A robust random feature method with Picard iteration for forward and inverse problems of nonlinear partial differential equations

Nonlinear partial differential equations (PDEs) arise ubiquitously in computational mechanics, requiring efficient and robust solvers for forward simulation and inverse identification. Physics-informed neural networks (PINNs) provide a flexible mesh-free framework, but their practical performance is often limited by slow and unstable training. The random feature method (RFM) offers an efficient alternative by recasting PDE solving as a convex linear least-squares problem, thereby avoiding gradient-based iterative optimization. 4existing RFMs for nonlinear PDEs commonly rely on stabilized Newton-type iterations, introducing substantial computational overhead through repeated Jacobian evaluations and line-search backtracking. In this work, we propose a robust RFM-Picard framework for forward and inverse problems governed by nonlinear PDEs. The method combines a space–time partition of unity with Jacobian-free Picard iteration, efficiently treating nonlinear terms without costly Jacobian-related matrix operations. To improve physical fidelity, an adaptive residual balancing strategy is introduced to strengthen the satisfaction of conservation laws and boundary conditions. The framework is assessed on representative nonlinear PDEs, including the nonlinear Helmholtz equation, the viscous Burgers’ equation, and the incompressible Navier–Stokes equations. Numerical results show that the method outperforms standard PINNs in both accuracy and computational efficiency, and achieves substantially higher efficiency than FEM at comparable accuracy. Compared with stabilized Newton-type RFM solvers, it achieves improved efficiency by eliminating Jacobian assembly and line-search overhead. For inverse problems, it remains robust under extremely sparse observations, enabling accurate reconstruction at computational costs comparable to forward simulations. These results demonstrate that RFM-Picard provides an efficient, robust, and accurate mesh-free RFM method for nonlinear PDEs.