mlsanityData to 7 Oct 2026

TransNet: Transferable Neural Networks for Partial Differential Equations

arXiv 2301.11701 ↗First posted 27 Jan 2023math.NA cs.LG cs.NAcited by 13

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Abstract (arXiv)

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to construct transferable neural feature spaces from purely function approximation perspectives without using PDE information. The construction of the feature space involves re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state of the art methods.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 28 of its 46 references that are in the map. First posted on arXiv 27 Jan 2023.

Builds on

28 of 46 references.

References · 28

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  4. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  5. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  7. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  8. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  9. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  10. 2017PDE-Net: Learning PDEs from Datacited by 304
  11. 2016Understanding Deep Neural Networks with Rectified Linear Unitscited by 253
  12. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
16 more
  1. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  2. 2019Complexity of Linear Regions in Deep Networkscited by 166
  3. 2017Bounding and Counting Linear Regions of Deep Neural Networkscited by 161
  4. 2015On the Equivalence between Kernel Quadrature Rules and Random Feature Expansionscited by 136
  5. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 122
  6. 2020Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyondcited by 107
  7. 2013On the number of response regions of deep feed forward networks with piece-wise linear activationscited by 94
  8. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
  9. 2020Transfer learning based multi-fidelity physics informed deep neural networkcited by 68
  10. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  11. 2021One-Shot Transfer Learning of Physics-Informed Neural Networkscited by 49
  12. 2019Nonlinear Approximation and (Deep) ReLU Networkscited by 45
  13. 2018Empirical Bounds on Linear Regions of Deep Rectifier Networkscited by 23
  14. 2021Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solvercited by 15
  15. 2022Random Feature Models for Learning Interacting Dynamical Systemscited by 10
  16. 2021Compositional Modeling of Nonlinear Dynamical Systems with ODE-based Random Featurescited by 8

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 6; 2026: 5.

13 papers.

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  1. 2025Orthogonal greedy algorithm for linear operator learning with shallow neural networkcited by 3
  2. 2025Solving Nonlinear PDEs with Sparse Radial Basis Function Networkscited by 3
  3. 2024Nonuniform random feature models using derivative informationcited by 2
  4. 2025A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problemscited by 2
  5. 2025Condition Numbers and Eigenvalue Spectra of Shallow Networks on Spherescited by 2
  6. 2024ε-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamicscited by 1
  7. 2025DeepONet Augmented by Randomized Neural Networks for Efficient Operator Learning in PDEscited by 1
  8. 2025SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Lawscited by 1
  9. 2026Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithmscited by 0
  10. 2026Solving and learning advective multiscale Darcian dynamics with the Neural Basis Methodcited by 0
  11. 2026MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functionscited by 0
  12. 2026Structure-Oriented Randomized Neural Networks for Poisson-Nernst-Planck and Poisson-Nernst-Planck-Navier-Stokes Systemscited by 0
1 more
  1. 2026Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domainscited by 0