Numerical approximation of partial differential equations by a variable projection method with artificial neural networks

We present a method for solving linear and nonlinear partial differential equations (PDE) based on the variable projection framework artificial neural networks. For PDEs, enforcing boundary/initial value problem collocation points gives rise to separable least squares about network coefficients. reformulate this by approach eliminate output-layer coefficients, leading reduced hidden-layer coefficients only. The is solved first determine then are computed method. that not separable, which precludes strategy such problems. To enable we linearize with Newton iteration, using particular linearization formulated in terms of updated approximation field. linearized system together Upon convergence neural-network provide representation solution field original problem. ample numerical examples demonstrate performance developed herein. smooth solutions, errors current decrease exponentially as number or increases. compare extensively extreme learning machine (ELM) from previous work. Under identical conditions configurations, exhibits an accuracy significantly superior ELM