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

arXiv: https://arxiv.org/abs/2210.04338 · HTML: https://mlsanity.com/paper/2210.04338/

## Key facts

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

## Builds on (most cited first, 20 of 20)

- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Distributed physics informed neural network for data-efficient solution to partial differential equations](https://mlsanity.com/paper/1907.08967/) (2019)
- [On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements](https://mlsanity.com/paper/2110.14121/) (2021)
- [Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Data](https://mlsanity.com/paper/2005.07916/) (2020)
- [Random Vector Functional Link Networks for Function Approximation on Manifolds](https://mlsanity.com/paper/2007.15776/) (2020)
- [A Modified Batch Intrinsic Plasticity Method for Pre-training the Random Coefficients of Extreme Learning Machines](https://mlsanity.com/paper/2103.08042/) (2021)
- [Train Like a (Var)Pro: Efficient Training of Neural Networks with Variable Projection](https://mlsanity.com/paper/2007.13171/) (2020)
- [Neural network augmented inverse problems for PDEs](https://mlsanity.com/paper/1712.09685/) (2017)
- [Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks](https://mlsanity.com/paper/2201.09989/) (2022)
- [Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine](https://mlsanity.com/paper/2204.11375/) (2022)
- [slimTrain -- A Stochastic Approximation Method for Training Separable Deep Neural Networks](https://mlsanity.com/paper/2109.14002/) (2021)

## Built on by (most cited first, 14 of 14)

- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)
- [Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.12360/) (2024)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [Learn Singularly Perturbed Solutions via Homotopy Dynamics](https://mlsanity.com/paper/2502.00488/) (2025)
- [Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Time](https://mlsanity.com/paper/2303.12245/) (2023)
- [DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Training](https://mlsanity.com/paper/2410.04344/) (2024)
- [Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations](https://mlsanity.com/paper/2603.01102/) (2026)
- [Error Analysis and Numerical Algorithm for PDE Approximation with Hidden-Layer Concatenated Physics Informed Neural Networks](https://mlsanity.com/paper/2406.06350/) (2024)
- [Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs](https://mlsanity.com/paper/2408.15393/) (2024)
- [Prescribed-Basis Coefficient-to-Coefficient Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2510.10350/) (2025)
- [Fast PINN Eigensolvers via Biconvex Reformulation](https://mlsanity.com/paper/2511.00792/) (2025)
- [RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservation](https://mlsanity.com/paper/2601.14794/) (2026)
- [Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics](https://mlsanity.com/paper/2603.01093/) (2026)
- [Structure-Oriented Randomized Neural Networks for Poisson-Nernst-Planck and Poisson-Nernst-Planck-Navier-Stokes Systems](https://mlsanity.com/paper/2606.19912/) (2026)

## 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.