# P2C2Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics

arXiv: https://arxiv.org/abs/2411.00040 · HTML: https://mlsanity.com/paper/2411.00040/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 47 references that are in the map.
- First posted on arXiv 29 Oct 2024.

## Builds on (most cited first, 30 of 34)

- [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 to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Eagle: Large-Scale Learning of Turbulent Fluid Dynamics with Mesh Transformers](https://mlsanity.com/paper/2302.10803/) (2023)
- [Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures](https://mlsanity.com/paper/2002.10253/) (2020)
- [Learning to correct spectral methods for simulating turbulent flows](https://mlsanity.com/paper/2207.00556/) (2022)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)

## Built on by (most cited first, 13 of 13)

- [PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems](https://mlsanity.com/paper/2505.01736/) (2025)
- [MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation](https://mlsanity.com/paper/2501.15987/) (2025)
- [PINP: Physics-Informed Neural Predictor with latent estimation of fluid flows](https://mlsanity.com/paper/2504.06070/) (2025)
- [Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learning](https://mlsanity.com/paper/2509.24332/) (2025)
- [StFT: Spatio-temporal Fourier Transformer for Long-term Dynamics Prediction](https://mlsanity.com/paper/2503.11899/) (2025)
- [Spectral-inspired Operator Learning with Limited Data and Unknown Physics](https://mlsanity.com/paper/2505.21573/) (2025)
- [An Attention-based Spatio-Temporal Neural Operator for Evolving Physics](https://mlsanity.com/paper/2506.11328/) (2025)
- [Learnable-Differentiable Finite Volume Solver for Accelerated Simulation of Flows](https://mlsanity.com/paper/2507.01975/) (2025)
- [Physics-based machine learning for mantle convection simulations](https://mlsanity.com/paper/2505.16041/) (2025)
- [OmniFluids: Physics Pre-trained Modeling of Fluid Dynamics](https://mlsanity.com/paper/2506.10862/) (2025)
- [CloDS: Visual-Only Unsupervised Cloth Dynamics Learning in Unknown Conditions](https://mlsanity.com/paper/2602.01844/) (2026)
- [A Multi-Resolution Finite-Volume Inspired Deep Learning Framework for Spatiotemporal Dynamics Prediction](https://mlsanity.com/paper/2607.00460/) (2026)
- [RECAST: A Machine-Learning Framework for Correction and Super-Resolution of Coarse-Grid PDE Solvers](https://mlsanity.com/paper/2608.11572/) (2026)

## Abstract (arXiv)

When solving partial differential equations (PDEs), classical numerical methods often require fine mesh grids and small time stepping to meet stability, consistency, and convergence conditions, leading to high computational cost. Recently, machine learning has been increasingly utilized to solve PDE problems, but they often encounter challenges related to interpretability, generalizability, and strong dependency on rich labeled data. Hence, we introduce a new PDE-Preserved Coarse Correction Network (P2C2Net) to efficiently solve spatiotemporal PDE problems on coarse mesh grids in small data regimes. The model consists of two synergistic modules: (1) a trainable PDE block that learns to update the coarse solution (i.e., the system state), based on a high-order numerical scheme with boundary condition encoding, and (2) a neural network block that consistently corrects the solution on the fly. In particular, we propose a learnable symmetric Conv filter, with weights shared over the entire model, to accurately estimate the spatial derivatives of PDE based on the neural-corrected system state. The resulting physics-encoded model is capable of handling limited training data (e.g., 3--5 trajectories) and accelerates the prediction of PDE solutions on coarse spatiotemporal grids while maintaining a high accuracy. P2C2Net achieves consistent state-of-the-art performance with over 50% gain (e.g., in terms of relative prediction error) across four datasets covering complex reaction-diffusion processes and turbulent flows.