mlsanityData to 7 Oct 2026

A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks

arXiv 2210.04338 ↗First posted 9 Oct 2022math.NA cs.LG cs.NAcited by 14

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Abstract (arXiv)

We present a method for computing the inverse parameters and the solution field to inverse parametric PDEs based on randomized neural networks. This extends the local extreme learning machine technique originally developed for forward PDEs to inverse problems. We develop three algorithms for training the neural network to solve the inverse PDE problem. The first algorithm (NLLSQ) determines the inverse parameters and the trainable network parameters all together by the nonlinear least squares method with perturbations (NLLSQ-perturb). The second algorithm (VarPro-F1) eliminates the inverse parameters from the overall problem by variable projection to attain a reduced problem about the trainable network parameters only. It solves the reduced problem first by the NLLSQ-perturb algorithm for the trainable network parameters, and then computes the inverse parameters by the linear least squares method. The third algorithm (VarPro-F2) eliminates the trainable network parameters from the overall problem by variable projection to attain a reduced problem about the inverse parameters only. It solves the reduced problem for the inverse parameters first, and then computes the trainable network parameters afterwards. VarPro-F1 and VarPro-F2 are reciprocal to each other in a sense. The presented method produces accurate results for inverse PDE problems, as shown by the numerical examples herein. For noise-free data, the errors for the inverse parameters and the solution field decrease exponentially as the number of collocation points or the number of trainable network parameters increases, and can reach a level close to the machine accuracy. For noisy data, the accuracy degrades compared with the case of noise-free data, but the method remains quite accurate. The presented method has been compared with the physics-informed neural network method.

Key facts. Cited by 14 papers in the map (data through 7 Oct 2026). Builds on 20 of its 72 references that are in the map. First posted on arXiv 9 Oct 2022.

Builds on

20 of 72 references.

References · 20

  1. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  2. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  3. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  4. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  5. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 78
  6. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  7. 2019DeepMoD: Deep learning for Model Discovery in noisy datacited by 70
  8. 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 66
  9. 2019Data-Driven Deep Learning of Partial Differential Equations in Modal Spacecited by 46
  10. 2018Data-driven discovery of PDEs in complex datasetscited by 42
  11. 2019Distributed physics informed neural network for data-efficient solution to partial differential equationscited by 26
  12. 2021On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elementscited by 25
8 more
  1. 2020Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Datacited by 19
  2. 2020Random Vector Functional Link Networks for Function Approximation on Manifoldscited by 19
  3. 2021A Modified Batch Intrinsic Plasticity Method for Pre-training the Random Coefficients of Extreme Learning Machinescited by 13
  4. 2020Train Like a (Var)Pro: Efficient Training of Neural Networks with Variable Projectioncited by 12
  5. 2017Neural network augmented inverse problems for PDEscited by 11
  6. 2022Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machinecited by 9
  7. 2022Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networkscited by 8
  8. 2021slimTrain -- A Stochastic Approximation Method for Training Separable Deep Neural Networkscited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 2; 2024: 5; 2025: 3; 2026: 4.

14 papers.

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  1. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  2. 2024Nonlinear Discrete-Time Observers with Physics-Informed Neural Networkscited by 8
  3. 2024Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiologycited by 3
  4. 2025Learn Singularly Perturbed Solutions via Homotopy Dynamicscited by 2
  5. 2023Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Timecited by 1
  6. 2024DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Trainingcited by 1
  7. 2026Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equationscited by 1
  8. 2024Error Analysis and Numerical Algorithm for PDE Approximation with Hidden-Layer Concatenated Physics Informed Neural Networkscited by 0
  9. 2024Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEscited by 0
  10. 2025Prescribed-Basis Coefficient-to-Coefficient Neural Operator for Partial Differential Equationscited by 0
  11. 2025Fast PINN Eigensolvers via Biconvex Reformulationcited by 0
  12. 2026RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservationcited by 0
2 more
  1. 2026Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristicscited by 0
  2. 2026Structure-Oriented Randomized Neural Networks for Poisson-Nernst-Planck and Poisson-Nernst-Planck-Navier-Stokes Systemscited by 0