Ill-posed inverse problems frequently arise in scientific and medical imaging, where recovering stable and high-fidelity solutions from incomplete or noisy data remains a central challenge. Motivated by this need, we propose a novel hybrid solver framework, the Neural-Enhanced Two-Step Modified Newton–Lavrentiev Method (NE-TSMNLM), which integrates deep neural corrections into the classical Two-Step Modified Newton–Lavrentiev Method for solving nonlinear inverse problems. Unlike black-box neural operators, our design preserves the convergence structure of the classical iteration while embedding neural modules for adaptive correction, regularization, and convergence prediction.
We establish theoretical guarantees on stability and convergence: under mild assumptions, the NE-TSMNLM method inherits the convergence of the classical TSMNLM and improves the effective convergence rate to with . This demonstrates the acceleration effect due to neural corrections, which has been theoretically proven.
We validate the proposed framework on synthetic and medical inverse problems, including low-dose Computed Tomography (CT) reconstruction, where NE-TSMNLM achieves a 50% radiation dose reduction while maintaining structural fidelity. Initial implementations show promising results with slight degradation (e.g., 17.3% error increase) due to untrained modules and data scarcity. We identify clear pathways for improvement using Transformer-based modules, residual-aware training, and scalable synthetic data.
These results position NE-TSMNLM as a structure-preserving neural framework with rigorous mathematical guarantees, bridging classical regularization theory and deep learning for stable, efficient, and interpretable scientific machine learning.
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