Parameter estimation and adaptive solution of the Leray-Burgers equation using physics-informed neural networks

Abstract

This study presents a unified framework that integrates physics-informed neural networks (PINNs) to address both the inverse and forward problems of the one-dimensional Leray-Burgers equation. First, we investigate the inverse problem by empirically determining a physically consistent range of the characteristic wavelength parameter α for which the Leray-Burgers solutions closely approximate those of the inviscid Burgers equation, using PINN-based computational experiments. Next, we solve the forward problem using a PINN architecture where α is dynamically optimized during training via a dedicated subnetwork, Alpha2Net. Crucially, Alpha2Net enforces α to remain within the inverse problem-derived bounds, ensuring physical fidelity while jointly optimizing network parameters (weights and biases). This integrated approach effectively captures complex dynamics, such as shock and rarefaction waves. This study also highlights the effectiveness and efficiency of the Leray-Burgers equation in real practical problems, specifically Traffic State Estimation. © 2025 The Authors