mlsanityData to 7 Oct 2026

Data-Driven Deep Learning of Partial Differential Equations in Modal Space

arXiv 1910.06948 ↗First posted 15 Oct 2019math.NA cs.LG cs.NAcited by 46

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Abstract (arXiv)

We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in infinite-dimensional space, maps the solution from a current time to a future time and completely characterizes the solution evolution of the underlying unknown PDE. Our recovery strategy relies on approximation of the evolution operator in a properly defined modal space, i.e., generalized Fourier space, in order to reduce the problem to finite dimensions. The finite dimensional approximation is then accomplished by training a deep neural network structure, which is based on residual network (ResNet), using the given data. Error analysis is provided to illustrate the predictive accuracy of the proposed method. A set of examples of different types of PDEs, including inviscid Burgers' equation that develops discontinuity in its solution, are presented to demonstrate the effectiveness of the proposed method.

Key facts. Cited by 46 papers in the map (data through 7 Oct 2026). First posted on arXiv 15 Oct 2019.

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Citing papers per year: 2021: 7; 2022: 11; 2023: 10; 2024: 10; 2025: 5; 2026: 3.

46 papers.

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  2. 2022Wavelet neural operator: a neural operator for parametric partial differential equationscited by 77
  3. 2021Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domainscited by 37
  4. 2023Physics informed WNOcited by 26
  5. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  6. 2023Learning Stochastic Dynamical System via Flow Map Operatorcited by 24
  7. 2024tLaSDI: Thermodynamics-informed latent space dynamics identificationcited by 23
  8. 2021Robust discovery of partial differential equations in complex situationscited by 21
  9. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  10. 2022Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation datacited by 18
  11. 2022WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimmingcited by 18
  12. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
34 more
  1. 2021The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimationcited by 14
  2. 2022A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networkscited by 14
  3. 2023Energy-Dissipative Evolutionary Deep Operator Neural Networkscited by 14
  4. 2022Online Weak-form Sparse Identification of Partial Differential Equationscited by 13
  5. 2023Fourier neural operator for real-time simulation of 3D dynamic urban microclimatecited by 10
  6. 2021DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle methodcited by 9
  7. 2022Deep Learning of Chaotic Systems from Partially-Observed Datacited by 9
  8. 2023Deep-OSG: Deep Learning of Operators in Semigroupcited by 9
  9. 2023Finite Expression Methods for Discovering Physical Laws from Datacited by 9
  10. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  11. 2025FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learningcited by 7
  12. 2023Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applicationscited by 6
  13. 2022Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learningcited by 5
  14. 2024Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systemscited by 5
  15. 2024Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?cited by 5
  16. 2025Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamicscited by 5
  17. 2022Modeling unknown dynamical systems with hidden parameterscited by 4
  18. 2024Adaptive deep density approximation for stochastic dynamical systemscited by 4
  19. 2024Physics-Constrained Learning for PDE Systems with Uncertainty Quantified Port-Hamiltonian Modelscited by 4
  20. 2025RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equationscited by 4
  21. 2023Discovering interpretable Lagrangian of dynamical systems from datacited by 3
  22. 2024Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Datacited by 3
  23. 2022Data-driven synchronization-avoiding algorithms in the explicit distributed structural analysis of soft tissuecited by 2
  24. 2022A variational neural network approach for glacier modelling with nonlinear rheologycited by 2
  25. 2024High Energy Density Radiative Transfer in the Diffusion Regime with Fourier Neural Operatorscited by 2
  26. 2026Semigroup Consistency as a Diagnostic for Learned Physics Simulatorscited by 2
  27. 2021Cell-average based neural network method for hyperbolic and parabolic partial differential equationscited by 1
  28. 2022Learning Fine Scale Dynamics from Coarse Observations via Inner Recurrencecited by 1
  29. 2023Learning Nonautonomous Systems via Dynamic Mode Decompositioncited by 1
  30. 2023Critical Sampling for Robust Evolution Operator Learning of Unknown Dynamical Systemscited by 0
  31. 2024Dynamic Gaussian Graph Operator: Learning parametric partial differential equations in arbitrary discrete mechanics problemscited by 0
  32. 2025BLADE: Bayesian Langevin Active Discovery with Replica Exchange for Identification of Complex Systemscited by 0
  33. 2026WGFINNs: Weak formulation-based GENERIC formalism informed neural networkscited by 0
  34. 2026Modeling Unknown Nonlocal PDE Systems via Flow Map Learningcited by 0