# Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D

arXiv: https://arxiv.org/abs/2012.11893 · HTML: https://mlsanity.com/paper/2012.11893/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2020.

## Built on by (most cited first, 27 of 27)

- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [MORPH: PDE Foundation Models with Arbitrary Data Modality](https://mlsanity.com/paper/2509.21670/) (2025)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data](https://mlsanity.com/paper/2206.09418/) (2022)
- [Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2405.14099/) (2024)
- [Pi-fusion: Physics-informed diffusion model for learning fluid dynamics](https://mlsanity.com/paper/2406.03711/) (2024)
- [A conditional latent autoregressive recurrent model for generation and forecasting of beam dynamics in particle accelerators](https://mlsanity.com/paper/2403.13858/) (2024)
- [PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations](https://mlsanity.com/paper/2507.02227/) (2025)
- [Metamizer: a versatile neural optimizer for fast and accurate physics simulations](https://mlsanity.com/paper/2410.19746/) (2024)
- [Towards Size-Independent Generalization Bounds for Deep Operator Nets](https://mlsanity.com/paper/2205.11359/) (2022)
- [ST-PINN: A Self-Training Physics-Informed Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2306.09389/) (2023)
- [Time-inversion of spatiotemporal beam dynamics using uncertainty-aware latent evolution reversal](https://mlsanity.com/paper/2408.07847/) (2024)
- [Accelerating hydrodynamic simulations of urban drainage systems with physics-guided machine learning](https://mlsanity.com/paper/2206.01538/) (2022)
- [RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network](https://mlsanity.com/paper/2212.02861/) (2022)
- [LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry](https://mlsanity.com/paper/2302.01518/) (2023)
- [Learning an Implicit Physics Model for Image-based Fluid Simulation](https://mlsanity.com/paper/2508.08254/) (2025)
- [Auxiliary-Tasks Learning for Physics-Informed Neural Network-Based Partial Differential Equations Solving](https://mlsanity.com/paper/2307.06167/) (2023)
- [Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp](https://mlsanity.com/paper/2310.05169/) (2023)
- [Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems](https://mlsanity.com/paper/2409.04067/) (2024)
- [Noisy PDE Training Requires Bigger PINNs](https://mlsanity.com/paper/2507.06967/) (2025)
- [Finite-difference-informed graph network for solving steady-state incompressible flows on block-structured grids](https://mlsanity.com/paper/2406.10534/) (2024)
- [Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditions](https://mlsanity.com/paper/2407.20741/) (2024)
- [Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations](https://mlsanity.com/paper/2603.23072/) (2026)
- [Shaping the Wind: Nested Potentials for Kinematically Admissible Urban Wind Prediction](https://mlsanity.com/paper/2610.07033/) (2026)

## Abstract (arXiv)

Physically plausible fluid simulations play an important role in modern computer graphics and engineering. However, in order to achieve real-time performance, computational speed needs to be traded-off with physical accuracy. Surrogate fluid models based on neural networks have the potential to achieve both, fast fluid simulations and high physical accuracy. However, these approaches rely on massive amounts of training data, require complex pipelines for training and inference or do not generalize to new fluid domains. In this work, we present significant extensions to a recently proposed deep learning framework, which addresses the aforementioned challenges in 2D. We go from 2D to 3D and propose an efficient architecture to cope with the high demands of 3D grids in terms of memory and computational complexity. Furthermore, we condition the neural fluid model on additional information about the fluid's viscosity and density which allows simulating laminar as well as turbulent flows based on the same surrogate model. Our method allows to train fluid models without requiring fluid simulation data beforehand. Inference is fast and simple, as the fluid model directly maps a fluid state and boundary conditions at a moment t to a subsequent fluid state at t+dt. We obtain real-time fluid simulations on a 128x64x64 grid that include various fluid phenomena such as the Magnus effect or Karman vortex streets and generalize to domain geometries not considered during training. Our method indicates strong improvements in terms of accuracy, speed and generalization capabilities over current 3D NN-based fluid models.