Residual cascade extreme learning machine for solving partial differential equations

Extreme learning machine (ELM) is a single-hidden-layer feedforward neural network trained using the Moore-Penrose pseudo-inverse. While faster than backpropagation-based networks, ELM suffers from ill-conditioning due to random hidden layer coefficients. This causes the solution that tends to prioritize large components, making it challenging to accurately represent smaller components, thereby hindering its application in solving complex problems. To address this, we propose residual cascade ELM (RC-ELM), a cascaded ELM architecture that mitigates ill-conditioning by passing residuals through multiple ELM blocks. Each block employs regularized singular value decomposition, and the final solution accumulates components obtained from each block. It achieves a cascading representation of the solution at different scales under limited machine precision, alleviating the impact of ill-conditioning on the solution. We applied RC-ELM to solve various partial differential equations (PDEs), including Helmholtz, diffusion, and Korteweg-de Vries equations. Numerical results show that RC-ELM significantly improves accuracy over classical ELM, with the minimum achievable solution error decreasing by about more than four orders of magnitude and approaching machine precision. Also, the method exhibits high spectral convergence and requires only one expensive pseudo-inverse computation, keeping additional costs minimal compared to classical ELM. Comparative analysis with Chebyshev spectral methods further highlights its potential for solving complex PDEs.