TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNs

TL;DR

Through incorporation of a shock-capturing loss function component as well as a parameter-dependent transform layer, the TGPT-PINN overcomes the limitations of linear model reduction in the transport-dominated regime.

Abstract

We introduce the Transformed Generative Pre-Trained Physics-Informed Neural Networks (TGPT-PINN) for accomplishing nonlinear model order reduction (MOR) of transport-dominated partial differential equations in an MOR-integrating PINNs framework. Building on the recent development of the GPT-PINN that is a network-of-networks design achieving snapshot-based model reduction, we design and test a novel paradigm for nonlinear model reduction that can effectively tackle problems with parameter-dependent discontinuities. Through incorporation of a shock-capturing loss function component as well as a parameter-dependent transform layer, the TGPT-PINN overcomes the limitations of linear model reduction in the transport-dominated regime. We demonstrate this new capability for nonlinear model reduction in the PINNs framework by several nontrivial parametric partial differential equations.

Figures (17)

• Figure 1: Shown on the left are $9$ snapshots of $u(x,\mu)$ with various $\mu$ values. On the right are the function at $\mu = 1.0$ and its three kinds of interpolations, the polynomial interpolation from snapshots with $\mu= 0.5, 0.85$ and $1.2$, the EIM approximation, and the TGPT-PINN interpolation.
• Figure 2: The GPT-PINN architecture chen2024gpt. A hyper-reduced network adaptively embeds pre-trained PINNs at the nodes of its sole hidden layer. It then allows a quick online generation of a surrogate solution at any given parameter value.
• Figure 3: The TGPT-PINN design schematic. For any given parameter value $\mu$, a $\mu$-dependent loss is constructed and the coefficients $c_j(\mu)$ and the weights and biases in $T_{\mu,\mu^i}$ are trained.
• Figure 4: Results from Section \ref{['sssec:function-example-1']} paired with rows 4 and 5 of Table \ref{['tab:tgpt_functions']}: Snapshots and EIM histories of convergence for functions with moving kinks. (a, b) $u(x,\mu)=\max(\sin(x+\mu),0)$; and (c, d) $u(x,\mu)=|x+\mu|$ .
• Figure 5: Results from Section \ref{['sssec:function-example-2']} paired with row 6 of Table \ref{['tab:tgpt_functions']}: TGPT-PINN histories of convergence when the number of neurons increases for functions with a moving discontinuity, $u(x, \mu)=\psi\left(\frac{x}{0.4+\mu}-1\right)$. (a) $\mu \in [0,1]$ with 101 equispaced points; (b) $\mu \in [-1, 1]$ with 201 equispaced points.