A Convex Quasilinearization Method for Solving Nonlinear PDEs with Physics-Informed Neural Networks

TL;DR

We propose a convex method (LiL-Q) that solves nonlinear PDEs by quasilinearization into linear subproblems, each solved via direct linear least-squares.

Problem

Standard PINNs suffer from slow convergence due to nonconvex gradient-based optimization, dependence on optimization tolerance, and require many parameters.

Approach

Bellman-Kalaba quasilinearization reduces the nonlinear PDE to a sequence of linear subproblems, each discretized by collocation onto a linear-in-parameters trial space (LiL) and solved via direct linear least-squares QR factorization. Trial spaces include random-feature extreme learning machines, spectral polynomial bases, and trigonometric expansions.

Results & Contribution

On seven benchmarks, the method converges in single-digit outer iterations in most cases, achieving machine precision in a single solve when the exact solution lies in the trial space. On Navier-Stokes benchmarks, it matches or exceeds published PINN solvers with up to two orders of magnitude fewer parameters, without gradient-based optimization.