Learning features from Newton's algorithm: a way to accelerate nonlinear parametrized PDE solvers

2026-07-30

AI Summary

This work addresses the sensitivity to initial guesses and high computational cost of Newton’s method for solving nonlinear parameterized partial differential equations. The authors propose a two-stage initialization strategy: first, by leveraging parameter sampling and a precomputed solution library, they construct two complementary feature spaces—solution manifold and corrected search directions—from discrete Newton trajectories; second, a regression model predicts a surrogate initial guess, which is then refined via lightweight GMRES-based residual minimization to yield a high-quality starting point. Operating under a weakly intrusive framework, this approach significantly accelerates high-fidelity Newton iterations, markedly reducing both iteration counts and total CPU time on benchmark PDE problems, outperforming existing methods that rely solely on surrogate-based initialization.

Abstract

It is well known that Newton's method converges faster when the initial guess is closer to a root of a system of nonlinear equations. In this paper, a two-stage Newton initial guess strategy is proposed by learning features from a parameter-space sampling and a database of precomputed solutions. The method uses discrete Newton trajectories to construct two complementary reduced spaces: a solution feature space, built from converged states, and a corrective search direction feature space, built from intermediate Newton increments. For an unseen parameter, a regression model is used to predict a surrogate solution approximation. Then, in a second step, a residual-minimizing correction is computed using a dedicated GMRES-based approach. The resulting state is then used as an initial guess for the high-fidelity Newton method, which completes convergence. The corrective step is computationally inexpensive since it only requires residual evaluations and the solution of a small least-squares problem. The methodology is weakly intrusive once the high-fidelity residual fields and a script-based programming interface are available. This strategy reduces the number of Newton iterations and decreases the overall CPU time. Numerical experiments on representative PDE problems show quantifiable speedups compared with standalone surrogate initialization. Significant speedups are observed. This generic approach can be applied to a broad class of large-scale nonlinear problems.

Problem

• nonlinear PDE solvers
• Newton's method
• initial guess
• computational acceleration
• parameterized problems

Innovation

• Newton's method
• reduced-order modeling
• surrogate initialization
• GMRES correction
• parameterized PDEs