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

2026-06-16

AI Summary

This work addresses the convergence difficulties and accuracy limitations of standard physics-informed neural networks (PINNs) when solving nonlinear partial differential equations, which stem from nonconvex optimization. The authors propose the LiL-Q method, which employs Bellman–Kalaba quasilinearization to decompose the original problem into a sequence of linear subproblems. These are discretized via collocation within a linear-in-learnable (LiL) trial function space—such as random features, spectral polynomials, or trigonometric bases—and solved directly through a single QR decomposition, thereby circumventing gradient-based optimization. For the first time within the PINN framework, this approach ensures convex optimization at each step and establishes a local Newton–Kantorovich convergence theory, elevating the accuracy ceiling from optimization tolerance to the best-approximation residual of the trial space. Across seven benchmark problems, LiL-Q converges in fewer than ten outer iterations, achieves machine precision when the exact solution lies in the trial space, and matches or exceeds state-of-the-art PINN performance on Navier–Stokes equations with less than 1% of the parameters.

Abstract

We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization. The trial space, which we term Linear-in-Learnables (LiL), comprises representations whose trainable parameters enter linearly, including random-feature extreme learning machines, spectral polynomial bases, and trigonometric expansions, each implemented as a physics-informed neural network. The method thus replaces the nonconvex gradient-based training that limits standard PINNs with a convex per-step solve. We establish local Newton-Kantorovich convergence of the outer iteration to a residual-limited neighborhood under an explicit smallness condition, with the limiting accuracy governed by the best-approximation residual of the trial space rather than by an optimization tolerance. The method, denoted LiL-Q, is assessed on seven benchmarks spanning scalar nonlinear PDEs (Bratu, viscous Burgers, Buckley-Leverett), coupled systems (plane-strain elasticity and the incompressible Navier-Stokes equations in two and three spatial dimensions), and steady-state Darcy flow with heterogeneous permeability. Across these problems, LiL-Q converges in single-digit outer iterations in most cases, even at the coarsest basis sizes and independent of the parameter count. When the exact solution lies in the span of the trial space, the method recovers it to machine precision in a single solve. On the Navier-Stokes benchmarks, it matches or exceeds published PINN solvers with up to two orders of magnitude fewer trainable parameters, without gradient-based optimization.

Problem

• nonlinear PDEs
• physics-informed neural networks
• convex optimization
• quasilinearization
• forward problem

Innovation

• Physics-Informed Neural Networks
• Quasilinearization
• Linear-in-Learnables
• Convex Optimization
• Nonlinear PDEs