Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations
This work addresses the problem of efficient and accurate bifurcation and stability analysis for researchers in computational physics and applied mathematics, though it is incremental as it extends existing neural network methods to new aspects of PDE analysis.
This research tackled the problem of analyzing steady states in nonlinear partial differential equations by proposing a neural network method combined with pseudo-arclength continuation for constructing bifurcation diagrams and solving eigenvalue problems for linear stability analysis. The results demonstrated that the neural network produced better solutions and more accurate bifurcation diagrams compared to a finite difference method, with reasonable computational times, as tested on the Bratu and Burgers equations.
This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams from parameterized nonlinear PDEs. Additionally, a neural network approach is also presented for solving eigenvalue problems to analyze solution linear stability, focusing on identifying the largest eigenvalue. The effectiveness of the proposed neural network is examined through experiments on the Bratu equation and the Burgers equation. Results from a finite difference method are also presented as comparison. Varying numbers of grid points are employed in each case to assess the behavior and accuracy of both the neural network and the finite difference method. The experimental results demonstrate that the proposed neural network produces better solutions, generates more accurate bifurcation diagrams, has reasonable computational times, and proves effective for linear stability analysis.