A Neural Network-Based Framework for Solving Large-Scale Inverse Problems

Abstract

Large-scale inverse problems represent a class of computational challenges that arise across scientific and engineering disciplines when estimating unknown system parameters from observed measurements. These problems are typically ill-posed and computationally demanding, often requiring sophisticated regularization techniques and iterative optimization methods. This paper presents a comprehensive neural network-based framework for efficiently solving large-scale inverse problems by leveraging recent advances in deep learning architectures, optimization strategies, and domain-specific knowledge incorporation. We examine fundamental theoretical concepts, including stability-accuracy trade-offs in neural networks for inverse problems, and demonstrate how deep learning approaches can not only accelerate solutions but also achieve superior performance compared to traditional optimization methods. The framework integrates physics-informed neural networks, generative adversarial networks, and differentiable simulation techniques to handle challenges such as local minima, chaotic behavior, and zero-gradient regions that plague conventional approaches. Through extensive analysis of applications in medical imaging, geophysical inversion, and partial differential equation (PDE)-based problems, we show that neural network methods can reduce computational costs by orders of magnitude while maintaining or even improving solution quality. The paper also addresses current limitations and outlines future research directions for enhancing the robustness, scalability, and theoretical understanding of neural network approaches for large-scale inverse problems.