mlsanityData to 7 Oct 2026

Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space

arXiv 2106.03603 ↗First posted 7 Jun 2021cs.LG cs.NA math.NAcited by 25

Not on the map. Why?

Abstract (arXiv)

We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the learning takes place in modal/Fourier space, the current method conducts the learning and modeling in physical space and uses measurement data as nodal values. We present a DNN structure that has a direct correspondence to the evolution operator of the underlying PDE, thus establishing the existence of the DNN model. The DNN model also does not require any geometric information of the data nodes. Consequently, a trained DNN defines a predictive model for the underlying unknown PDE over structureless grids. A set of examples, including linear and nonlinear scalar PDE, system of PDEs, in both one dimension and two dimensions, over structured and unstructured grids, are presented to demonstrate the effectiveness of the proposed DNN modeling. Extension to other equations such as differential-integral equations is also discussed.

Key facts. Cited by 25 papers in the map (data through 7 Oct 2026). Builds on 16 of its 25 references that are in the map. First posted on arXiv 7 Jun 2021.

Builds on

16 of 25 references.

References · 16

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  4. 2017PDE-Net: Learning PDEs from Datacited by 304
  5. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  6. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  7. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  8. 2018Data Driven Governing Equations Approximation Using Deep Neural Networkscited by 94
  9. 2018Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systemscited by 88
  10. 2017Extracting Sparse High-Dimensional Dynamics from Limited Datacited by 50
  11. 2019Data-Driven Deep Learning of Partial Differential Equations in Modal Spacecited by 46
  12. 2019NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Datacited by 42
4 more
  1. 2018Numerical Aspects for Approximating Governing Equations Using Datacited by 21
  2. 2020Learning reduced systems via deep neural networks with memorycited by 20
  3. 2019Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Datacited by 18
  4. 2020Data-driven learning of non-autonomous systemscited by 7

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 8; 2023: 4; 2024: 6; 2025: 4; 2026: 3.

25 papers.

Sort:

  1. 2023Learning Stochastic Dynamical System via Flow Map Operatorcited by 24
  2. 2022Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation datacited by 18
  3. 2022WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimmingcited by 18
  4. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
  5. 2022Deep Learning of Chaotic Systems from Partially-Observed Datacited by 9
  6. 2023Deep-OSG: Deep Learning of Operators in Semigroupcited by 9
  7. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  8. 2022Semi-analytic PINN methods for singularly perturbed boundary value problemscited by 6
  9. 2022Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learningcited by 5
  10. 2024Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systemscited by 5
  11. 2025Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamicscited by 5
  12. 2024Adaptive deep density approximation for stochastic dynamical systemscited by 4
13 more
  1. 2024Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Datacited by 3
  2. 2025Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation lawscited by 3
  3. 2022Data-driven synchronization-avoiding algorithms in the explicit distributed structural analysis of soft tissuecited by 2
  4. 2024A TVD neural network closure and application to turbulent combustioncited by 2
  5. 2022Learning Fine Scale Dynamics from Coarse Observations via Inner Recurrencecited by 1
  6. 2026Parametric Hyperbolic Conservation Laws: A Unified Framework for Conservation, Entropy Stability, and Hyperbolicitycited by 1
  7. 2022Unsupervised Legendre-Galerkin Neural Network for Singularly Perturbed Partial Differential Equationscited by 0
  8. 2023Critical Sampling for Robust Evolution Operator Learning of Unknown Dynamical Systemscited by 0
  9. 2023A Spectral Approach for Learning Spatiotemporal Neural Differential Equationscited by 0
  10. 2024Entropy stable conservative flux form neural networkscited by 0
  11. 2025BLADE: Bayesian Langevin Active Discovery with Replica Exchange for Identification of Complex Systemscited by 0
  12. 2026Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory datacited by 0
  13. 2026Modeling Unknown Nonlocal PDE Systems via Flow Map Learningcited by 0