence for Fisher's equation: log-scale training loss $\mathcal{L}$ (left) and predicted wave speed $\omega$ (right) compared to the exact wave speed (dashed).
  • Figure 3: Training behavior on the original (orange) and restricted (green) domains for Fisher's equation: log-scale training loss $\mathcal{L}$ (left) and predicted wave speed $\omega$ (right) compared to the exact wave speed (dashed).
  • Figure 4: Solution pipeline of the proposed scaling PINN framework: a scaling transformation, approximation of the scaled equation using a trained PINN solver, and an inverse transformation.
  • Figure 5: Solution profiles, from left to right, for Fisher's equation at $T = 0.002$, NWS equation $(q=2)$ at $T = 0.002$, Zeldovich equation at $T = 0.006$, and bistable equation $(a=0.2)$ at $T = 0.005$. The dashed black line is the exact solution.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 4.1: Universal approximation theorem Cybenko
  • Remark 4.2
  • Lemma 4.3: Universal approximation under constraints
  • proof
  • Theorem 4.4: Universal approximation for traveling wave solutions
  • proof