A Discrete-Time Random Feature Method for Nonlinear Evolution Equations with Implicit-Explicit Runge--Kutta Time Stepping

TL;DR

This paper introduces a discrete-time random feature method combined with an IMEX-RK scheme for solving nonlinear PDEs, achieving high accuracy and efficiency compared to existing approaches.

Contribution

The paper develops a novel discrete-time random feature method with linear least-squares formulation and third-order IMEX-RK time stepping for nonlinear PDEs.

Findings

Achieves relative $L^2$-errors of order $10^{-6}$.

Demonstrates convergence rates consistent with third-order IMEX scheme.

Outperforms IMEX-PINN in accuracy and computational efficiency.

Abstract

We study a discrete-time random feature method for nonlinear, time-dependent partial differential equations. In contrast to continuous-time formulations that treat time as an additional input variable, the method advances the solution step by step, with each time level computed from previously available states. The spatial solution at each step is represented in the random feature trial space, and the time discretization is given by an implicit-explicit Runge--Kutta (IMEX-RK, 4 stages, third-order) scheme. After splitting the operator into linear and nonlinear parts, each stage admits a linear least-squares formulation, which avoids nonlinear least-squares solves. We also derive a global error estimate for the fully discrete method, separating the contributions of the stage-wise RFM approximation, perturbations in the least-squares coefficients, and the temporal discretization. Numerical experiments for the Allen--Cahn, Burgers, Korteweg--De Vries, and Cahn--Hilliard equations show relative $L^2$-errors of order $10^{-6}$ and convergence rates consistent with the third-order IMEX scheme. A comparison with an IMEX-PINN variant shows that the proposed method achieves higher accuracy at substantially lower computational cost.