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

arXiv: https://arxiv.org/abs/2409.04067 · HTML: https://mlsanity.com/paper/2409.04067/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 52 references that are in the map.
- First posted on arXiv 6 Sep 2024.

## Builds on (most cited first, 20 of 20)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Hamiltonian Monte Carlo for Hierarchical Models](https://mlsanity.com/paper/1312.0906/) (2013)
- [VarNet: Variational Neural Networks for the Solution of Partial Differential Equations](https://mlsanity.com/paper/1912.07443/) (2019)
- [DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretization](https://mlsanity.com/paper/2005.08357/) (2020)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D](https://mlsanity.com/paper/2012.11893/) (2020)
- [Hybrid FEM-NN models: Combining artificial neural networks with the finite element method](https://mlsanity.com/paper/2101.00962/) (2021)
- [Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control](https://mlsanity.com/paper/2105.14094/) (2021)
- [Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies](https://mlsanity.com/paper/2306.17648/) (2023)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Finite Element Method-enhanced Neural Network for Forward and Inverse Problems](https://mlsanity.com/paper/2205.08321/) (2022)
- [Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics](https://mlsanity.com/paper/2207.09908/) (2022)

## Built on by (most cited first, 2 of 2)

- [Deep operator network models for predicting post-burn contraction](https://mlsanity.com/paper/2411.14555/) (2024)
- [Preconditioned Physics-Informed Neural Operator Training](https://mlsanity.com/paper/2609.36216/) (2026)

## Abstract (arXiv)

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.