An Incomplete Constraint Method with Two Feedforward Neural Networks for Solving Linear Partial Differential Equations - Lyu, Yanxia; Li, Xianchi (Journal of Applied Nonlinear Dynamics, 2026)
This paper proposes an incomplete constraint (IC) method, together with extreme learning machines (ELM) and multilayer perceptrons (MLP), to solve linear partial differential equations (PDEs). Unlike the physics-informed neural networks (PINN) approach, the IC method does not enforce strict adherence to initial and boundary conditions within the neural network architecture, which simplifies formulation and enhances accuracy. Specifically, by employing trial functions generated via ELM/MLP and spatiotemporal sampling-based configuration points, PDEs are discretized into weak optimization problems described as nonlinear algebraic equations. And then, optimal parameters are determined through nonlinear optimization algorithms and iterative. Finally, for a comprehensive error distribution analysis, five numerical examples are provided to verify the validity and efficiency of the proposed method. Numerical results demonstrate it has lower error levels and/or computational costs than PINN and radial basis collocation methods.