Eig-PIELM: A Mesh-Free Approach for Efficient Eigen-Analysis with Physics-Informed Extreme Learning Machines

In this work, a novel Eig-PIELM framework is proposed that extends physics-informed extreme learning machine for an efficient and accurate solution of linear eigenvalue problems. The method reformulates the governing differential equations into a compact algebraic system solvable in a single step. Boundary conditions are enforced exactly via an algebraic projection onto the boundary-admissible subspace, eliminating the computational overhead of penalty parameters, and backpropagation while preserving the computational advantages of extreme learning machines. The proposed framework is mesh-free and yields both eigenvalues and mode shapes simultaneously in one linear solve. The robustness and accuracy of the proposed framework is demonstrated through a range of benchmark problems. We believe that the mesh-free nature, solution structure and accuracy of Eig-PIELM makes it particularly valuable for parametric studies in mechanical, acoustic, and electromechanical systems where rapid frequency spectrum analysis is critical.

💡 Research Summary

The paper introduces Eig‑PIELM, a novel mesh‑free framework that extends the Physics‑Informed Extreme Learning Machine (PIELM) to solve linear eigenvalue problems efficiently and accurately. Traditional eigenvalue solvers rely on discretizing the governing differential operators with finite differences, finite elements, or spectral methods, followed by iterative algorithms such as QR, power iteration, or Krylov subspace methods. Recent attempts to use Physics‑Informed Neural Networks (PINNs) for eigenproblems suffer from non‑convex optimization, sensitivity to hyper‑parameters, and a spectral bias that hampers high‑frequency mode recovery.

PIELM mitigates the training cost by freezing a single hidden layer and solving for output weights via a linear least‑squares problem. However, in eigenvalue problems the residual depends simultaneously on the eigenvalue λ and the output weights β, breaking the single‑step solution property. Eig‑PIELM overcomes this by (1) enforcing boundary conditions exactly through an algebraic projection. A linear map T_b, constructed from the boundary operator B and the basis functions, re‑parameterizes the coefficients as β = T_b y, restricting the search to a boundary‑admissible subspace. This eliminates penalty terms and guarantees that the boundary conditions are satisfied a priori. (2) Formulating the interior residual r_PDE_i(β,λ) = Lϕ_i^T β – λ ϕ_i^T β and expanding the loss J(β,λ) = ½ β^T (A – λP + λ²G) β, where A, P, and G are symmetric matrices built from the differential operator applied to the frozen features. Stationarity with respect to β and λ yields a coupled quadratic‑linear system that, after applying T_b, collapses to a single symmetric generalized eigenvalue problem S_red y = λ G_red y with G_red positive‑definite.

The resulting problem can be solved with any standard eigensolver (e.g., LAPACK’s symmetric generalized eigenvalue routine) in one shot, delivering both eigenvalues and eigenvectors without iterative optimization or back‑propagation. Because the method is mesh‑free, only interior and boundary collocation points are needed; the hidden features can be random, polynomial, or Bernstein basis functions. The computational cost scales with the cube of the number of hidden neurons (typically a few dozen to a few hundred), making the approach suitable for real‑time or many‑query scenarios.

Numerical experiments cover (i) transverse vibration of Euler‑Bernoulli beams with simply supported, fixed‑fixed, and fixed‑free boundary conditions, and (ii) the 2‑D Helmholtz equation on a rectangular domain with homogeneous Neumann boundaries. Using Bernstein polynomials with N_ϕ = 24 hidden neurons and 50 000 interior collocation points, the method recovers the first five eigenfrequencies with absolute errors ranging from 10⁻¹⁶ to 10⁻¹⁰ rad/s—essentially machine precision. Mode shapes match analytical solutions visually and quantitatively. Computation times are on the order of 0.16–0.20 s on a standard laptop CPU, representing a speed‑up of two to three orders of magnitude compared with PINN‑based eigenvalue solvers that require thousands of training epochs.

Key advantages of Eig‑PIELM include exact boundary enforcement without penalty parameters, elimination of hyper‑parameter tuning, a single linear solve for multiple eigenpairs, and mesh‑free flexibility that easily adapts to higher‑order differential operators and mixed boundary conditions. Limitations involve the memory footprint of the Gram matrices for very large hidden layers and potential conditioning issues for extremely high‑dimensional problems, which the authors suggest could be addressed by adaptive basis selection or dimensionality‑reduction techniques.

In conclusion, Eig‑PIELM provides a powerful, accurate, and ultra‑fast tool for eigenvalue analysis across mechanical, acoustic, and electromechanical domains. Its ability to deliver full spectra in a single linear solve makes it especially attractive for parametric studies, design optimization, and real‑time monitoring where rapid frequency analysis is critical. Future work may extend the framework to nonlinear eigenproblems, coupled multiphysics systems, and three‑dimensional geometries, further broadening its impact in computational science and engineering.