Corrigendum to “Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations” [Appl. Math. Comput. 483 (2024) 128985] (Applied Mathematics and Computation (2024) 483, (S0096300324004466), (10.1016/j.amc.2024.128985))

While writing our paper [1], we did not adequately discuss the work of Fabiani et al. [2]. Their study investigated extreme learning machines (ELM) combined with pseudo-arclength continuation for bifurcation analysis. We also notice another research employing machine learning for bifurcation analysis, proposed by Galaris et al. [3]. However, their research used feed-forward neural networks and random projection networks to address inverse problems. Specifically, they trained these models using data from Lattice Boltzmann model simulations and subsequently constructed bifurcation diagrams based on the obtained machine learning models. Regarding the work by Fabiani et al. [2], we provide the following remarks: 1. Fabiani et al. [2] were the first to combine a pseudo arclength continuation with ELM. Our approach [1] for constructing bifurcation diagrams builds upon this contribution.2. Regarding simulations, we acknowledge that the equations studied by Fabiani et al. [2] served as excellent benchmark problems for testing continuation methods. We used the same equations in our research.3. In addition to the results we presented in our paper [1], we note that Fabiani et al. [2] also provided comprehensive results demonstrating that ELM can outperform traditional methods, such as finite difference and finite element methods, in terms of accuracy and computational times.4. In [1], we refer to an incomplete bifurcation in the two-dimensional Bratu equation as one that does not sufficiently show the upper branch. However, based on our correspondence with the authors of [2], they confirmed that they had produced the complete bifurcation diagram for the two-dimensional Bratu equation, although it was not shown in their paper because they chose to focus on the turning point.We also have several additional notes that support our paper [1]: 1. We conjecture that if a problem can be solved with ELM, then the problem can also be solved by any neural network. However, it is not immediately apparent that if an ELM can be combined with a pseudo arclength continuation for bifurcation analysis, then any neural network can also be combined with a pseudo arclength continuation for bifurcation analysis. The reason is as follows. Once the inner weights and biases of ELM are fixed, the model becomes (see Eq. (2) in Fabiani et al. [2]): [Formula presented] Here, the model is simplified, as it becomes a linear combination of [Formula presented], for [Formula presented]. This simplicity in approximation implies that the success of such a model, when combined with a pseudo arclength continuation, is insufficient to infer that other more complex neural networks (which require complete updates of all weights and biases) can also be effectively combined with a pseudo arclength continuation. Later, in our research [1], we successfully demonstrated that general neural networks can be combined with a pseudo arclength continuation for bifurcation analysis. We showed that despite entirely manipulating both the inner and outer weights and biases (see Eqs. (42), (43), and (44) in our paper [1]), the shape of the approximate solutions obtained from neural networks remains well preserved across many continuation iterations.2. In addition, the set [Formula presented] in Eq. (1) serves as a basis for constructing the approximate solution u. From this perspective, ELM in Eq. (1) behaves less like other neural networks and more like Fourier series or similar function approximations. For example, in the one-dimensional problems, [Formula presented] can be replaced with sine and cosine functions (as in the Fourier series), leading to: [Formula presented] This approach can be extended to solve two-dimensional problems similarly. This perspective suggests that instead of directly concluding the applicability of a pseudo arclength continuation with neural networks from the results of Fabiani et al. [2], it is more appropriate to infer that other function approximations representable by Eq. (1) can also be combined with a pseudo arclength continuation. For curiosity, we have implemented a combination of the Fourier series with a pseudo arclength continuation for the one- and two-dimensional Bratu equations, which allowed us to compute complete bifurcation diagrams. These results will be reported elsewhere.3. The implementation of combining neural networks with a pseudo arclength continuation is harder compared to combining ELM, which is a linear combination of [Formula presented], [Formula presented], with a pseudo arclength continuation.4. We employed predictor and continuation equations different from those used by Fabiani et al. [2]. Our continuation equation, incorporating [Formula presented], is easier to interpret for the discussed equations. In our implementation, δ (see Eq. (37) in [1]) exactly represents the arclength in the bifurcation diagram plane ([Formula presented] vs μ).5. We also proposed using neural networks for linear stability analysis.