RDNN+: A Robust Neural Network Framework for Solving Nonlinear Partial Differential Equations

Abstract

Physics-informed neural networks (PINNs) have become a pertinent direction of the mesh-free approach to the solution of nonlinear partial differential equations (PDEs); nevertheless, they deteriorate greatly when it comes to sharp gradients, discontinuities, and delayed dynamics. Specifically, the smoothness bias of neural networks' inherent nature and the consistent application of physical constraints restrict the capability of conventional PINNs to be accurate and robust to shock-dominated systems or memory-dependent systems. In an effort to overcome these problems, the current paper presents RDNN+, a physics-based learning framework combining hybrid physics to solve nonlinear PDEs with both discontinuities and time-delays effects. The given method combines together discontinuity-conscious learning strategy that decomposes the space of solutions adaptively into the smooth and non-smooth subsets and a delay-differential formulation directly incorporated into the physics-based loss. This is a directional application of physical restraints and is better at convergence stability and an increase in shock resolution. RDNN+ is proven to be effective on the viscous Burgers equation in both regular and delay-dependent conditions. According to quantitative assessments, global relative error reduction and better localization of shock fronts are achieved with the hybrid PINNs than with conventional PINNs and hybrid baselines, especially at higher delay parameters. These findings demonstrate the prospect of RDNN+ as a powerful and generalizable model of application in scientific machine learning to discontinuous and memory-based physical systems.