Fast PINN Eigensolvers via Biconvex Reformulation

TL;DR

This paper presents a reformulated PINN method for eigenvalue problems that significantly accelerates convergence by transforming the problem into a biconvex optimization, achieving speeds up to 500 times faster than traditional PINN training.

Contribution

It introduces a biconvex reformulation of PINNs for eigenvalue problems, enabling fast, provably convergent alternating convex search with analytical updates.

Findings

PINN-ACS achieves high accuracy in eigenpair computation.

Convergence speed is up to 500 times faster than gradient-based PINNs.

The method is validated through numerical experiments.

Abstract

Eigenvalue problems have a distinctive forward-inverse structure and are fundamental to characterizing a system's thermal response, stability, and natural modes. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative for solving such problems but are often orders of magnitude slower than classical numerical schemes. In this paper, we introduce a reformulated PINN approach that casts the search for eigenpairs as a biconvex optimization problem, enabling fast and provably convergent alternating convex search (ACS) over eigenvalues and eigenfunctions using analytically optimal updates. Numerical experiments show that PINN-ACS attains high accuracy with convergence speeds up to 500$\times$ faster than gradient-based PINN training. We release our codes at https://github.com/NeurIPS-ML4PS-2025/PINN_ACS_CODES.