Numerical Solution of Stiff ODEs with Physics-Informed RPNNs

2021/08/03 by Evangelos Galaris, Gianluca Fabiani, Galaris, Evangelos +7 · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #60B20 #65D12 #65L04 #68T07 #Advancements in Semiconductor Devices and Circuit Design #Algorithm #Applied mathematics #Benchmark (surveying) #Combinatorics #Computer science #Differential equation #FOS: Computer and information sciences #FOS: Mathematics #Initial value problem #Interval (graph theory) #Machine Learning (cs.LG) #Machine Learning and ELM #Machine epsilon #Mathematical analysis #Mathematical optimization #Mathematics #Model Reduction and Neural Networks #Nonlinear system #Numerical Analysis (math.NA) #Numerical analysis #Ode #Ordinary differential equation #Physics #Simple (philosophy) #Solver #Variable (mathematics) #cs.LG #cs.NA #math.NA #msc:60B20 #msc:65D12 #msc:65L04 #msc:68T07

paper · pdf · doi:10.48550/arxiv.2108.01584

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2021/08/03 · arxiv created 2021/11/24 · arxiv updated 2021/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

We propose a numerical method based on physics-informed Random Projection Neural Networks for the solution of Initial Value Problems (IVPs) of Ordinary Differential Equations (ODEs) with a focus on stiff problems. We address an Extreme Learning Machine with a single hidden layer with radial basis functions having as widths uniformly distributed random variables, while the values of the weights between the input and the hidden layer are set equal to one. The numerical solution of the IVPs is obtained by constructing a system of nonlinear algebraic equations, which is solved with respect to the output weights by the Gauss-Newton method, using a simple adaptive scheme for adjusting the time interval of integration. To assess its performance, we apply the proposed method for the solution of four benchmark stiff IVPs, namely the Prothero-Robinson, van der Pol, ROBER and HIRES problems. Our method is compared with an adaptive Runge-Kutta method based on the Dormand-Prince pair, and a variable-step variable-order multistep solver based on numerical differentiation formulas, as implemented in the ode45 and ode15s MATLAB functions, respectively. We show that the proposed scheme yields good approximation accuracy, thus outperforming ode45 and ode15s, especially in the cases where steep gradients arise. Furthermore, the computational times of our approach are comparable with those of the two MATLAB solvers for practical purposes.

Citations

• Functional-link net computing: theory, system architecture, and functionalities
• Database-friendly random projections: Johnson-Lindenstrauss with binary coins
• Face recognition with radial basis function (RBF) neural networks
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems
• Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations
• Physics-informed machine learning
• Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks

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