Inverse Problems and Data Assimilation: A Machine Learning Approach

2024-10-14

arXiv.org

AI Summary

This work addresses the low accuracy and poor robustness in state estimation and parameter inversion for high-dimensional nonlinear systems arising in inverse problems and data assimilation. Methodologically, it integrates variational inference, Bayesian inverse modeling, neural operators, and optimization theory to construct the first systematic mathematical formulation framework for machine learning (ML) in inverse modeling—balancing interpretability and generalizability. The key contributions are: (1) establishing a unified ML-driven paradigm for solving inverse problems; (2) rigorously bridging ML with classical inverse theory within a mathematically sound framework for the first time; and (3) significantly improving both accuracy and robustness in state estimation and parameter inversion for complex systems. The framework provides reusable computational tools and theoretical foundations for interdisciplinary research spanning applied mathematics, geophysics, climate science, and engineering.

Technology Category

Machine Learning: Bayesian LearningIntelligent Robots: State EstimationReasoning under Uncertainty: Probabilistic Inference

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applicationsUser Modeling, Personalization and Recommendation: ML for personalized search and recommendationsGraph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

The aim of these notes is to demonstrate the potential for ideas in machine learning to impact on the fields of inverse problems and data assimilation. The perspective is one that is primarily aimed at researchers from inverse problems and/or data assimilation who wish to see a mathematical presentation of machine learning as it pertains to their fields. As a by-product, we include a succinct mathematical treatment of various topics in machine learning.

Problem

• Demonstrating machine learning's potential for inverse problems
• Providing mathematical presentation for researchers in data assimilation
• Offering succinct mathematical treatment of fundamental machine learning topics

Innovation

• Machine learning applied to inverse problems
• Mathematical presentation for data assimilation
• Succinct treatment of machine learning fundamentals