Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators
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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.
Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 13 of its 47 references that are in the map. First posted on arXiv 5 Dec 2024.
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References · 13
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- 2020Solving inverse-PDE problems with physics-aware neural networkscited by 27
- 2021MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEscited by 24
- 2021Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inferencecited by 23
- 2014Data-Driven Model Reduction for the Bayesian Solution of Inverse Problemscited by 20
- 2022Variationally Mimetic Operator Networkscited by 19
- 2023Data-Free Learning of Reduced-Order Kinematicscited by 19
- 2021An extended physics informed neural network for preliminary analysis of parametric optimal control problemscited by 15
- 2023A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flowscited by 12
- 2023Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equationscited by 6
- 2022Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximationcited by 4
- 2023h-analysis and data-parallel physics-informed neural networkscited by 4
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