# DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretization

arXiv: https://arxiv.org/abs/2005.08357 · HTML: https://mlsanity.com/paper/2005.08357/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2020.

## Built on by (most cited first, 30 of 35)

- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
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- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs](https://mlsanity.com/paper/2110.01601/) (2021)
- [Hard Encoding of Physics for Learning Spatiotemporal Dynamics](https://mlsanity.com/paper/2105.00557/) (2021)
- [Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous media](https://mlsanity.com/paper/2203.01514/) (2022)
- [A composable machine-learning approach for steady-state simulations on high-resolution grids](https://mlsanity.com/paper/2210.05837/) (2022)
- [A Thermal Machine Learning Solver For Chip Simulation](https://mlsanity.com/paper/2209.04741/) (2022)
- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)
- [Generating Physical Dynamics under Priors](https://mlsanity.com/paper/2409.00730/) (2024)
- [Distributed Multigrid Neural Solvers on Megavoxel Domains](https://mlsanity.com/paper/2104.14538/) (2021)
- [Diffusion model based data generation for partial differential equations](https://mlsanity.com/paper/2306.11075/) (2023)
- [A Latent space solver for PDE generalization](https://mlsanity.com/paper/2104.02452/) (2021)
- [LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry](https://mlsanity.com/paper/2302.01518/) (2023)
- [ParticleWNN: a Novel Neural Networks Framework for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.12433/) (2023)
- [Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics](https://mlsanity.com/paper/2311.12915/) (2023)
- [When Geoscience Meets Generative AI and Large Language Models: Foundations, Trends, and Future Challenges](https://mlsanity.com/paper/2402.03349/) (2024)
- [A composable autoencoder-based iterative algorithm for accelerating numerical simulations](https://mlsanity.com/paper/2110.03780/) (2021)
- [EPINN-NSE: Enhanced Physics-Informed Neural Networks for Solving Navier-Stokes Equations](https://mlsanity.com/paper/2304.03689/) (2023)
- [Finite Volume Features, Global Geometry Representations, and Residual Training for Deep Learning-based CFD Simulation](https://mlsanity.com/paper/2311.14464/) (2023)
- [Geometry encoding for numerical simulations](https://mlsanity.com/paper/2104.07792/) (2021)
- [Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems](https://mlsanity.com/paper/2409.04067/) (2024)
- [Data-Driven Adaptive Gradient Recovery for Unstructured Finite Volume Computations](https://mlsanity.com/paper/2507.16571/) (2025)
- [MRF-PINN: A Multi-Receptive-Field convolutional physics-informed neural network for solving partial differential equations](https://mlsanity.com/paper/2209.03151/) (2022)
- [NLP Inspired Training Mechanics For Modeling Transient Dynamics](https://mlsanity.com/paper/2211.02716/) (2022)
- [Physics-informed Neural Network Combined with Characteristic-Based Split for Solving Navier-Stokes Equations](https://mlsanity.com/paper/2304.10717/) (2023)

## Abstract (arXiv)

Over the last few decades, existing Partial Differential Equation (PDE) solvers have demonstrated a tremendous success in solving complex, non-linear PDEs. Although accurate, these PDE solvers are computationally costly. With the advances in Machine Learning (ML) technologies, there has been a significant increase in the research of using ML to solve PDEs. The goal of this work is to develop an ML-based PDE solver, that couples important characteristics of existing PDE solvers with ML technologies. The two solver characteristics that have been adopted in this work are: 1) the use of discretization-based schemes to approximate spatio-temporal partial derivatives and 2) the use of iterative algorithms to solve linearized PDEs in their discrete form. In the presence of highly non-linear, coupled PDE solutions, these strategies can be very important in achieving good accuracy, better stability and faster convergence. Our ML-solver, DiscretizationNet, employs a generative CNN-based encoder-decoder model with PDE variables as both input and output features. During training, the discretization schemes are implemented inside the computational graph to enable faster GPU computation of PDE residuals, which are used to update network weights that result into converged solutions. A novel iterative capability is implemented during the network training to improve the stability and convergence of the ML-solver. The ML-Solver is demonstrated to solve the steady, incompressible Navier-Stokes equations in 3-D for several cases such as, lid-driven cavity, flow past a cylinder and conjugate heat transfer.