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Fast PINN Eigensolvers via Biconvex Reformulation

Akshay Sai Banderwaar, Abhishek Gupta

TL;DR

The paper addresses the slow convergence of Physics-Informed Neural Networks (PINNs) for solving differential eigenvalue problems by introducing a biconvex reformulation that trains only the output layer $W^{[L]}$ while keeping the hidden layers fixed. This makes the loss convex in each block (in $W^{[L]}$ for fixed eigenvalue $\lambda_i$, and in $\lambda_i$ for fixed $W^{[L]}$), enabling alternating convex search (ACS) with analytically solvable subproblems via the Moore–Penrose pseudoinverse (and regularized variants). Theoretical guarantees (Theorem 1) ensure monotonic convergence of the loss, and practical results show speedups up to $500\times$ over gradient-based PINN training across problems including Euler buckling, Helmholtz on an L-shaped domain, and biharmonic plate vibrations. The approach demonstrates high-accuracy eigenpairs with substantial computational gains, and lays groundwork for extending ACS to nonlinear operators and broader inverse problems.

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.

Fast PINN Eigensolvers via Biconvex Reformulation

TL;DR

The paper addresses the slow convergence of Physics-Informed Neural Networks (PINNs) for solving differential eigenvalue problems by introducing a biconvex reformulation that trains only the output layer while keeping the hidden layers fixed. This makes the loss convex in each block (in for fixed eigenvalue , and in for fixed ), enabling alternating convex search (ACS) with analytically solvable subproblems via the Moore–Penrose pseudoinverse (and regularized variants). Theoretical guarantees (Theorem 1) ensure monotonic convergence of the loss, and practical results show speedups up to over gradient-based PINN training across problems including Euler buckling, Helmholtz on an L-shaped domain, and biharmonic plate vibrations. The approach demonstrates high-accuracy eigenpairs with substantial computational gains, and lays groundwork for extending ACS to nonlinear operators and broader inverse problems.

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 faster than gradient-based PINN training. We release our codes at https://github.com/NeurIPS-ML4PS-2025/PINN_ACS_CODES.

Paper Structure

This paper contains 11 sections, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Methodology
  4. PINN Loss for Eigenvalue Problems
  5. Biconvex Reformulation
  6. Alternating Convex Search
  7. Numerical Results
  8. Euler's Buckling
  9. Helmholtz Equation on L-shaped Domain
  10. Biharmonic Equation for Thin Plates
  11. Conclusions

Figures (3)

  • Figure 1: Illustrative plots for Euler's buckling show that PINN-ACS leads to monotonic convergence as predicted by Theorem 1. Gradient descent using Adam yoo2025physicskingma2014adam oscillates due to conflicting loss terms. Tabulated average runtimes indicate substantial speedups with ACS alongside higher accuracy.
  • Figure 2: PINN-ACS closely approximates the ground truth (fine-mesh finite difference solutions) for the Laplace operator. Gradient descent results are omitted due to convergence difficulties within a reasonable time, especially for higher-order modes, consistent with observations in yoo2025physics. In contrast, PINN-ACS achieves monotonic loss convergence in agreement with theory.
  • Figure 3: Convergence trends of PINN-ACS eigenvalues to the ground truth for free vibrations of a thin plate timoshenko1959theory. Gradient descent results excluded as it fails to converge reliably for higher modes.