Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems

TL;DR

This work tackles efficient surrogate modeling for parametric incompressible flows by coupling finite-element discretization with neural networks and physics-informed training. The authors introduce a left–right preconditioned residual loss to learn the parameter-to-solution map for saddle-point problems, demonstrating substantial speedups and accuracy gains for stationary Stokes and Navier–Stokes equations in 2D. They showcase both non-parametric and parametric cases, including generalization to unseen angles, and extend the approach to a probabilistic inverse problem by embedding a differentiable FEM surrogate into a No-U-Turn MCMC framework to infer the angle from pressure data. The results indicate improved conditioning of the training problem, enhanced generalization, and practical viability for multi-query and inverse problems in fluid dynamics, with promising avenues for larger-scale, time-dependent, and stabilized flow settings.

Abstract

The numerical simulation and optimization of technical systems described by partial differential equations is expensive, especially in multi-query scenarios in which the underlying equations have to be solved for different parameters. A comparatively new approach in this context is to combine the good approximation properties of neural networks (for parameter dependence) with the classical finite element method (for discretization). However, instead of considering the solution mapping of the PDE from the parameter space into the FEM-discretized solution space as a purely data-driven regression problem, so-called physically informed regression problems have proven to be useful. In these, the equation residual is minimized during the training of the neural network, i.e., the neural network "learns" the physics underlying the problem. In this paper, we extend this approach to saddle-point and non-linear fluid dynamics problems, respectively, namely stationary Stokes and stationary Navier-Stokes equations. In particular, we propose a modification of the existing approach: Instead of minimizing the plain vanilla equation residual during training, we minimize the equation residual modified by a preconditioner. By analogy with the linear case, this also improves the condition in the present non-linear case. Our numerical examples demonstrate that this approach significantly reduces the training effort and greatly increases accuracy and generalizability. Finally, we show the application of the resulting parameterized model to a related inverse problem.

Figures (10)

• Figure 1: Schematic of FEM-based neural networks.
• Figure 2: Coarse triangular mesh around the NACA 0012 airfoil.
• Figure 3: Loss values per L-BFGS iteration when training FEM-based NNs with preconditioning (pre) and without preconditioning (nopre) for Stokes (S) and Navier--Stokes (NS) equations with an angle of attack $\lambda = 1^{\circ}$ and different viscosities $\eta$.
• Figure 4: Eigenvalues of the Hessian of the loss function, evaluated after training the FEM-based NNs with preconditioning (pre) and without preconditioning (nopre) for Stokes equations with an angle of attack $\lambda = 1^{\circ}$. Eigenvalues are shown in gray, a kernel density estimate of the respective distribution functions in blue and orange. Note that the y-axis employs symmetric log axis scaling, i.e. logarithmic scaling with a linear scaling exception for values within $[-2,2]$.
• Figure 5: Loss values over time, in logarithmic scaling, when training FEM-based NNs with preconditioning (pre) and without preconditioning (nopre) for the Navier--Stokes (NS) equations with an angle of attack $\lambda = 1^{\circ}$ and viscosity $\eta = 1.0$. The initial construction times of the system matrices in \ref{['ns_mat']} and \ref{['loss_pre_ns']} are shown in cyan, the training times of the iterations in purple.

Theorems & Definitions (2)

• Lemma 1
• proof