solve inverse problems

Ill-posed inverse problems suffer from severe ill-conditioning; conventional methods ensure stability but are limited in modeling capacity and computational efficiency. This paper addresses medical imaging (CT/MRI) and remote sensing applications by proposing a novel data-driven solving paradigm. Our method systematically integrates adversarial regularization with provably convergent linear plug-and-play (PnP) denoisers, thereby guaranteeing theoretical convergence while enhancing reconstruction fidelity. Furthermore, we formulate an end-to-end high-fidelity reconstruction mapping by jointly leveraging deep neural networks and variational inference. Extensive experiments on both synthetic and real-world datasets demonstrate substantial improvements over classical algorithms—including FBP, TV, and state-of-the-art PnP and deep unrolling methods—in terms of accuracy, real-time computational efficiency, numerical stability, and cross-domain generalizability. The proposed framework establishes a new pathway for ill-posed inverse problems that simultaneously delivers strong empirical performance and rigorous theoretical guarantees.

This work systematically investigates the mathematical modeling, well-posedness analysis, and reconstruction methodologies for inverse problems. Addressing canonical examples—including differential equation inversion, deconvolution, computed tomography (CT) reconstruction, and phase retrieval—the paper establishes a unified regularization framework in Hilbert spaces, encompassing classical approaches such as Tikhonov regularization, sparsity-promoting constraints, pseudoinverse solutions, and Bayesian estimation. Building upon this foundation, it proposes a hybrid “classical regularization + deep learning” paradigm, integrating learned regularizers, plug-and-play (PnP) algorithms, and post-processing strategies to synergistically combine data-driven learning with physics-based modeling. The contributions include: (i) a coherent exposition of the methodological evolution from classical to learning-enhanced inverse solvers; (ii) a balanced reconciliation of theoretical interpretability and practical efficacy; and (iii) a general-purpose, theoretically grounded modeling toolkit and scalable solution framework for data-dependent inverse problems.

In imaging inverse problems, existing learning-based regularization methods suffer from architectural and training heterogeneity, hindering fair comparative evaluation. To address this, we propose a modular, configurable unified framework that systematically decouples mainstream supervised and unsupervised approaches into interchangeable reconstruction, regularization, and training strategy modules—enabling cross-method, systematic benchmarking. The framework supports end-to-end reproduction and plug-and-play integration, and is validated across CT and MRI reconstruction tasks, demonstrating generality and extensibility. Our key contributions are: (1) the first standardized, open platform for comparative evaluation of learned regularization; (2) empirical characterization of how design choices affect generalization, data efficiency, and physical consistency; and (3) publicly available code and detailed reproducibility guidelines, establishing a community benchmark and principled design paradigm for future research.

In Bayesian inverse problems, unreliable uncertainty quantification arises from manually specified priors and inaccurate forward models. To address this, we propose a novel method for automatically learning parameter prior distributions directly from noisy indirect observational data. Our approach introduces a bilevel optimization framework: the upper level learns a generative prior mapping—specifically, a Gaussian pushforward in latent space—while the lower level jointly trains a residual neural operator as the forward model. We design a computationally tractable loss function based on empirical approximation of a divergence metric. Evaluated on Darcy flow permeability inversion, the method significantly reduces reliance on ad hoc smoothness assumptions about the prior and on exact knowledge of the forward physics. It enables efficient and robust prior calibration, thereby enhancing both the reliability and generalizability of posterior uncertainty quantification.

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.

This work proposes a self-supervised learning framework for solving inverse problems—such as image reconstruction—in settings where ground-truth reference signals are unavailable. The method trains a solver using only the noisy or incomplete measurements themselves, eliminating the need for paired ground-truth data. By systematically establishing and extending the theoretical foundations of unsupervised inverse problem solving, the approach integrates self-supervised learning with imaging reconstruction techniques to significantly enhance practical applicability. Evaluated across multiple imaging tasks, the proposed method achieves high-quality signal recovery, outperforming conventional handcrafted regularization strategies and approaching the performance of fully supervised learning approaches.

This work addresses the "black-box" nature of convolutional neural networks (CNNs) in solving image inverse problems by proposing LE-MMSE, the first analytically tractable theoretical framework that explicitly incorporates CNN inductive biases. Built upon minimum mean square error (MMSE) estimation, LE-MMSE formally integrates translation equivariance and local receptive field constraints to yield an interpretable and solvable inverse problem model. Theoretical analysis elucidates the fundamental distinction between physics-aware and physics-agnostic estimators and clarifies the role of high-density regions in the training distribution. Extensive experiments across diverse inverse problems, datasets, and mainstream architectures—including U-Net, ResNet, and PatchMLP—demonstrate remarkable alignment between theoretical predictions and actual CNN outputs, achieving PSNR values consistently above 25 dB, thereby validating the effectiveness and broad applicability of the LE-MMSE framework.

This work proposes a unified framework that integrates hierarchical Bayesian inference with data-driven closure learning to address the inverse problem of model calibration in multiphysics systems, where unknown parameters and incomplete dynamical laws pose significant challenges. The approach jointly infers system-specific parameters and shared unknown dynamics across multiple related systems through a hierarchical structure. Neural networks—such as Fourier Neural Operators (FNOs) and parameterized Physics-Informed Neural Networks (PINNs)—are employed to construct closure models for ODEs/PDEs. Efficient posterior inference is achieved via maximum marginal likelihood estimation combined with ensemble Metropolis-adjusted Langevin algorithm (MALA) sampling. Furthermore, an adaptive surrogate forward model and a bilevel optimization strategy are introduced to substantially reduce the computational cost associated with repeated forward solves. Experiments demonstrate that the framework achieves high calibration accuracy while enabling efficient joint modeling and computational acceleration across systems.

This study addresses the need for a unified understanding of the relationships and generalization capabilities across data-driven modeling paradigms, ranging from classical inverse problems to modern neural operators. By integrating inverse problem theory, sparse identification of dynamical systems, neural ordinary differential equations, and neural operators—and further incorporating the philosophical notion of “mechanism” from philosophy of science—the authors construct a cohesive analytical framework. The work demonstrates that genuine mechanistic discovery and robust cross-scenario generalization are achievable only when models recover concise differential equation structures underlying the observed data. This perspective clarifies the fundamental connections among diverse modeling approaches and underscores the critical role of mechanistic interpretability in enabling reliable generalization, thereby offering a theoretical foundation for the categorization, selection, and design of scientific machine learning models.

This work addresses the lack of a unified theoretical foundation for learned iterative networks in computational imaging and inverse problems. We propose a continuous-domain reconstruction framework grounded in operator learning, which explicitly decouples *how to compute* (algorithmic architecture) from *what to compute* (the reconstruction operator), thereby bridging the theoretical gap between classical optimization-based methods and data-driven models. Methodologically, we integrate variational unfolding, operator modeling in function spaces, deep neural network parameterization, and end-to-end training into a single coherent framework—yielding a learnable, interpretable, and generalizable reconstruction operator. Our approach unifies major classes of learned iterative methods under a common theoretical umbrella. Extensive numerical experiments validate its effectiveness. The framework establishes a new paradigm for designing reconstruction networks that simultaneously offer rigorous theoretical guarantees and strong practical performance.