Correction: WF-PINNs: solving forward and inverse problems of burgers equation with steep gradients using weak-form physics-informed neural networks

Correction to: Scientific Reports https://doi.org/10.1038/s41598-025-24427-4, published online 18 November 2025

The original version of this Article contained an error in Equation 2, where the expression “$\frac{\partial u}{\partial t}$” was incorrectly stated as “$\partial u\partial t$”. As a result,

$$\begin{aligned} \int _{0}^{\infty }{\int _{-\infty }^{\infty }{\left({\partial u}{\partial t}+u\frac{\partial u}{\partial x}-v\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}\right)}}\phi \,dxdt=0. \end{aligned}$$

now reads:

$$\begin{aligned} \int _{0}^{\infty }{\int _{-\infty }^{\infty }{\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}-v\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}\right)}}\phi \,dxdt=0. \end{aligned}$$

Furthermore, Figure 8 contained errors, where panel (a) was duplicated from Figure 4, and where panel (b) and panel (c) were duplicated from Figure 3 panels (a) and (b). The original Figure 8 and accompanying legend appear below.

Fig. 8

A comparison of the results from the WF-PINNs and PINNs for solving the Burgers’ equation at different time instants. (a) $t=0.$ (b) $t=0.2.$ (c) $t=0.4.$ (d) $t=0.6.$ (e) $t=0.8.$ (f) $t=1.0$.

In addition, Figure 9 was a duplication of Figure 5. The original Figure 9 and accompanying legend appear below.

Fig. 9

Solutions of the Burgers equation using different PINN methods.

The original Article has been corrected.