# NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition

arXiv: https://arxiv.org/abs/2302.10255 · HTML: https://mlsanity.com/paper/2302.10255/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 20 Feb 2023.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2011.04602/) (2020)
- [LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data](https://mlsanity.com/paper/2206.09418/) (2022)
- [Physics-informed Deep Super-resolution for Spatiotemporal Data](https://mlsanity.com/paper/2208.01462/) (2022)

## Built on by (most cited first, 11 of 11)

- [RealPDEBench: A Benchmark for Complex Physical Systems with Real-World Data](https://mlsanity.com/paper/2601.01829/) (2026)
- [Deciphering and integrating invariants for neural operator learning with various physical mechanisms](https://mlsanity.com/paper/2311.14361/) (2023)
- [PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations](https://mlsanity.com/paper/2507.02227/) (2025)
- [PGODE: Towards High-quality System Dynamics Modeling](https://mlsanity.com/paper/2311.06554/) (2023)
- [PAPM: A Physics-aware Proxy Model for Process Systems](https://mlsanity.com/paper/2407.05232/) (2024)
- [Spectral-inspired Operator Learning with Limited Data and Unknown Physics](https://mlsanity.com/paper/2505.21573/) (2025)
- [A novel Fourier neural operator framework for classification of multi-sized images: Application to three dimensional digital porous media](https://mlsanity.com/paper/2402.11568/) (2024)
- [How to Re-enable PDE Loss for Physical Systems Modeling Under Partial Observation](https://mlsanity.com/paper/2412.09116/) (2024)
- [A Knowledge-Guided Cross-Modal Feature Fusion Model for Local Traffic Demand Prediction](https://mlsanity.com/paper/2509.06976/) (2025)
- [Stable spectral neural operator for learning stiff PDE systems from limited data](https://mlsanity.com/paper/2512.11686/) (2025)
- [GlueNN: gluing patchwise analytic solutions with neural networks](https://mlsanity.com/paper/2601.05889/) (2026)

## Abstract (arXiv)

Neural networks have shown great potential in accelerating the solution of partial differential equations (PDEs). Recently, there has been a growing interest in introducing physics constraints into training neural PDE solvers to reduce the use of costly data and improve the generalization ability. However, these physics constraints, based on certain finite dimensional approximations over the function space, must resolve the smallest scaled physics to ensure the accuracy and stability of the simulation, resulting in high computational costs from large input, output, and neural networks. This paper proposes a general acceleration methodology called NeuralStagger by spatially and temporally decomposing the original learning tasks into several coarser-resolution subtasks. We define a coarse-resolution neural solver for each subtask, which requires fewer computational resources, and jointly train them with the vanilla physics-constrained loss by simply arranging their outputs to reconstruct the original solution. Due to the perfect parallelism between them, the solution is achieved as fast as a coarse-resolution neural solver. In addition, the trained solvers bring the flexibility of simulating with multiple levels of resolution. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations, which leads to an additional 10~100× speed-up. Moreover, the experiment also shows that the learned model could be well used for optimal control.