Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach

Abstract

We address a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solution is sought in the function space spanned by physics-informed random projection neural networks (PI-RPNNs), and discretized via a collocation approach. These are single-hidden-layer networks with randomly sampled and fixed a priori hidden-layer weights; only the linear output layer weights are optimized, reducing training to a single least-squares solve. This linear output structure enables the direct and explicit formulation of the eigenvalue problem governing the linear stability of stationary solutions. This takes a generalized eigenvalue form, which naturally separates the physical domain interior dynamics from the algebraic constraints imposed by boundary conditions, at no additional training cost and without requiring additional PDE solves. However, the random projection collocation matrix is inherently numerically rank-deficient, rendering naive eigenvalue computation unreliable and contaminating the true eigenvalue spectrum with spurious near-zero modes. To overcome this limitation, we introduce a matrix-free shift-invert Krylov-Arnoldi method that operates directly in weight space, avoiding explicit inversion of the numerically rank-deficient collocation matrix and enabling the reliable computation of several leading eigenpairs of the physical Jacobian - the discretized Frechet derivative of the PDE operator with respect to the solution field, whose eigenvalue spectrum determines linear stability. We further prove that the PI-RPNN-based generalized eigenvalue problem is almost surely regular, guaranteeing solvability with standard eigensolvers, and that the singular values of the random projection collocation matrix decay exponentially for analytic activation functions.

Figures (6)

• Figure 1: Numerical results for the one-dimensional Liouville-Bratu-Gelfand problem (\ref{['eq:Bratu']}). (a) Bifurcation diagram; (b) Dominant eigenvalue of the corresponding generalized eigenproblem. (c) Co-existing solution profiles at $p=3$ corresponding to the lower-stable branch and to the upper-unstable branch. (d) Eigenfunction $\phi$ associated with the dominant eigenvalue for the two coexisting solutions at $p=3$. The lower (stable) branch has dominant eigenvalue $\lambda_1 \approx -4.64$, while the upper (unstable) branch has dominant eigenvalue $\lambda_1 \approx 7.01$.
• Figure 2: Demonstrating the impact of collocation matrix rank-deficiency and the efficacy of the shift-invert formulation. Results for the 1D Liouville-Bratu-Gelfand problem at $p=3$. (a) Singular values of the random projection basis matrix $\Psi \in \mathbb{R}^{50 \times 101}$, exhibiting an empirically observed exponential decay with rate $\sigma_n \approx O(5^{-n/2})$. (b) The dominant eigenvectors computed via finite differences (FD, reference), the proposed shift-invert PI-RPNN method, and a "naive" PI-RPNN approach (computing the physical Jacobian direcly as $J_u=J_w\cdot \Psi^{\dagger}$, and then solving the generelized eigenvalue problem), are visually indistinguishable. (c) Corresponding leading eigenvalues. While all three methods recover the correct dominant eigenvalue, the "naive" approach also produces a cluster of spurious near-zero eigenvalues originating from the truncated singular modes of $\Psi$. (d) Spurious eigenmodes from the "naive" approach for the stable (upper panel) and unstable (lower panel) branches. Shown are ten representative eigenvectors corresponding to near-zero eigenvalues from the cluster in (c).
• Figure 3: Numerical results for the two-dimensional Liouville-Bratu-Gelfand problem (\ref{['eq:Bratu']}). (a) Bifurcation diagram. (b) Dominant eigenvalue of the corresponding generalized eigenproblem. The saddle-node, $p_c \approx 6.80736$, marks the onset of instability for the upper solution branch with positive dominant eigenvalues. (c) Co-existing solution profiles at $p=6$ corresponding to the lower-stable branch and to the upper-unstable branch. (d) Eigenvector, $\phi$, corresponding to the dominant eigenvalue for the two co-existing solutions at $p=6$.
• Figure 4: Numerical results for the FitzHugh-Nagumo problem (\ref{['eq:FHN_PDE']}). (a)-(b) Bifurcation diagrams, showing the dependence of $<u>$ and $<v>$ on the parameter $\varepsilon$. The open circle at $\varepsilon_c \approx 0.9446$ marks the saddle point at which stability switches from the upper stable branch to the lower unstable branch. The black cross at $\varepsilon_H \approx 0.0184$ denotes a Hopf point. (c) Solution profiles of components $u$ and $v$ for $\varepsilon$ values on both sides of the Hopf point: the top subfigure depicts the unstable solution at $\varepsilon=0.01$, and the bottom subfigure displays the stable solution profiles at $\varepsilon=0.03$. The circled lines correspond to numerical solutions computed using finite differences with the same number of equidistant collocation points; (d) Dominant eigenvalues at $\varepsilon=0.01$ and $0.03$ (top and bottom subfigures). At $\varepsilon=0.01$, a pair of complex eigenvalues with positive real parts quantifies the instability of the steady-state solution. At $\varepsilon=0.03$, the corresponding complex pair has negative real parts.
• Figure 5: Coexisting steady-state solutions of the FitzHugh-Nagumo problem (\ref{['eq:FHN_PDE']}) at $\varepsilon=0.8$.(a) The top subfigure shows the solution on the upper stable branch, with negative dominant eigenvalue ($\lambda \approx -0.0495$). The bottom subfigure displays the solution on the lower unstable branch, characterized by a positive dominant eigenvalue ($\lambda \approx 0.1266$). (b) The right panel depicts the corresponding dominant eigenvectors for the $u$ and $v$ components (top: stable branch, bottom: unstable branch). The circled curves in panels (a) and (b) correspond to the finite-difference reference solutions.

Theorems & Definitions (8)

• Proposition 1: Almost sure regularity of the generalized pencil in the PI-RPNN basis
• proof : Sketch of the proof
• Proposition 2: Asymptotic singular value decay of the collocation matrix
• proof : Sketch of the proof
• Remark 3.1
• Remark 3.2
• proof
• proof