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. .

翻译：随机特征法是一种基于机器学习的无网格框架，已成为求解复杂区域上偏微分方程的有前景的替代方案。然而，对于大规模三维非线性问题，要获得高精度通常需要在每个子域内布置大量配置点和随机特征，这会导致形成规模极大且病态的非线性最小二乘系统。为克服这些挑战，我们提出了两种随机化牛顿型求解器。第一种是带右预条件处理的不精确牛顿法，该方法利用随机化雅可比矩阵压缩和QR分解构建高效预条件子，显著降低条件数。每个牛顿步通过LSQR近似求解，并结合无导数线搜索以确保残差下降和稳定收敛。在此框架基础上，我们进一步提出了一种自适应多步不精确预条件牛顿法。该方法中，预条件化雅可比矩阵在多个内迭代中重复使用，同时通过预设的最大内迭代次数与自适应早停准则共同决定当前预条件子能否在后续外迭代中保留。这些机制有效避免了冗余计算并增强了鲁棒性。在具有复杂几何结构的三维稳态与二维时变偏微分方程上的大量数值实验证实了所提求解器的显著有效性。与传统离散化技术及近期基于机器学习的方法相比，这些方法始终能实现显著的精度提升与鲁棒收敛，从而确立了随机特征法与IPN/AMIPN相结合作为大规模非线性偏微分方程高效求解框架的地位。