# Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators

arXiv: https://arxiv.org/abs/2412.04409 · HTML: https://mlsanity.com/paper/2412.04409/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 47 references that are in the map.
- First posted on arXiv 5 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Solving inverse-PDE problems with physics-aware neural networks](https://mlsanity.com/paper/2001.03608/) (2020)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference](https://mlsanity.com/paper/2110.07653/) (2021)
- [Data-Driven Model Reduction for the Bayesian Solution of Inverse Problems](https://mlsanity.com/paper/1403.4290/) (2014)
- [Variationally Mimetic Operator Networks](https://mlsanity.com/paper/2209.12871/) (2022)
- [Data-Free Learning of Reduced-Order Kinematics](https://mlsanity.com/paper/2305.03846/) (2023)
- [An extended physics informed neural network for preliminary analysis of parametric optimal control problems](https://mlsanity.com/paper/2110.13530/) (2021)
- [A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flows](https://mlsanity.com/paper/2310.04690/) (2023)
- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)
- [Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximation](https://mlsanity.com/paper/2210.14764/) (2022)
- [h-analysis and data-parallel physics-informed neural networks](https://mlsanity.com/paper/2302.08835/) (2023)
- [Electrical Impedance Tomography with Deep Calder'on Method](https://mlsanity.com/paper/2304.09074/) (2023)

## Built on by (most cited first, 1 of 1)

- [Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks](https://mlsanity.com/paper/2602.14757/) (2026)

## Abstract (arXiv)

We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an autoencoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train an operator network to map the expansion coefficients representing the boundary data to the finite element (FE) solution of the PDE. Finally, we connect the autoencoder's decoder to the operator network which enables us to solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method (FEM) in the linear setting and establish an optimal error estimate in the H1-norm. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.