Physics-informed MeshGraphNets (PI-MGNs): Neural finite element solvers for non-stationary and nonlinear simulations on arbitrary meshes

TL;DR

This work introduces physics-informed MeshGraphNets (PI-MGNs), a hybrid framework that embeds FEM-based physics into Graph Neural Surrogates to solve non-stationary, nonlinear PDEs on arbitrary meshes. By integrating a FEM-inspired loss with MGNs and augmenting with time-bundling and global features, PI-MGNs achieve fast, differentiable, and generalizable solvers that require little or no simulation data for training. Across 2D and 3D heat-transport problems with varying geometries and material distributions, PI-MGNs match or outperform data-driven baselines while significantly outpacing conventional FEM in speed, especially on large meshes. The results highlight substantial potential for rapid design optimization and multi-physics surrogate modeling in engineering applications, with avenues for Bayesian extensions and broader nonlinearities.

Abstract

Engineering components must meet increasing technological demands in ever shorter development cycles. To face these challenges, a holistic approach is essential that allows for the concurrent development of part design, material system and manufacturing process. Current approaches employ numerical simulations, which however quickly becomes computation-intensive, especially for iterative optimization. Data-driven machine learning methods can be used to replace time- and resource-intensive numerical simulations. In particular, MeshGraphNets (MGNs) have shown promising results. They enable fast and accurate predictions on unseen mesh geometries while being fully differentiable for optimization. However, these models rely on large amounts of expensive training data, such as numerical simulations. Physics-informed neural networks (PINNs) offer an opportunity to train neural networks with partial differential equations instead of labeled data, but have not been extended yet to handle time-dependent simulations of arbitrary meshes. This work introduces PI-MGNs, a hybrid approach that combines PINNs and MGNs to quickly and accurately solve non-stationary and nonlinear partial differential equations (PDEs) on arbitrary meshes. The method is exemplified for thermal process simulations of unseen parts with inhomogeneous material distribution. Further results show that the model scales well to large and complex meshes, although it is trained on small generic meshes only.

Figures (10)

• Figure 1: Given a solution $T_v^n$ and additional properties of the time step $t^n$, a graph $G^n$ is assembled and passed to the pimgn, which output a graph $G^n_{\text{out}}$. This graph contains the approximations $\tilde{T}_v^{n+\tilde{n}}$ of the next time steps $t^{n+\tilde{n}}$, where $\tilde{n} = 1, ..., N_{\text{TB}}$. Using the fem, an element- and test function-wise error is calculated and summed up to a test function-wise error. Finally, the loss is calculated as the mse of all errors to update the pimgn using backpropagation.
• Figure 2: Schematic overview of the experiments. Figure a) shows an example of the L-shaped domain of a 2D linear heat diffusion experiment and Figure b) the initialization of an convex polygon with $7$ random points of the nonlinear experiment. Figure c) depicts a hollow cylinder of the 3D experiment with a heating boundary condition inside the cylinder (red arrows) and an adiabatic boundary condition at the outer surface (red coils).
• Figure 3: Comparison of the pimgn approximation to the fem solution at the last time step $t=1s$ for an unseen problem of the 2D linear experiment of \ref{['subsec:exp_small_meshes']}. \ref{['fig:UHDL']} a) shows the randomly sampled initial condition on a new L-shaped mesh geometry. The relative error of the pimgn in \ref{['fig:UHDL']} b) measures the deviation of the pimgn approximation in \ref{['fig:UHDL']} c) from the ground truth in \ref{['fig:UHDL']} d).
• Figure 4: The pimgn solution is compared to the fem for an nonlinear problem of the experiment of \ref{['subsec:exp_small_meshes']} after $t=0.1s$. The geometry and material distribution (cf. \ref{['fig:UHDNL']} a)) of the part was unseen during training. \ref{['fig:UHDNL']} c) and \ref{['fig:UHDNL']} d) show the approximation of the pimgn respectively the solution of the fem. \ref{['fig:UHDNL']} d) depicts the relative error between both solutions.
• Figure 5: The 3D experiment of \ref{['subsec:exp_small_meshes']} with mixed boundary conditions. \ref{['fig:UHDL3D']} a) shows the solution of the pimgn at the last time step $t=1s$ and \ref{['fig:UHDL3D']} b) the ground truth fem solution. \ref{['fig:UHDL3D']} c) depicts the relative error of \ref{['fig:UHDL3D']} a) compared to \ref{['fig:UHDL3D']} b). A quarter of the 3D hollow cylinders is cut off for visualization in \ref{['fig:UHDL3D']} a), b) and c).