Robust and efficient solvers for nonlinear partial differential equations based on random feature method

TL;DR

The paper addresses the challenge of solving large-scale nonlinear PDEs discretized via the mesh-free Random Feature Method on complex geometries, where classical solvers struggle with ill-conditioned Jacobians. It introduces two randomized Newton-type solvers, IPN and AMIPN, which use randomized right preconditioning and inner-loop reuse of the preconditioned Jacobian to stabilize and speed up convergence, combined with derivative-free line searches. When integrated with RFM (via partition of unity and local random features), these solvers deliver spectral-like convergence and robust performance across 3D steady-state and 2D time-dependent PDEs, outperforming FEM, FDM, PINN, and WAN baselines in accuracy and robustness. The methods show strong potential for large-scale, complex PDE simulations, with future work aimed at rigorous convergence analysis and broader applicability to nonlinear PDE classes.

Abstract

The random feature method (RFM), a mesh-free machine learning-based framework, has emerged as a promising alternative for solving PDEs on complex domains. However, for large three-dimensional nonlinear problems, attaining high accuracy typically requires domain partitioning with many collocation points and random features per subdomain, which leads to extremely large and ill-conditioned nonlinear least-squares systems. To overcome these challenges, we propose two randomized Newton-type solvers. The first is an inexact Newton method with right preconditioning (IPN), in which randomized Jacobian compression and QR factorization are used to construct an efficient preconditioner that substantially reduces the condition number. Each Newton step is then approximately solved by LSQR, and a derivative-free line search is incorporated to ensure residual reduction and stable convergence. Building upon this framework, we further develop an adaptive multi-step inexact preconditioned Newton method (AMIPN). In this approach, the preconditioned Jacobian is reused across multiple inner iterations, while a prescribed maximum number of inner iterations together with an adaptive early-stopping criterion determines whether the current preconditioner can be retained in subsequent outer iterations. These mechanisms effectively avoid redundant computations and enhance robustness. Extensive numerical experiments on both three-dimensional steady-state and two-dimensional time-dependent PDEs with complex geometries confirm the remarkable effectiveness of the proposed solvers. Compared with classical discretization techniques and recent machine-learning-based approaches, the methods consistently deliver substantial accuracy improvements and robust convergence, thereby establishing the RFM combined with IPN/AMIPN as an efficient framework for large-scale nonlinear PDEs. .

Figures (9)

• Figure 1: Left: WAN (Relative $L^2$ vs. training); Right: PINN (Relative $L^2$ and loss vs. training).
• Figure 2: Absolute error on the $x$-$y$ plane at times $t = 0, 0.20, 0.40, 0.60, 0.80, \text{and} 1.00$ for the Allen--Cahn equation \ref{['eq:allen_cahn_movinghole']}
• Figure 3: The evolution of the zero level sets of $F_1(x,y,t)$ and $F_2(x,y,t)$, where the corresponding excised regions first merge and then split over time.
• Figure 4: Absolute error on the spatial cross-sections at $t = 0$, $0.41$, $0.80$, $1.20$, $1.59$, and $2.00$ for the Klein-Gordon equation \ref{['eq:KG_system']}
• Figure 5: Absolute errors of $u$ and $v$ at different time instances $t=0,~0.35,~0.70,~1.00$. The white region indicates the moving hole $\Omega(t)$ transported by the prescribed velocity field.

Theorems & Definitions (2)

• Remark 2.1
• Remark 4.1