EGPT-PINN: Entropy-enhanced Generative Pre-Trained Physics Informed Neural Networks for parameterized nonlinear conservation laws

TL;DR

EGPT-PINN addresses parameterized nonlinear conservation laws with shocks by reformulating the PDE in characteristic form and enforcing physics-informed, entropy-aware losses that include Rankine–Hugoniot constraints. It combines a nonlinear parameter-dependent transform layer with a separable offline-online training strategy and an offline greedy snapshot selection to capture parameter-dependent discontinuities efficiently. The method extends TGPT-PINN with an entropy-aware full-order model (EPINN) and demonstrates that, for inviscid Burgers' and 1D Euler equations, it can achieve forward and inverse problem accuracy comparable to full EPINN while using only a handful of neurons and significantly reducing training time. This results in robust shock-capturing, rapid online inference, and effective inverse problem solving, suggesting broad practical impact for parametric hyperbolic PDEs in science and engineering.

Abstract

We propose an entropy-enhanced Generative Pre-Trained Physics-Informed Neural Network with a transform layer (EGPT-PINN) for solving parameterized nonlinear conservation laws. The EGPT-PINN extends the traditional physics-informed neural networks and its recently proposed generative pre-trained strategy for linear model reduction to nonlinear model reduction and shock-capturing domains. By utilizing an adaptive meta-network, a simultaneously trained transform layer, entropy enhancement strategies, implementable shock interaction analysis, and a separable training process, the EGPT-PINN efficiently captures complex parameter-dependent shock formations and interactions. Numerical results of EGPT-PINN applied to the families of inviscid Burgers' equation and the Euler equations, parameterized by their initial conditions, demonstrate the robustness and accuracy of the proposed technique. It accurately solves the viscosity solution via very few neurons without leveraging any {\it a priori} knowledge of the equations or its initial condition. Moreover, via a simple augmentation of the loss function by model-data mismatch, we demonstrate the robustness of EGPT-PINN in solving inverse problems more accurately than the vanilla and entropy-enhanced versions of PINN.

Figures (12)

• Figure 1: The EGPT-PINN design schematic for Burgers' (top) and Euler (bottom) equations. For any given parameter value $\mu$, a $\mu$-dependent loss is constructed and the coefficients $\{c_i(\mu)\}_{i=1}^n$ and the weights and biases in $\{\mathcal{T}_{\mu,\mu^i}\}_{i=1}^n$ are trained.
• Figure 2: Results of Case ${\rm B_{1S}}$. (a) Full EPINN loss at $\mu = 1.0$ and comparison with the exact solution; (b) One-neuron EGPT-PINN loss and comparison with the exact solution at $\mu = 0.75$; (c) One-neuron EGPT-PINN loss and comparison with the exact solution at $\mu = 1.75$.
• Figure 3: Error test across the parameter domain for case ${\rm B_{1S}}$ (a) and ${\rm B_{R}}$ (b).
• Figure 4: Results of Case ${\rm B_{2S}}$. (a-c) Two-neurons EGPT-PINN results for $\mu = (0.75,1.6)$; (d-f) Two-neurons EGPT-PINN results for $\mu =(0.85, 1.84)$.
• Figure 5: Results of Case ${\rm B_{Sm}}$. (a, d) Full EPINN results for $\mu = (1.0, 0)$ , and the one-neuron EGPT-PINN results for $\mu = (0.75, 0.1)$ (b, e), $\mu = (1.25, 0.4)$ (c, f).