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

math.NAMar 16, 2026

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

Authors: 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.

Figures & tables

	$R(u)$	Parameter range
Fisher	$\rho\,u(1-u)$	–
NWS	$\rho\,u(1-u^{q})$	$q>0$
Zeldovich	$\rho\,u^{2}(1-u)$	–
bistable	$\rho\,u(1-u)(u-a)$	$0<a<1$

Table 1. Reaction terms

R(u)

for Fisher’s, NWS, Zeldovich, and bistable equations.

	$c$	$u(\mathbf{x},t)$
Fisher	$5\sqrt{\frac{\rho D}{6}}$	$\frac{1}{\left\{1+\exp{\left[\sqrt{\frac{\rho}{6D}}(\mathbf{n}\cdot\mathbf{x}-ct)\right]}\right\}^{2}}$
NWS	$\frac{q+4}{\sqrt{2q+4}}\sqrt{\rho D}$	$\left\{\frac{1}{2}+\frac{1}{2}\tanh{\left[-\frac{q}{2\sqrt{2q+4}}\sqrt{\frac{\rho}{D}}(\mathbf{n}\cdot\mathbf{x}-ct)\right]}\right\}^{\frac{2}{q}}$
Zeldovich	$\sqrt{\frac{\rho D}{2}}$	$\frac{1}{1+\exp{\left[\sqrt{\frac{\rho}{2D}}(\mathbf{n}\cdot\mathbf{x}-ct)\right]}}$
bistable	$-(1+a)\sqrt{\frac{\rho D}{2}}$	$\frac{1+a}{2}+\frac{1-a}{2}\tanh{\left[\frac{1-a}{4}\sqrt{\frac{2\rho}{D}}(\mathbf{n}\cdot\mathbf{x}-ct)\right]}$

Table 2. Special wave speed

c

and its corresponding closed-form traveling wave solution

u(\mathbf{x},t)

for Fisher’s, NWS, Zeldovich, and bistable equations.

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).

		Fisher	NWS $(q=2)$	Zeldovich	bistable $(a=0.2)$
$\|c-\omega\|$	original	1.31e-4 (8.82e-5)	2.94e-4 (2.39e-4)	4.14e-5 (2.89e-5)	4.23e-5 (4.95e-5)
$\|c-\omega\|$	restricted	1.91e-6 (1.54e-6)	1.98e-6 (1.16e-6)	1.73e-6 (2.24e-6)	1.09e-6 (6.92e-7)

Table 3. Absolute error in predicted wave speed on original and restricted domains.

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).

$q$	3	4
$\|c-\omega\|$	1.50e-6	3.63e-6

Table 4. Absolute error in the predicted wave speed for NWS equation

(q=3,4)

on the restricted domain.

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.

	variable	$\rho$
		$1$	$10^{2}$	$10^{4}$	$10^{6}$
Fisher	$x,y$	$[-5,25]$	$[-1,5]$	$[-1,5]$	$[-1,5]$
	$t$	$[0,10]$	$[0,0.2]$	$[0,0.02]$	$[0,0.002]$
NWS	$x,y$	$[-5,25]$	$[-1,5]$	$[-1,5]$	$[-1,5]$
	$t$	$[0,10]$	$[0,0.2]$	$[0,0.02]$	$[0,0.002]$
Zeldovich	$x,y$	$[-5,25]$	$[-1,5]$	$[-1,5]$	$[-1,5]$

Table 5. Spatial and temporal domains for each reaction-diffusion equation and each value of

\rho

		$\rho$
		$1$	$10^{2}$	$10^{4}$	$10^{6}$
Fisher	original	5.21e-5 (2.41e-5)	6.60e-5 (3.46e-5)	1.76e-4 (1.18e-4)	5.55e-4 (3.75e-4)
	restricted	5.33e-6 (1.52e-6)	7.52e-6 (1.77e-6)	6.49e-6 (1.54e-6)	1.04e-5 (5.36e-6)
NWS	original	1.13e-4 (8.40e-5)	1.54e-4 (1.20e-4)	4.75e-4 (3.86e-4)	1.50e-3 (1.22e-3)
	restricted	6.02e-6 (2.37e-6)	6.14e-6 (2.21e-6)	5.67e-6 (2.05e-6)	1.12e-5 (5.89e-6)
Zeldovich	original	6.55e-5 (2.44e-5)	7.69e-5 (3.71e-5)	2.02e-4 (1.39e-4)	6.37e-4 (4.45e-4)

Table 6.

L_{2}

error for one-dimensional Fisher’s, NWS

(q=2)

, Zeldovich and bistable

(a=0.2)

equations.

		$\rho$
		$1$	$10^{2}$	$10^{4}$	$10^{6}$
Fisher	original	1.80e-4 (9.90e-5)	3.27e-4 (2.07d-4)	3.14e-3 (2.12e-3)	3.06e-2 (2.07e-2)
	restricted	1.82e-5 (5.70e-6)	1.91e-5 (5.13e-6)	5.02e-5 (3.60e-5)	4.47e-4 (3.60e-4)
NWS	original	5.72e-4 (3.74e-4)	1.14e-3 (7.62e-4)	1.15e-2 (7.72e-3)	1.12e-1 (7.35e-2)
	restricted	1.39e-5 (5.52e-6)	1.42e-5 (5.37e-6)	8.10e-5 (4.83e-5)	8.01e-4 (4.83e-4)
Zeldovich	original	2.42e-4 (1.27e-4)	4.39e-4 (2.82e-4)	4.30e-3 (2.95e-3)	4.29e-2 (2.94e-2)

Table 7.

L_{\infty}

error for one-dimensional Fisher’s, NWS

(q=2)

, Zeldovich and bistable

(a=0.2)

equations.

$q$	norm	$\rho$
		$1$	$10^{2}$	$10^{4}$	$10^{6}$
3	$L_{2}$	6.39e-6	5.78e-6	4.39e-6	8.81e-6
	$L_{\infty}$	1.23e-5	1.44e-5	6.15e-5	6.00e-4
4	$L_{2}$	1.36e-5	1.26e-5	8.48e-6	2.08e-5
	$L_{\infty}$	3.67e-5	3.67e-5	1.69e-4	1.53e-3

Table 8.

L_{2}

and

L_{\infty}

errors for NWS equation

(q=3,4)

using scaled TW-PINNs on the restricted domain.

Figure 5. Solution profiles, from left to right, for Fisher’s equation at

T=0.002

, NWS equation

(q=2)

T=0.002

, Zeldovich equation at

T=0.006

, and bistable equation

(a=0.2)

T=0.005

. The dashed black line is the exact solution.

Figure 6. Evolution of log-scale maximum errors over time, from left to right, for Fisher’s, NWS

(q=2)

, Zeldovich, and bistable

(a=0.2)

equations. The rows correspond to

\rho=1,10^{2},10^{4},

and

10^{6}

from top to bottom.

		$\mathbf{n}$	$\rho$
			$1$	$10^{2}$	$10^{4}$	$10^{6}$
Fisher	original	(1,1)	5.07e-5 (2.63e-5)	6.80e-5 (3.82e-5)	1.91e-4 (1.28e-4)	5.35e-4 (3.61e-4)
		(1,3)	5.02e-5 (2.46e-5)	6.58e-5 (3.57e-5)	1.81e-4 (1.21e-4)	5.67e-4 (3.82e-4)
	restricted	(1,1)	5.12e-6 (1.50e-6)	6.50e-6 (1.36e-6)	6.36e-6 (1.48e-6)	9.95e-6 (5.18e-6)
		(1,3)	5.14e-6 (1.50e-6)	6.94e-6 (1.53e-6)	6.36e-6 (1.47e-6)	1.04e-5 (5.55e-6)
NWS	original	(1,1)	1.20e-4 (8.98e-5)	1.65e-4 (1.29e-4)	5.60e-4 (4.54e-4)	5.28e-3 (4.17e-3)

Table 9.

L_{2}

error for two-dimensional Fisher’s, NWS

(q=2)

, Zeldovich and bistable

(a=0.2)

equations.

		$\mathbf{n}$	$\rho$
			$1$	$10^{2}$	$10^{4}$	$10^{6}$
Fisher	original	(1,1)	1.80e-4 (9.90e-5)	3.26e-4 (2.07e-4)	3.10e-3 (2.09e-3)	2.45e-2 (1.66e-2)
		(1,3)	1.80e-4 (9.90e-5)	3.27e-4 (2.07e-4)	3.14e-3 (2.12e-3)	2.96e-2 (2.00e-2)
	restricted	(1,1)	1.84e-5 (5.43e-6)	1.91e-5 (5.13e-6)	4.97e-5 (3.57e-5)	3.55e-4 (2.86e-4)
		(1,3)	1.84e-5 (5.49e-6)	1.91e-5 (5.13e-6)	4.98e-5 (3.57e-5)	4.31e-4 (3.49e-4)
NWS	original	(1,1)	5.21e-4 (4.05e-4)	1.04e-3 (8.29e-4)	1.04e-2 (8.42e-3)	9.92e-2 (7.69e-2)

Table 10.

L_{\infty}

error for two-dimensional Fisher’s, NWS

(q=2)

, Zeldovich and bistable

(a=0.2)

equations.

Figure 7. Absolute error between the exact and scaled TW-PINN solutions in the filled contour plot, 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 $T=0.005$ with $\rho=10^{4}$ (top) and $\rho=10^{6}$ (bottom).

Figure 8. Comparison of the structure of the wave layer in wave-PINN (left), scaled TW-PINN (middle) and scaled gTW-PINN (right).

	norm	$\rho$
		$1$	$10^{2}$	$10^{4}$	$10^{6}$
wave-PINN	$L_{2}$	2.23e-3 (1.67e-3)	3.11e-3 (2.35e-3)	9.72e-3 (7.32e-3)	2.37e-2 (1.44e-2)
	$L_{\infty}$	8.95e-3 (6.75e-3)	1.77e-2 (1.34e-2)	1.71e-1 (1.26e-1)	7.64e-1 (2.94e-1)
scaled TW-PINN	$L_{2}$	5.21e-5 (2.41e-5)	6.60e-5 (3.46e-5)	1.76e-4 (1.18e-4)	5.55e-4 (3.75e-4)
	$L_{\infty}$	1.80e-4 (9.90e-5)	3.27e-4 (2.07e-4)	3.14e-3 (2.12e-3)	3.06e-2 (2.07e-2)

Table 11.

L_{2}

and

L_{\infty}

errors of wave-PINN and scaled TW-PINN for Fisher’s equation.

Figure 9. Solution profiles (top) and log-scale pointwise errors (bottom) for Fisher’s equation with

\rho=10^{6}

at the initial time

t=0

(left) and the final time

T=0.002

(right) approximated by wave-PINN (cyan) and scaled TW-PINN (red). The dashed black line is the exact solution.

Figure 10. Solution profiles for Fisher’s equation (

\rho=10^{2}

) subject to the initial condition (i) the step function (left), (ii) ( 16 ) with

\lambda=2

(middle) and (iii) ( 16 ) with

\lambda=\tfrac{1}{2}

(right) at the final time

T=0.3

approximated by scaled gTW-PINN (red). The dashed black line is the reference solution.

Figure 11. Solution profiles for Fisher’s equation (

\rho=10^{4}

) subject to the initial condition (i) the step function (left), (ii) ( 16 ) with

\lambda=2

(middle) and (iii) ( 16 ) with

\lambda=\tfrac{1}{2}

(right) at the final time

T=0.03

approximated by scaled gTW-PINN (red). The dashed black line is the reference solution.

Reaction-Diffusion Systems

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Apr 20, 2026J/K move · Enter open · S save

Physics-Informed Neural Networks for Biological $2\mathrm{D}{+}t$ Reaction-Diffusion Systems

William Lavery, Jodie A. Cochrane, Christian Olesen, Dagim S. Tadele+2

Dept. of Information Technology Uppsala University Uppsala, Sweden · Dept. of Medical Genetics Oslo University Hospital Oslo, Norway · Dept. of Translational Hematology and Oncology Research, Cleveland Clinic, Cleveland, USA+1

Dynamical Systems Reaction-Diffusion Systems PDE Discovery Physics-Informed ML

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Physics-informed neural networks (PINNs) provide a powerful framework for learning governing equations of dynamical systems from data. Biologically-informed neural networks (BINNs) are a variant of PINNs that preserve the known differential operator structure (e.g., reaction-diffusion) while learning constitutive terms via trainable neural subnetworks, enforced through soft residual penalties. Existing BINN studies are limited to

1\mathrm{D}{+}t

reaction-diffusion systems and focus on forward prediction, using the governing partial differential equation as a regulariser rather than an explicit identification target. Here, we extend BINNs to

2\mathrm{D}{+}t

systems within a PINN framework that combines data preprocessing, BINN-based equation learning, and symbolic regression post-processing for closed-form equation discovery. We demonstrate the framework's real-world applicability by learning the governing equations of lung cancer cell population dynamics from time-lapse microscopy data, recovering

2\mathrm{D}{+}t

reaction-diffusion models from experimental observations. The proposed framework is readily applicable to other spatio-temporal systems, providing a practical and interpretable tool for fast analytic equation discovery from data.

Abstract

Reaction-Diffusion Systems

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