mlsanityData to 7 Oct 2026

A physics-informed operator regression framework for extracting data-driven continuum models

arXiv 2009.11992 ↗First posted 25 Sep 2020physics.comp-ph cs.LG cs.NAcited by 60

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Abstract (arXiv)

The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space, allowing a characterization of differential operators while providing structure which may be used to impose biases related to symmetry, isotropy, and conservation form. We demonstrate the effectiveness of our framework for a variety of physics, including local and nonlocal diffusion processes and single and multiphase flows. For the flow physics we demonstrate this approach leads to a learned operator that generalizes to system characteristics not included in the training sets, such as variable particle sizes, densities, and concentration.

Key facts. Cited by 60 papers in the map (data through 7 Oct 2026). First posted on arXiv 25 Sep 2020.

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Citing papers per year: 2021: 8; 2022: 19; 2023: 10; 2024: 11; 2025: 11; 2026: 1.

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  1. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  2. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  3. 2022Message Passing Neural PDE Solverscited by 368
  4. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  5. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  6. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 97
  7. 2022Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transportcited by 92
  8. 2024Operator Learning: Algorithms and Analysiscited by 84
  9. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 76
  10. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  11. 2024On conditional diffusion models for PDE simulationscited by 56
  12. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
38 more
  1. 2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 53
  2. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  3. 2022SVD Perspectives for Augmenting DeepONet Flexibility and Interpretabilitycited by 35
  4. 2021Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Datacited by 31
  5. 2025DoMINO: A Decomposable Multi-scale Iterative Neural Operator for Modeling Large Scale Engineering Simulationscited by 26
  6. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  7. 2022Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learningcited by 24
  8. 2022Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEscited by 24
  9. 2023A Neural PDE Solver with Temporal Stencil Modelingcited by 22
  10. 2024An operator learning perspective on parameter-to-observable mapscited by 22
  11. 2022Guiding continuous operator learning through Physics-based boundary constraintscited by 21
  12. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  13. 2022Fully probabilistic deep models for forward and inverse problems in parametric PDEscited by 19
  14. 2023Deep Equilibrium Based Neural Operators for Steady-State PDEscited by 19
  15. 2022IAE-Net: Integral Autoencoders for Discretization-Invariant Learningcited by 15
  16. 2023Energy-Dissipative Evolutionary Deep Operator Neural Networkscited by 14
  17. 2023Finite Element Operator Network for Solving Elliptic-type parametric PDEscited by 14
  18. 2022Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEscited by 11
  19. 2023Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layerscited by 11
  20. 2023Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equationscited by 10
  21. 2022A composable machine-learning approach for steady-state simulations on high-resolution gridscited by 8
  22. 2025Structure-Preserving Digital Twins via Conditional Neural Whitney Formscited by 8
  23. 2025The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operatorscited by 8
  24. 2022Semi-supervised Learning of Partial Differential Operators and Dynamical Flowscited by 6
  25. 2022Separable PINN: Mitigating the Curse of Dimensionality in Physics-Informed Neural Networkscited by 5
  26. 2024Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformationscited by 5
  27. 2025Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Modelscited by 5
  28. 2025SciML Agents: Write the Solver, Not the Solutioncited by 5
  29. 2023An Enhanced V-cycle MgNet Model for Operator Learning in Numerical Partial Differential Equationscited by 4
  30. 2023Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Datacited by 4
  31. 2021A composable autoencoder-based iterative algorithm for accelerating numerical simulationscited by 3
  32. 2024Up-sampling-only and Adaptive Mesh-based GNN for Simulating Physical Systemscited by 3
  33. 2022Error-in-variables modelling for operator learningcited by 2
  34. 2024Neural Operator-Based Proxy for Reservoir Simulations Considering Varying Well Settings, Locations, and Permeability Fieldscited by 2
  35. 2024Stratospheric aerosol source inversion: Noise, variability, and uncertainty quantificationcited by 2
  36. 2025Physics-informed deep learning for infectious disease forecastingcited by 2
  37. 2022Data-driven soliton mappings for integrable fractional nonlinear wave equations via deep learning with Fourier neural operatorcited by 1
  38. 2025A Neural Operator based Hybrid Microscale Model for Multiscale Simulation of Rate-Dependent Materialscited by 1

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