mlsanityData to 7 Oct 2026

NeuralPDE: Modelling Dynamical Systems from Data

arXiv 2111.07671 ↗First posted 15 Nov 2021cs.LGcited by 9

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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.

Key facts. Cited by 9 papers in the map (data through 7 Oct 2026). Builds on 19 of its 34 references that are in the map. First posted on arXiv 15 Nov 2021.

Builds on

19 of 34 references.

References · 19

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2016Identity Mappings in Deep Residual Networkscited by 1,960
  3. 2015Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcastingcited by 1,428
  4. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  5. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  6. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  7. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  8. 2020AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradientscited by 282
  9. 2020WeatherBench: A benchmark dataset for data-driven weather forecastingcited by 272
  10. 2015A Critical Review of Recurrent Neural Networks for Sequence Learningcited by 223
  11. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  12. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
7 more
  1. 2020Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Predictioncited by 137
  2. 2020Learning continuous-time PDEs from sparse data with graph neural networkscited by 53
  3. 2018Data-driven discovery of PDEs in complex datasetscited by 42
  4. 2019Learning Dynamical Systems from Partial Observationscited by 36
  5. 2021Finite Volume Neural Network: Modeling Subsurface Contaminant Transportcited by 14
  6. 2020DeepSTCL: A Deep Spatio-temporal ConvLSTM for Travel Demand Predictioncited by 10
  7. 2019A Distributed Neural Network Architecture for Robust Non-Linear Spatio-Temporal Predictioncited by 7

Built on by

Cited by year

Citing papers per year: 2022: 3; 2023: 2; 2024: 3; 2025: 1.

9 papers.

  1. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  2. 2022Recognition Models to Learn Dynamics from Partial Observations with Neural ODEscited by 8
  3. 2023DynaBench: A benchmark dataset for learning dynamical systems from low-resolution datacited by 8
  4. 2024A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solidscited by 4
  5. 2025Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation lawscited by 3
  6. 2022NeuralSI: Structural Parameter Identification in Nonlinear Dynamical Systemscited by 1
  7. 2023TaylorPDENet: Learning PDEs from non-grid Datacited by 0
  8. 2024GrINd: Grid Interpolation Network for Scattered Observationscited by 0
  9. 2024Entropy stable conservative flux form neural networkscited by 0