# NeuralPDE: Modelling Dynamical Systems from Data

arXiv: https://arxiv.org/abs/2111.07671 · HTML: https://mlsanity.com/paper/2111.07671/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 15 Nov 2021.

## Builds on (most cited first, 19 of 19)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [WeatherBench: A benchmark dataset for data-driven weather forecasting](https://mlsanity.com/paper/2002.00469/) (2020)
- [A Critical Review of Recurrent Neural Networks for Sequence Learning](https://mlsanity.com/paper/1506.00019/) (2015)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Learning continuous-time PDEs from sparse data with graph neural networks](https://mlsanity.com/paper/2006.08956/) (2020)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Learning Dynamical Systems from Partial Observations](https://mlsanity.com/paper/1902.11136/) (2019)
- [Finite Volume Neural Network: Modeling Subsurface Contaminant Transport](https://mlsanity.com/paper/2104.06010/) (2021)
- [DeepSTCL: A Deep Spatio-temporal ConvLSTM for Travel Demand Prediction](https://mlsanity.com/paper/2009.00096/) (2020)
- [A Distributed Neural Network Architecture for Robust Non-Linear Spatio-Temporal Prediction](https://mlsanity.com/paper/1912.11141/) (2019)

## Built on by (most cited first, 9 of 9)

- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)
- [DynaBench: A benchmark dataset for learning dynamical systems from low-resolution data](https://mlsanity.com/paper/2306.05805/) (2023)
- [A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solids](https://mlsanity.com/paper/2408.15408/) (2024)
- [Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws](https://mlsanity.com/paper/2507.01795/) (2025)
- [NeuralSI: Structural Parameter Identification in Nonlinear Dynamical Systems](https://mlsanity.com/paper/2208.12771/) (2022)
- [TaylorPDENet: Learning PDEs from non-grid Data](https://mlsanity.com/paper/2306.14511/) (2023)
- [GrINd: Grid Interpolation Network for Scattered Observations](https://mlsanity.com/paper/2403.19570/) (2024)
- [Entropy stable conservative flux form neural networks](https://mlsanity.com/paper/2411.01746/) (2024)

## Abstract (arXiv)

Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are still very limited, as they do not exploit the knowledge about the dynamical nature of the system, require extensive prior knowledge about the governing equations or are limited to linear or first-order equations. In this work we make the observation that the Method of Lines used to solve PDEs can be represented using convolutions which makes convolutional neural networks (CNNs) the natural choice to parametrize arbitrary PDE dynamics. We combine this parametrization with differentiable ODE solvers to form the NeuralPDE Model, which explicitly takes into account the fact that the data is governed by differential equations. We show in several experiments on toy and real-world data that our model consistently outperforms state-of-the-art models used to learn dynamical systems.