Stabilizing and Solving Inverse Problems using Data and Machine Learning

2024-12-05

arXiv.org

AI Summary

This work addresses the inverse problem of nonlinear partial differential equations (PDEs) under unknown boundary conditions, aiming to reconstruct the PDE solution from collective boundary measurements (i.e., ensembles of typical solutions) and a single interior-domain observation. To overcome the challenge of integrating heterogeneous multi-source data—where conventional methods fail—we propose an end-to-end framework coupling Proper Orthogonal Decomposition (POD)-based compression, autoencoder-driven nonlinear dimensionality reduction, and physics-constrained neural operators. This is the first approach jointly modeling POD-dominant mode compression, nonlinear latent-space parameterization, and physically consistent operator learning to achieve implicit regularization and stable inversion. We theoretically establish optimal H¹-norm convergence rates for linear cases. Numerical experiments demonstrate that our method significantly outperforms classical regularization techniques in accuracy for nonlinear inverse problems.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

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 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 in the linear setting and establish an optimal error estimate in the $H^1$-norm. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.

Problem

• Reconstruct nonlinear PDE solutions
• Leverage collective boundary data
• Stabilize and solve inverse problems

Innovation

• Proper Orthogonal Decomposition compression
• Autoencoder for low-dimensional structure
• Operator network mapping to PDE solution