Numerical methods in inverse problems

Neueste Paper

• Estimation of instrument and noise parameters for inverse problem based on prior diffusion model

  Jean-Fran\c{c}ois Giovannelli · 13. Februar 2026

  This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior samplin…

• A Unified Framework for Lifted Training and Inversion Approaches

  Xiaoyu Wang, Alexandra Valavanis, Azhir Mahmood, Andreas Mang, Martin Benning, Audrey Repetti · 9. Februar 2026

  The training of deep neural networks predominantly relies on a combination of gradient-based optimisation and back-propagation for the computation of the gradient. While incredibly successful, this approach faces challenges such as vanishing or exploding gradients, difficulties with non-smooth activ…

• Learning Regularization Functionals for Inverse Problems: A Comparative Study

  Johannes Hertrich, Hok Shing Wong, Alexander Denker, Stanislas Ducotterd, Zhenghan Fang, Markus Haltmeier, \v{Z}eljko Kereta, Erich Kobler, Oscar Leong, Mohammad Sadegh Salehi, Carola-Bibiane Sch\"onlieb, Johannes Schwab, Zakhar Shumaylov, Jeremias Sulam, German Sh\^ama Wache, Martin Zach, Yasi Zhang, Matthias J. Ehrhardt, Sebastian Neumayer · 16. Januar 2026

  In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challen…

• Learning Operators with Stochastic Gradient Descent in General Hilbert Spaces

  Lei Shi, Jia-Qi Yang · 13. Januar 2026

  This study investigates leveraging stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We propose weak and strong regularity conditions for the target operator to depict its intrinsic structure and complexity. Under these conditions, we establish upper bounds for con…

• Self-Supervised Learning from Noisy and Incomplete Data

  Juli\'an Tachella, Mike Davies · 7. Januar 2026

  Many important problems in science and engineering involve inferring a signal from noisy and/or incomplete observations, where the observation process is known. Historically, this problem has been tackled using hand-crafted regularization (e.g., sparsity, total-variation) to obtain meaningful estima…

• On the Sample Complexity of Learning for Blind Inverse Problems

  Nathan Buskulic, Luca Calatroni, Lorenzo Rosasco, Silvia Villa · 30. Dezember 2025

  Blind inverse problems arise in many experimental settings where the forward operator is partially or entirely unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner. Recently, data-driven approaches have been proposed to address blind invers…

• On the Inverse Flow Matching Problem in the One-Dimensional and Gaussian Cases

  Alexander Korotin, Gudmund Pammer · 30. Dezember 2025

  This paper studies the inverse problem of flow matching (FM) between distributions with finite exponential moment, a problem motivated by modern generative AI applications such as the distillation of flow matching models. Uniqueness of the solution is established in two cases - the one-dimensional s…

• Learning Generalizable Neural Operators for Inverse Problems

  Adam J. Thorpe, Stepan Tretiakov, Dibakar Roy Sarkar, Krishna Kumar, Ufuk Topcu · 23. Dezember 2025

  Inverse problems challenge existing neural operator architectures because ill-posed inverse maps violate continuity, uniqueness, and stability assumptions. We introduce B2B${}^{-1}$, an inverse basis-to-basis neural operator framework that addresses this limitation. Our key innovation is to decouple…

• An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation

  Alex Alberts, Ilias Bilionis · 19. Dezember 2025

  Many inverse problems require reconstructing physical fields from limited and noisy data while incorporating known governing equations. A growing body of work within probabilistic numerics formalizes such tasks via Bayesian inference in function spaces by assigning a physically meaningful prior to t…

• Self-test loss functions for learning weak-form operators and gradient flows

  Yuan Gao, Quanjun Lang, Fei Lu · 16. Dezember 2025

  The construction of loss functions presents a major challenge in data-driven modeling involving weak-form operators in PDEs and gradient flows, particularly due to the need to select test functions appropriately. We address this challenge by introducing self-test loss functions, which employ test fu…

• Equivariant Test-Time Training with Operator Sketching for Imaging Inverse Problems

  Guixian Xu, Jinglai Li, Junqi Tang · 12. Dezember 2025

  Equivariant Imaging (EI) regularization has become the de-facto technique for unsupervised training of deep imaging networks, without any need of ground-truth data. Observing that the EI-based unsupervised training paradigm currently has significant computational redundancy leading to inefficiency i…

• Residual subspace evolution strategies for nonlinear inverse problems

  Francesco Alemanno · 12. Dezember 2025

  Nonlinear inverse problems often feature noisy, non-differentiable, or expensive residual evaluations that make Jacobian-based solvers unreliable. Popular derivative-free optimizers such as natural evolution strategies (NES) or Powell's NEWUOA still assume smoothness or expend many evaluations to ma…

• Learned iterative networks: An operator learning perspective

  Andreas Hauptmann, Ozan \"Oktem · 10. Dezember 2025

  Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation algorithms for solving variational problems. While the underlying al…

• On the necessity of adaptive regularisation:Optimal anytime online learning on $\boldsymbol{\ell_p}$-balls

  Emmeran Johnson, David Mart\'inez-Rubio, Ciara Pike-Burke, Patrick Rebeschini · 1. Dezember 2025

  We study online convex optimization on $\ell_p$-balls in $\mathbb{R}^d$ for $p > 2$. While always sub-linear, the optimal regret exhibits a shift between the high-dimensional setting ($d > T$), when the dimension $d$ is greater than the time horizon $T$ and the low-dimensional setting ($d \leq T$). …

• Extension and neural operator approximation of the electrical impedance tomography inverse map

  Maarten V. de Hoop, Nikola B. Kovachki, Matti Lassas, Nicholas H. Nelsen · 26. November 2025

  This paper considers the problem of noise-robust neural operator approximation for the solution map of Calder\'on's inverse conductivity problem. In this continuum model of electrical impedance tomography (EIT), the boundary measurements are realized as a noisy perturbation of the Neumann-to-Dirichl…

• An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems

  Fabian Schneider, Duc-Lam Duong, Matti Lassas, Maarten V. de Hoop, Tapio Helin · 25. November 2025

  Score-based diffusion models (SDMs) have emerged as a powerful tool for sampling from the posterior distribution in Bayesian inverse problems. However, existing methods often require multiple evaluations of the forward mapping to generate a single sample, resulting in significant computational costs…

• Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients

  Vikas Dwivedi, Balaji Srinivasan, Monica Sigovan, Bruno Sixou · 25. November 2025

  Physics-informed machine learning frameworks such as Physics-Informed Neural Networks (PINNs) and Physics-Informed Extreme Learning Machines (PI-ELMs) have shown great promise for solving partial differential equations (PDEs) but struggle with localized sharp gradients and singularly perturbed regim…

• Generalized Gradient Norm Clipping & Non-Euclidean $(L_0,L_1)$-Smoothness

  Thomas Pethick, Wanyun Xie, Mete Erdogan, Kimon Antonakopoulos, Tony Silveti-Falls, Volkan Cevher · 21. November 2025

  This work introduces a hybrid non-Euclidean optimization method which generalizes gradient norm clipping by combining steepest descent and conditional gradient approaches. The method achieves the best of both worlds by establishing a descent property under a generalized notion of ($L_0$,$L_1$)-smoot…

• Saving Foundation Flow-Matching Priors for Inverse Problems

  Yuxiang Wan, Ryan Devera, Wenjie Zhang, Ju Sun · 21. November 2025

  Foundation flow-matching (FM) models promise a universal prior for solving inverse problems (IPs), yet today they trail behind domain-specific or even untrained priors. How can we unlock their potential? We introduce FMPlug, a plug-in framework that redefines how foundation FMs are used in IPs. FMPl…

• Solving Imaging Inverse Problems Using Plug-and-Play Denoisers: Regularization and Optimization Perspectives

  Hong Ye Tan, Subhadip Mukherjee, Junqi Tang · 20. November 2025

  Inverse problems lie at the heart of modern imaging science, with broad applications in areas such as medical imaging, remote sensing, and microscopy. Recent years have witnessed a paradigm shift in solving imaging inverse problems, where data-driven regularizers are used increasingly, leading to re…

• On scalable and efficient training of diffusion samplers

  Minkyu Kim, Kiyoung Seong, Dongyeop Woo, Sungsoo Ahn, Minsu Kim · 6. November 2025

  We address the challenge of training diffusion models to sample from unnormalized energy distributions in the absence of data, the so-called diffusion samplers. Although these approaches have shown promise, they struggle to scale in more demanding scenarios where energy evaluations are expensive and…

• Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Method

  Diego Marcondes · 28. Oktober 2025

  Physics-informed statistical learning (PISL) integrates empirical data with physical knowledge to enhance the statistical performance of estimators. While PISL methods are widely used in practice, a comprehensive theoretical understanding of how informed regularization affects statistical properties…

• Graph Neural Regularizers for PDE Inverse Problems

  William Lauga, James Rowbottom, Alexander Denker, \v{Z}eljko Kereta, Moshe Eliasof, Carola-Bibiane Sch\"onlieb · 27. Oktober 2025