Inverse Problems

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Inverse problems work backward from observations to estimate the hidden image, physical state, or system parameters that produced them, often when measurements are sparse, noisy, or ambiguous. Methods range from classical regularization and optimization to physics-informed neural networks, learned operators, and generative priors such as diffusion and flow models; Bayesian methods can also represent multiple plausible solutions and their uncertainty. Applications include medical and computational imaging, seismic exploration, spectroscopy, and identifying material or dynamical properties from sensor data. A central challenge is balancing measurement consistency with useful prior information: learned priors can recover detail but may invent unsupported structure, especially when the forward model or data differs from training, so identifiability, stability, and calibrated uncertainty remain important.

Latest papers

246 papers30 papers in the last four weeks, up 36% on the four weeks before. 0.2% of all new papers.

Mar 3, 2026

Inverse Reconstruction of Shock Time Series from Shock Response Spectrum Curves using Machine Learning

The shock response spectrum (SRS) is widely used to characterize the response of single-degree-of-freedom (SDOF) systems to transient accelerations. Because the mapping from acceleration time history to SRS is nonlinear and many-to-one, reconstructing time-domain signals from a target spectrum is inherently ill-posed. Conventional approaches address this problem through iterative optimization, typically representing signals as sums of exponentially decayed sinusoids, but these methods are computationally expensive and constrained by predefined basis functions. We propose a conditional variational autoencoder (CVAE) that learns a data-driven inverse mapping from SRS to acceleration time series. Once trained, the model generates signals consistent with prescribed target spectra without requiring iterative optimization. Experiments demonstrate improved spectral fidelity relative to classical techniques, strong generalization to unseen spectra, and inference speeds three to six orders of magnitude faster. These results establish deep generative modeling as a scalable and efficient approach for inverse SRS reconstruction.

Adam Watts, Andrew Jeon, Destry Newton +1

Los Alamos National Laboratory · University of Rochester

Inverse Problems

Mar 3, 2026

Infinite dimensional generative sensing

Deep generative models have become a standard for modeling priors for inverse problems, going beyond classical sparsity-based methods. However, existing theoretical guarantees are mostly confined to finite-dimensional vector spaces, creating a gap when the physical signals are modeled as functions in Hilbert spaces. This work presents a rigorous framework for generative compressed sensing in Hilbert spaces. We extend the notion of local coherence in an infinite-dimensional setting, to derive optimal, resolution-independent sampling distributions. Thanks to a generalization of the Restricted Isometry Property, we show that stable recovery holds when the number of measurements is proportional to the prior's intrinsic dimension (up to logarithmic factors), independent of the ambient dimension. Finally, numerical experiments on the Darcy flow equation validate our theoretical findings and demonstrate that in severely undersampled regimes, employing lower-resolution generators acts as an implicit regularizer, improving reconstruction stability.

Paolo Angella, Vito Paolo Pastore, Matteo Santacesaria

Compressed Sensing Inverse Problems

Mar 3, 2026

Neural Electromagnetic Fields for High-Resolution Material Parameter Reconstruction

Creating functional Digital Twins, simulatable 3D replicas of the real world, is a central challenge in computer vision. Current methods like NeRF produce visually rich but functionally incomplete twins. The key barrier is the lack of underlying material properties (e.g., permittivity, conductivity). Acquiring this information for every point in a scene via non-contact, non-invasive sensing is a primary goal, but it demands solving a notoriously ill-posed physical inversion problem. Standard remote signals, like images and radio frequencies (RF), deeply entangle the unknown geometry, ambient field, and target materials. We introduce NEMF, a novel framework for dense, non-invasive physical inversion designed to build functional digital twins. Our key insight is a systematic disentanglement strategy. NEMF leverages high-fidelity geometry from images as a powerful anchor, which first enables the resolution of the ambient field. By constraining both geometry and field using only non-invasive data, the original ill-posed problem transforms into a well-posed, physics-supervised learning task. This transformation unlocks our core inversion module: a decoder. Guided by ambient RF signals and a differentiable layer incorporating physical reflection models, it learns to explicitly output a continuous, spatially-varying field of the scene's underlying material parameters. We validate our framework on high-fidelity synthetic datasets. Experiments show our non-invasive inversion reconstructs these material maps with high accuracy, and the resulting functional twin enables high-fidelity physical simulation. This advance moves beyond passive visual replicas, enabling the creation of truly functional and simulatable models of the physical world.

Zhe Chen, Peilin Zheng, Wenshuo Chen +3

Xidian University · The Hong Kong University of Science and Technology

Materials Property Prediction Inverse Problems Digital Twins Physics-Informed ML

Mar 2, 2026

Constrained Particle Seeking: Solving Diffusion Inverse Problems with Just Forward Passes

Diffusion models have gained prominence as powerful generative tools for solving inverse problems due to their ability to model complex data distributions. However, existing methods typically rely on complete knowledge of the forward observation process to compute gradients for guided sampling, limiting their applicability in scenarios where such information is unavailable. In this work, we introduce Constrained Particle Seeking (CPS), a novel gradient-free approach that leverages all candidate particle information to actively search for the optimal particle while incorporating constraints aligned with high-density regions of the unconditional prior. Unlike previous methods that passively select promising candidates, CPS reformulates the inverse problem as a constrained optimization task, enabling more flexible and efficient particle seeking. We demonstrate that CPS can effectively solve both image and scientific inverse problems, achieving results comparable to gradient-based methods while significantly outperforming gradient-free alternatives. Code is available at https://github.com/deng-ai-lab/CPS.

Hongkun Dou, Zike Chen, Zeyu Li +3

Beihang University · Beijing Zhongguancun Academy

Diffusion-Based Inverse Problems Inverse Problems Image Inverse Problems Diffusion Guidance Constrained Generative Modeling

Mar 2, 2026

FAST-DIPS: Adjoint-Free Analytic Steps and Hard-Constrained Likelihood Correction for Diffusion-Prior Inverse Problems

Training-free diffusion priors enable inverse-problem solvers without retraining, but for nonlinear forward operators data consistency often relies on repeated derivatives or inner optimization/MCMC loops with conservative step sizes, incurring many iterations and denoiser/score evaluations. We propose a training-free solver that replaces these inner loops with a hard measurement-space feasibility constraint (closed-form projection) and an analytic, model-optimal step size, enabling a small, fixed compute budget per noise level. Anchored at the denoiser prediction, the correction is approximated via an adjoint-free, ADMM-style splitting with projection and a few steepest-descent updates, using one VJP and either one JVP or a forward-difference probe, followed by backtracking and decoupled re-annealing. We prove local model optimality and descent under backtracking for the step-size rule, and derive an explicit KL bound for mode-substitution re-annealing under a local Gaussian conditional surrogate. We also develop a latent variant and a one-parameter pixel

\rightarrow

latent hybrid schedule. Experiments achieve competitive PSNR/SSIM/LPIPS with up to 19.5

\times

speedup, without hand-coded adjoints or inner MCMC.

Minwoo Kim, Seunghyeok Shin, Hongki Lim

Department of Electrical and Computer Engineering, Inha University Incheon, 22212, South Korea

Diffusion-Based Inverse Problems Inverse Problems Diffusion Model Inference Acceleration Constrained Optimization Differentiable Optimization

Mar 2, 2026

PhysFormer: A Physics-Embedded Generative Model for Physically Self-Consistent Spectral Synthesis

In scientific and engineering domains, modeling high-dimensional complex systems governed by partial differential equations (PDEs) remains challenging in terms of physical consistency and numerical stability. However, existing approaches, such as physics-informed neural networks (PINNs), typically rely on known physical fields or coefficients and enforce physical constraints via external loss functions, which can lead to training instability and make it difficult to handle high-dimensional or unobservable scenarios. To this end, we propose PhysFormer, a generative modeling framework that is self-consistent at both the data and physical levels. PhysFormer leverages a low-dimensional, physically interpretable latent space to learn key physical quantities directly from data without requiring known high-dimensional physical field parameters, and embeds the physical process of radiative flux generation within the network to ensure the physical consistency of the generated spectra. In high-dimensional, degenerate inversion tasks, PhysFormer constrains generation within physical limits and enhances spectral fidelity and inversion stability under varying signal-to-noise ratios (SNRs). More broadly, this approach shifts the physical processes from external loss functions into the generative mechanism itself, providing a physically consistent generative modeling paradigm for complex systems involving unknown or unobservable physical quantities.

Siqi Wang, Mengmeng Zhang, Yude Bu +1

School of Mathematics and Statistics, Shandong University, Weihai, 264209, Shandong, China · Weihai Institute for Interdisciplinary Research, Shandong University, Weihai 264209, Shandong, China

Radiative Transfer Physics-Informed Generative Modeling Constrained Generative Modeling Inverse Problems

Feb 28, 2026

Dual-space posterior sampling for Bayesian inference in constrained inverse problems

Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy, incomplete data or inherent non-uniqueness. A prominent example is full waveform inversion (FWI), which estimates Earth's subsurface properties by fitting seismic measurements subject to the wave equation, where ill-conditioning stems from noisy, band-limited, finite-aperture measurements and complex geological structures. A Bayesian framework describes the solution more comprehensively: instead of a single estimate, a posterior distribution of plausible solutions characterizes the non-uniqueness and can be sampled to quantify uncertainty. However, no clear procedure exists for translating hard physical constraints, such as the wave equation, into priors amenable to existing sampling techniques. We address this by sampling the posterior in the dual space via an augmented Lagrangian formulation, which converts hard constraints into penalties suited to sampling algorithms while enforcing them progressively through multiplier updates, so they are satisfied in the limit. We integrate the alternating direction method of multipliers (ADMM) with Stein variational gradient descent (SVGD), a particle-based sampler: the constraint is relaxed at each iteration and the multiplier updates progressively enforce its satisfaction. This enables posterior sampling under hard constraints while inheriting the favorable conditioning of dual-space solvers, where partial constraint relaxation permits productive updates even when the current model is far from the true solution. We validate the method on a stylized Rosenbrock conditional inference problem and on frequency-domain FWI for a Gaussian anomaly model and the Marmousi II benchmark, demonstrating physically consistent uncertainty estimates and posterior contraction with increasing data coverage.

Ali Siahkoohi, Kamal Aghazade, Ali Gholami

Department of Computer Science, University of Central Florida, Orlando, FL, USA · Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland

Bayesian Inverse Problems Inverse Problems Posterior Sampling PDE Inverse Problems

Feb 27, 2026

DiffSOS: Acoustic Conditional Diffusion Model for Speed-of-Sound Reconstruction in Ultrasound Computed Tomography

Accurate Speed-of-Sound (SoS) reconstruction from acoustic waveforms is a cornerstone of ultrasound computed tomography (USCT), enabling quantitative velocity mapping that reveals subtle anatomical details and pathological variations often invisible in conventional imaging. However, practical utility is hindered by the limitations of existing algorithms; traditional Full Waveform Inversion (FWI) is computationally intensive, while current deep learning approaches tend to produce oversmoothed results lacking fine details. We propose DiffSOS, a conditional diffusion model that directly maps acoustic waveforms to SoS maps. Our framework employs a specialized acoustic ControlNet to strictly ground the denoising process in physical wave measurements. To ensure structural consistency, we optimize a hybrid loss function that integrates noise prediction, spatial reconstruction, and noise frequency content. To accelerate inference, we employ stochastic Denoising Diffusion Implicit Model (DDIM) sampling, achieving near real-time reconstruction with only 10 steps. Crucially, we exploit the stochastic generative nature of our framework to estimate pixel-wise uncertainty, providing a measure of reliability that is often absent in deterministic approaches. Evaluated on the OpenPros USCT benchmark, DiffSOS significantly outperforms state-of-the-art networks, achieving an average Multi-scale Structural Similarity of 0.957. Our approach provides high-fidelity SoS maps with a principled measure of confidence, facilitating safer and faster clinical interpretation.

Yujia Wu, Shuoqi Chen, Shiru Wang +3

Thayer School of Engineering, Dartmouth College, Hanover, NH, USA · NVIDIA Corp

Ultrasound Imaging Inverse Problems Diffusion-Based Inverse Problems

Feb 27, 2026

Improving Full Waveform Inversion in Large Model Era

Full Waveform Inversion (FWI) is a highly nonlinear and ill-posed problem that aims to recover subsurface velocity maps from surface-recorded seismic waveforms data. Existing data-driven FWI typically uses small models, as available datasets have limited volume, geological diversity, and spatial extent, leading to substantial concerns about overfitting. Although they perform well on synthetic datasets, current methods fail to generalize to more realistic geological structures. In this work, we show that a model trained entirely on simulated and relatively simple data can generalize remarkably well to challenging and unseen geological benchmarks. We provide a working recipe that tames a billion-parameter model for FWI through coordinated scaling across three axes: model capacity, data diversity, and training strategy. Our model achieves state-of-the-art performance on OpenFWI and significantly narrows the generalization gap in data-driven FWI. Across six challenging geophysical benchmarks, including Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP, Sigsbee, and SEAM Phase I, it infers complex structures absent from the training set and delivers significant performance improvements (SSIM from 0.5844 to 0.7669). Overall, our results demonstrate that with an appropriate scaling strategy, large models trained on simple synthetic data can achieve substantial generalization to more complex and realistic geological structures.

Yinan Feng, Peng Jin, Yuzhe Guo +2

School of Data Science and Society University of North Carolina at Chapel Hill Chapel Hill, NC, USA · The Pennsylvania State University State College, PA, USA · Google DeepMind

Inverse Problems OOD Generalization Neural Network Generalization

Feb 26, 2026

Advancing accelerator virtual beam diagnostics through latent evolution modeling: an integrated solution to forward, inverse, tuning, and UQ problems

Virtual beam diagnostics relies on computationally intensive beam dynamics simulations where high-dimensional charged particle beams evolve through the accelerator. We propose Latent Evolution Model (LEM), a hybrid machine learning framework with an autoencoder that projects high-dimensional phase spaces into lower-dimensional representations, coupled with transformers to learn temporal dynamics in the latent space. This approach provides a common foundational framework addressing multiple interconnected challenges in beam diagnostics. For forward modeling, a Conditional Variational Autoencoder (CVAE) encodes 15 unique projections of the 6D phase space into a latent representation, while a transformer predicts downstream latent states from upstream inputs. For inverse problems, we address two distinct challenges: (a) predicting upstream phase spaces from downstream observations by utilizing the same CVAE architecture with transformers trained on reversed temporal sequences along with aleatoric uncertainty quantification, and (b) estimating RF settings from the latent space of the trained LEM using a dedicated dense neural network that maps latent representations to RF parameters. For tuning problems, we leverage the trained LEM and RF estimator within a Bayesian optimization framework to determine optimal RF settings that minimize beam loss. This paper summarizes our recent efforts and demonstrates how this unified approach effectively addresses these traditionally separate challenges.

Mahindra Rautela, Alexander Scheinker

Los Alamos National Laboratory, Los Alamos, NM, US

Latent Dynamics Modeling Inverse Problems

Feb 25, 2026

Learning to reconstruct from saturated data: audio declipping and high-dynamic range imaging

Learning based methods are now ubiquitous for solving inverse problems, but their deployment in real-world applications is often hindered by the lack of ground truth references for training. Recent self-supervised learning strategies offer a promising alternative, avoiding the need for ground truth. However, most existing methods are limited to linear inverse problems. This work extends self-supervised learning to the non-linear problem of recovering audio and images from clipped measurements, by assuming that the signal distribution is approximately invariant to changes in amplitude. We provide sufficient conditions for learning to reconstruct from saturated signals alone and a self-supervised loss that can be used to train reconstruction networks. Experiments on both audio and image data show that the proposed approach is almost as effective as fully supervised approaches, despite relying solely on clipped measurements for training.

Victor Sechaud, Laurent Jacques, Patrice Abry +1

CNRS and ENS de Lyon, Laboratoire de physique, Lyon, France · UCLouvain, ICTEAM, Louvain-la-Neuve, Belgium · CIFAR Fellow

Inverse Problems Self-Supervised Learning

Feb 23, 2026

GSNR: Graph Smooth Null-Space Representation for Inverse Problems

Inverse problems in imaging are ill-posed, leading to infinitely many solutions consistent with the measurements due to the non-trivial null-space of the sensing matrix. Common image priors promote solutions on the general image manifold, such as sparsity, smoothness, or score function. However, as these priors do not constrain the null-space component, they can bias the reconstruction. Thus, we aim to incorporate meaningful null-space information in the reconstruction framework. Inspired by smooth image representation on graphs, we propose Graph-Smooth Null-Space Representation (GSNR), a mechanism that imposes structure only into the invisible component. Particularly, given a graph Laplacian, we construct a null-restricted Laplacian that encodes similarity between neighboring pixels in the null-space signal, and we design a low-dimensional projection matrix from the

p

-smoothest spectral graph modes (lowest graph frequencies). This approach has strong theoretical and practical implications: i) improved convergence via a null-only graph regularizer, ii) better coverage, how much null-space variance is captured by

p

modes, and iii) high predictability, how well these modes can be inferred from the measurements. GSNR is incorporated into well-known inverse problem solvers, e.g., PnP, DIP, and diffusion solvers, in four scenarios: image deblurring, compressed sensing, demosaicing, and image super-resolution, providing consistent improvement of up to 4.3 dB over baseline formulations and up to 1 dB compared with end-to-end learned models in terms of PSNR.

Romario Gualdrón-Hurtado, Roman Jacome, Rafael S. Suarez +1

Universidad Industrial de Santander, Colombia, 680002

Inverse Problems Image Inverse Problems Graph Signal Processing Compressed Sensing

Feb 18, 2026

Separating Oblivious and Adaptive Models of Variable Selection

Sparse recovery is among the most well-studied problems in learning theory and high-dimensional statistics. In this work, we investigate the statistical and computational landscapes of sparse recovery with

\ell_\infty

error guarantees. This variant of the problem is motivated by variable selection tasks, where the goal is to estimate the support of a

k

-sparse signal in

\mathbb{R}^d

. Our main contribution is a provable separation between the oblivious (for each'') and *adaptive* (for all'') models of

\ell_\infty

sparse recovery. We show that under an oblivious model, the optimal

\ell_\infty

error is attainable in near-linear time with

\approx k\log d

samples, whereas in an adaptive model,

\gtrsim k^2

samples are necessary for any algorithm to achieve this bound. This establishes a surprising contrast with the standard

\ell_2

setting, where

\approx k \log d

samples suffice even for adaptive sparse recovery. We conclude with a preliminary examination of a partially-adaptive model, where we show nontrivial variable selection guarantees are possible with

\approx k\log d

measurements.

Ziyun Chen, Jerry Li, Kevin Tian +1

University of Washington · University of Texas at Austin

Sparse Recovery Feature Selection Statistical Learning Theory Inverse Problems

Feb 17, 2026

Guided Diffusion by Optimized Loss Functions on Relaxed Parameters for Inverse Material Design

Inverse design problems are common in engineering and materials science. The forward direction, i.e., computing output quantities from design parameters, typically requires running a numerical simulation, such as a FEM, as an intermediate step, which is an optimization problem by itself. In many scenarios, several design parameters can lead to the same or similar output values. For such cases, multi-modal probabilistic approaches are advantageous to obtain diverse solutions. A major difficulty in inverse design stems from the structure of the design space, since discrete parameters or further constraints disallow the direct use of gradient-based optimization. To tackle this problem, we propose a novel inverse design method based on diffusion models. Our approach relaxes the original design space into a continuous grid representation, where gradients can be computed by implicit differentiation in the forward simulation. A diffusion model is trained on this relaxed parameter space in order to serve as a prior for plausible relaxed designs. Parameters are sampled by guided diffusion using gradients that are propagated from an objective function specified at inference time through the differentiable simulation. A design sample is obtained by backprojection into the original parameter space. We develop our approach for a composite material design problem where the forward process is modeled as a linear FEM problem. We evaluate the performance of our approach in finding designs that match a specified bulk modulus. We demonstrate that our method can propose multiple diverse designs within 1% relative error margin from medium to high target bulk moduli in 2D and 3D settings. We also demonstrate that the material density of generated samples can be minimized simultaneously by using a multi-objective loss function.

Jens U. Kreber, Christian Weißenfels, Joerg Stueckler

Intelligent Perception in Technical Systems Group, University of Augsburg, Germany · Faculty of Mathematics, Natural Science and Engineering, University of Augsburg, Germany · Centre for Advanced Analytics and Predictive Sciences, University of Augsburg, Germany

Inverse Design Inverse Problems Materials Science Diffusion Model Guidance

Feb 10, 2026

Stability and Concentration in Nonlinear Inverse Problems with Block-Structured Parameters: Lipschitz Geometry, Identifiability, and an Application to Gaussian Splatting

We develop an operator-theoretic framework for stability and statistical concentration in nonlinear inverse problems with block-structured parameters. Under a unified set of assumptions combining blockwise Lipschitz geometry, local identifiability, and sub-Gaussian noise, we establish deterministic stability inequalities, global Lipschitz bounds for least-squares misfit functionals, and nonasymptotic concentration estimates. These results yield high-probability parameter error bounds that are intrinsic to the forward operator and independent of any specific reconstruction algorithm. As a concrete instantiation, we verify that the Gaussian Splatting rendering operator satisfies the proposed assumptions and derive explicit constants governing its Lipschitz continuity and resolution-dependent observability. This leads to a fundamental stability--resolution tradeoff, showing that estimation error is inherently constrained by the ratio between image resolution and model complexity. Overall, the analysis characterizes operator-level limits for a broad class of high-dimensional nonlinear inverse problems arising in modern imaging and differentiable rendering.

Joe-Mei Feng, Hsin-Hsiung Kao

Computer Science and Information Engineering, Tamkang University, New Taipei City, Taiwan (R.O.C.) · Department of Information Management, Central Police University, Taoyuan, Taiwan

Inverse Rendering Parameter Identifiability Gaussian Splatting Inverse Problems

Feb 9, 2026

Trajectory Stitching for Solving Inverse Problems with Flow-Based Models

Flow-based generative models have emerged as powerful priors for solving inverse problems. One option is to directly optimize the initial latent code (noise), such that the flow output solves the inverse problem. However, this requires backpropagating through the entire generative trajectory, incurring high memory costs and numerical instability. We propose MS-Flow, which represents the trajectory as a sequence of intermediate latent states rather than a single initial code. By enforcing the flow dynamics locally and coupling segments through trajectory-matching penalties, MS-Flow alternates between updating intermediate latent states and enforcing consistency with observed data. This reduces memory consumption while improving reconstruction quality. We demonstrate the effectiveness of MS-Flow over existing methods on image recovery and inverse problems, including inpainting, super-resolution, and computed tomography.

Alexander Denker, Zeljko Kereta, Carola-Bibiane Schönlieb +1

Department of Computer Science, University College London · Department of Applied Mathematics and Theoretical Physics, University of Cambridge.

Generative Flow Networks Flow-Based Generative Modeling Inverse Problems Image Inverse Problems

Feb 3, 2026

Variational Sparse Paired Autoencoders (vsPAIR) for Inverse Problems and Uncertainty Quantification

Inverse problems are fundamental to many scientific and engineering disciplines; they arise when one seeks to reconstruct hidden, underlying quantities from noisy measurements. Many applications demand not just point estimates but interpretable uncertainty. Providing fast inference alongside uncertainty estimates remains challenging yet desirable in numerous applications. We propose the Variational Sparse Paired Autoencoder (vsPAIR) to address this challenge. The architecture pairs a standard VAE encoding observations with a sparse VAE encoding quantities of interest (QoI), connected through a learned latent mapping. The variational structure enables uncertainty estimation, the paired architecture encourages interpretability by anchoring QoI representations to clean data, and sparse encodings provide structure by concentrating information into identifiable factors rather than diffusing across all dimensions. To validate the effectiveness of our proposed architecture, we conduct experiments on blind inpainting, computed tomography (CT), and initial-condition inference for the heat equation, demonstrating that vsPAIR is a capable inverse problem solver that can provide interpretable and structured uncertainty estimates.

Jack Michael Solomon, Rishi Leburu, Matthias Chung

Emory University Department of Mathematics

Variational Autoencoders Uncertainty Quantification Autoencoders Sparse Autoencoders Inverse Problems

Jan 29, 2026

Inverse Problems Conditioned on Observation Ensembles: Applications and Methods

We introduce a new multivariate statistical problem that we refer to as the Ensemble-conditioned Inverse Problem (EIP). The aim of EIP is to invert for an ensemble that is distributed according to the pushforward of a prior under a forward process. In high energy physics (HEP), this is related to a widely known problem called unfolding, which aims to reconstruct the true physics distribution from observations that are distorted by detector effects. The EIP also arises in full waveform inversion (FWI) and inverse imaging with unknown priors. We propose non-iterative inference-time methods that construct posterior samplers based on a new class of conditional generative models, which we call ensemble inverse generative models. For the posterior modeling, these models additionally use the ensemble information contained in the observation set on top of single observations. Unlike existing methods, our proposed methods avoid explicit and iterative use of the forward model at inference time via training across several sets of truth-observation pairs that are consistent with the same forward model, but originate from a wide range of priors. We empirically demonstrate that this training procedure can implicitly encode the likelihood model, enabling direct posterior inference for unseen priors to some degree. We benchmark the proposed method on several synthetic and real datasets in inverse imaging, HEP, and FWI. Our code is available at https://github.com/ZhengyanHuan/EIP.

Zhengyan Huan, Camila Pazos, Martin Klassen +3

Department of Electrical and Computer Engineering, Tufts University, Medford, Massachusetts · The NSF AI Institute for Artificial Intelligence and Fundamental Interactions · Department of Physics and Astronomy, Tufts University, Medford, Massachusetts +1

Posterior Sampling Inverse Problems Image Inverse Problems

Jan 12, 2026

Fundamental Recovery Bounds for SPAD Signals under Stationary Flux

Single-photon avalanche diodes (SPADs) record light as a discrete stream of individual detections. The signal is stochastic. Its statistical structure depends on the sensor's operation mode: binary detection in fixed bins, timestamped detection in fixed bins, or free-running timestamped detection. We derive the likelihood score function for each of these three passive modes. From this single object, stem both fundamental limits of recovery (Cramer-Rao bounds) and practical recovery algorithms based on diffusion posterior sampling. The paper further generalizes fundamental limits to Bayesian Cramer-Rao lower bounds. This generalization makes use of a learned approximation of the score function of signal priors. In prior art, analyses and diffusion-based reconstruction for SPAD data have treated individual modes in isolation. Our unified treatment shows a qualitative high-flux gap between modes: binary counts saturate exponentially, while timestamped modes degrade only linearly. We further extend diffusion posterior sampling, previously restricted to binary SPAD data, to a full timestamped case using the suitable score function. We demonstrate experimentally that matching the score to the operation mode is beneficial for high-fidelity reconstruction. By tying the recovery bounds and diffusion to the score function, this work aims to establish a common foundation for both asking what is recoverable in single-photon sensing, and building methods that approach the bound.

Lior Dvir, Nadav Torem, Mohit Gupta +1

Viterbi Faculty of Electrical and Computer Engineering, Technion-Israel Institute of Technology, Haifa 3200003, Israel · Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI 53706, USA

Bayesian Inverse Problems Diffusion Posterior Sampling Inverse Problems

Dec 29, 2025

Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they encode priors and yield efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., plug-and-play methods with learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferentials of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton--Jacobi partial differential equations (HJ PDEs) to develop deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present numerical results in dimensions up to

64

, where the recovered prior is evaluated in a single forward pass.

Oluwatosin Akande, Gabriel P. Langlois, Akwum Onwunta

Industrial and Systems Engineering, Lehigh University, 200 West Packer Avenue, Bethlehem, PA 18015, USA · Department of Mathematics, University of Illinois Urbana-Champaign, Chicago, IL, USA

Inverse Problems

Dec 5, 2025

InverseCrafter: Efficient Video ReCapture as a Latent Domain Inverse Problem

Recent approaches in controllable novel view video generation often rely on fine-tuning pre-trained Video Diffusion Models (VDMs). This dominant paradigm is computationally expensive and frequently suffers from catastrophic forgetting of the model's original generative priors. To address this challenge, here we propose InverseCrafter, a VDM training-free framework that reformulates novel view video generation as an inpainting-based inverse problem in the latent space, eliminating the need for any annotated 4D training data. The core of our method is to establish operator equivalence by employing a lightweight latent mask encoder to define a latent-domain masking operation via a continuous, multi-channel representation. This principled representation faithfully models the forward process in the latent domain, enabling efficient, backpropagation-free solvers while bypassing the costly bottleneck of repeated VAE operations. InverseCrafter achieves high-fidelity, spatio-temporally coherent novel view synthesis with near-zero additional inference overhead and excels at general-purpose video inpainting and editing by fully preserving the pre-trained VDM's generative capabilities.

Yeobin Hong, Suhyeon Lee, Hyungjin Chung +1

Graduate School of AI, KAIST, South Korea · EverEx, South Korea · Korea University, South Korea

Camera-Controlled Video Generation Video Diffusion Models Dynamic Novel View Synthesis Inverse Problems

Dec 2, 2025

Flexible Gravitational-Wave Parameter Estimation with Transformers

Gravitational-wave data analysis relies on accurate and efficient methods to extract physical information from noisy detector signals, yet the increasing rate and complexity of observations represent a growing challenge. Deep learning provides a powerful alternative to traditional inference, but existing neural models typically lack the flexibility to handle variations in data analysis settings. Such variations accommodate imperfect observations or are required for specialized tests, and could include changes in detector configurations, overall frequency ranges, or localized cuts. We introduce a flexible transformer-based architecture paired with a training strategy that enables adaptation to diverse analysis settings at inference time. Applied to parameter estimation, we demonstrate that a single flexible model, called Dingo-T1, can (i) analyze 48 gravitational-wave events from the third LIGO-Virgo-KAGRA Observing Run under a wide range of analysis configurations, (ii) enable systematic studies of how detector and frequency configurations impact inferred posteriors, and (iii) perform inspiral-merger-ringdown consistency tests probing general relativity. Dingo-T1 also improves median sample efficiency on real events from a baseline of 1.4% to 4.2%. Our approach thus demonstrates flexible and scalable inference with a principled framework for handling missing or incomplete data, key capabilities for current and next-generation observatories.

Annalena Kofler, Maximilian Dax, Stephen R. Green +6

Max Planck Institute for Intelligent Systems, Max-Planck-Ring 4, 72076 Tübingen, Germany · ELLIS Institute Tübingen, Maria-von-Linden-Straße 2, 72076 Tübingen, Germany · Tübingen AI Center, Maria-von-Linden-Straße 1, 72076 Tübingen, Germany +4

Transformer Inference Gravitational-Wave Astronomy Bayesian Inference Inverse Problems

Nov 1, 2025

Physics-Informed Neural Networks for Speech Production

The analysis of speech production based on physical models of the vocal folds and vocal tract is essential for studies on vocal-fold behavior and linguistic research. This paper proposes a speech production analysis method using physics-informed neural networks (PINNs). The networks are trained directly on the governing equations of vocal-fold vibration and vocal-tract acoustics. Vocal-fold collisions introduce nondifferentiability and vanishing gradients, challenging phenomena for PINNs. We demonstrate, however, that introducing a differentiable approximation function enables the analysis of vocal-fold vibrations within the PINN framework. The period of self-excited vocal-fold vibration is generally unknown. We show that by treating the period as a learnable network parameter, a periodic solution can be obtained. Furthermore, by implementing the coupling between glottal flow and vocal-tract acoustics as a hard constraint, glottis-tract interaction is achieved without additional loss terms. We confirmed the method's validity through forward and inverse analyses, demonstrating that the glottal flow rate, vocal-fold vibratory state, and subglottal pressure can be simultaneously estimated from speech signals. Notably, the same network architecture can be applied to both forward and inverse analyses, highlighting the versatility of this approach. The proposed method inherits the advantages of PINNs, including mesh-free computation and the natural incorporation of nonlinearities, and thus holds promise for a wide range of applications.

Kazuya Yokota, Ryosuke Harakawa, Masaaki Baba +1

Department of Mechanical Engineering, Nagaoka University of Technology, 1603-1, Kamitomioka, Nagaoka, Niigata, Japan · Department of Electrical, Electronics and Information Engineering, Nagaoka University of Technology

Speech Processing Physics-Informed ML Inverse Problems

Oct 28, 2025

Robustness to Sparse Adversarial Corruption in Arbitrary Linear Measurements: Beyond Exact Recovery

Recovery from linear measurements under sparse adversarial corruption is typically formulated as an exact-recovery problem: one seeks structural conditions on

\mathbf{A}

(e.g., restricted isometry property) guaranteeing unique recovery of

\mathbf{x}^\star

from

\mathbf{y} = \mathbf{A}\mathbf{x}^\star + \mathbf{e}

with

\|\mathbf{e}\|_0 \leq q

. However, these guarantees provide no guidance once exact recovery fails. This limitation obscures simple robustness phenomena -- for instance, repeated rows in

\mathbf{A}

can preserve nontrivial information about

\mathbf{x}^\star

under sparse corruption. In this paper, we study what information about

\mathbf{x}^\star

can be uniformly recovered from

\mathbf{y} = \mathbf{A}\mathbf{x}^\star + \mathbf{e}

for arbitrary

\mathbf{A}\in\mathbb{R}^{m\times n}

and any

q

-sparse

\mathbf{e}

. We show that the robust information is precisely

\mathbf{x}^\star + \ker(\mathbf{U})

, where

\mathbf{U}

is the orthogonal projection onto the intersection of rowspaces of all submatrices of

\mathbf{A}

obtained by deleting

2q

rows. This clarifies how the row structure of

\mathbf{A}

governs whether a

q

-sparse corruption allows exact, partial, or only trivial recovery. We further prove every

\mathbf{x}

minimizing

\|\mathbf{y} - \mathbf{A} \mathbf{x}\|_0

belongs to

\mathbf{x}^\star + \ker(\mathbf{U})

, yielding a constructive approach to recover this set. For i.i.d. Gaussian matrices, we establish a sharp phase transition between exact and trivial recovery. We sketch two applications: robust network tomography and signal reconstruction from oversampled DCT.

Vishal Halder, Alexandre Reiffers-Masson, Abdeldjalil Aïssa-El-Bey +1

IMT Atlantique, Lab-STICC, CNRS UMR 6285, Brest, France · Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India

Sparse Recovery Adversarial Robustness Corruption Robustness Inverse Problems

Oct 6, 2025

Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions

Data-driven inverse optimization for mixed-integer linear programs (MILPs), which seeks to learn an objective function and constraints consistent with observed decisions, is important for building accurate mathematical models in a variety of domains, including power systems and scheduling. However, to the best of our knowledge, existing data-driven inverse optimization methods primarily focus on learning objective functions under known constraints, and learning both objective functions and constraints from data for MILPs remains largely unexplored. In this paper, we propose a two-stage approach for a class of inverse optimization problems in which the objective is a linear combination of given feature functions and the constraints are parameterized by unknown functions and thresholds. Our method first learns the constraints and then, conditioned on the learned constraints, estimates the objective-function weights. On the theoretical side, we provide finite-sample guarantees for solving the proposed inverse optimization problem. To this end, we develop statistical learning tools for pseudo-metric spaces under sub-Gaussian assumptions and use them to derive a learning-theoretic framework for inverse optimization with both unknown objectives and constraints. On the experimental side, we demonstrate that our method successfully solves inverse optimization problems on scheduling instances formulated as ILPs with up to 100 decision variables.

Akira Kitaoka

Mixed-Integer Linear Programming Learning to Optimize Inverse Problems Integer Linear Programming

Oct 2, 2025

MACS: Measurement-Aware Consistency Sampling for Inverse Problems

Diffusion models have emerged as powerful generative priors for solving inverse imaging problems. However, their practical deployment is hindered by the substantial computational cost of slow, multi-step sampling. Although Consistency Models (CMs) address this limitation by enabling high-quality generation in only one or a few steps, their direct application to inverse problems has remained largely unexplored. This paper introduces a modified consistency sampling framework specifically designed for inverse problems. The proposed approach regulates the sampler's stochasticity through a measurement-consistency mechanism that leverages the degradation operator, thereby enforcing fidelity to the observed data while preserving the computational efficiency of consistency-based generation. Comprehensive experiments on the Fashion-MNIST and LSUN Bedroom datasets demonstrate consistent improvements across both perceptual and pixel-level metrics, including the Fréchet Inception Distance (FID), Kernel Inception Distance (KID), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM), compared with baseline consistency and diffusion-based sampling methods. The proposed method achieves competitive or superior reconstruction quality with only a small number of sampling steps.

Amirreza Tanevardi, Pooria Abbas Rad Moghadam, Seyed Mohammad Eshtehardian +2

Department of Electrical Engineering Sharif University of Technology Tehran, Iran · Electronics Research Institute Sharif University of Technology Tehran, Iran

Diffusion Model Sampling Inverse Problems Consistency Models Image Inverse Problems

Sep 27, 2025

Patch Rebirth: Fast and Transferable Model Inversion of Vision Transformers

Model inversion is a widely adopted technique in data-free learning that reconstructs synthetic inputs from a pretrained model through iterative optimization, without access to original training data. Unfortunately, its application to state-of-the-art Vision Transformers (ViTs) poses a major computational challenge, due to their expensive self-attention mechanisms. To address this, Sparse Model Inversion (SMI) was proposed to improve efficiency by pruning and discarding seemingly unimportant patches, which were even claimed to be obstacles to knowledge transfer. However, our empirical findings suggest the opposite: even randomly selected patches can eventually acquire transferable knowledge through continued inversion. This reveals that discarding any prematurely inverted patches is inefficient, as it suppresses the extraction of class-agnostic features essential for knowledge transfer, along with class-specific features. In this paper, we propose Patch Rebirth Inversion (PRI), a novel approach that incrementally detaches the most important patches during the inversion process to construct sparse synthetic images, while allowing the remaining patches to continue evolving for future selection. This progressive strategy not only improves efficiency, but also encourages initially less informative patches to gradually accumulate more class-relevant knowledge, a phenomenon we refer to as the Re-Birth effect, thereby effectively balancing class-agnostic and class-specific knowledge. Experimental results show that PRI achieves up to 10x faster inversion than standard Dense Model Inversion (DMI) and 2x faster than SMI, while consistently outperforming SMI in accuracy and matching the performance of DMI.

Seongsoo Heo, Dong-Wan Choi

Inha University

Vision Transformer Inverse Problems Data-Efficient Learning

Sep 23, 2025

A Gradient Flow Approach to Solving Inverse Problems with Latent Diffusion Models

Solving ill-posed inverse problems requires powerful and flexible priors. We propose leveraging pretrained latent diffusion models for this task through a new training-free approach, termed Diffusion-regularized Wasserstein Gradient Flow (DWGF). Specifically, we formulate the posterior sampling problem as a Wasserstein gradient flow in the latent space of an expected negative log posterior objective, regularized by a Kullback-Leibler divergence to the diffusion prior. We demonstrate the performance of our method on standard benchmarks using StableDiffusion (Rombach et al., 2022) as the prior.

Tim Y. J. Wang, O. Deniz Akyildiz

Department of Mathematics Imperial College London

Wasserstein Gradient Flows Bayesian Inverse Problems Diffusion-Based Inverse Problems Diffusion Posterior Sampling Inverse Problems

Jul 3, 2025

NeuralDMD: Interpretable Neural Representation of Dynamics from Sparse and Noisy Measurements

Many challenges in scientific imaging involve solving ill-posed inverse problems, where the goal is to recover spatio-temporal fields from indirect, noisy, and highly sparse measurements - often without access to ground truth data or reliable simulators. To address this challenging scenario, we present NeuralDMD, an interpretable, untrained (per-instance) reconstruction framework that combines neural implicit representations with Dynamic Mode Decomposition (DMD) to reconstruct continuous spatio-temporal dynamics directly from measurements. NeuralDMD parameterizes DMD modes as continuous neural fields, and imposes a low-rank linear dynamics prior with spectral time evolution to enforce temporal continuity. This formulation enables both forecasting under sparsity, and yields interpretable modes and spectra. We find that NeuralDMD outperforms baselines on a wide variety of tasks: from weather data assimilation from sparse station observations to interferometric (Fourier domain) observations of Sagittarius A, the black hole at the center of our galaxy. Moreover, NeuralDMD remains stable when extrapolating into the future. While this framework is most naturally suited to linear dynamics, we show that it can be applied to nonlinear regimes, though with extrapolation performance that degrades with increasing nonlinearity. Together, these results show that NeuralDMD enables interpretable reconstruction and forecasting of spatio-temporal dynamics directly from sparse and indirect measurements without relying on numerical simulators or training data.

Ali SaraerToosi, Renbo Tu, Esther Y. H. Lin +2

University of Toronto, Toronto, ON, Canada · Vector Institute, Toronto, ON, Canada · NVIDIA Corporation, USA

Spatiotemporal Forecasting Inverse Problems Dynamic Mode Decomposition

May 30, 2025

Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments

Rapid urbanization demands efficient monitoring of turbulent wind and pollutant dispersion, yet existing reconstruction and sensor placement strategies fail under realistic sparsity constraints. Here, we introduce Diff--SPORT, a diffusion-based framework that combines a generative diffusion prior with maximum a posteriori inference and Shapley-value attribution for high-fidelity flow reconstruction and optimal sensor placement. By training a diffusion prior model once over a domain, Diff--SPORT enables non-linear optimal sensor placement and near-real-time flow reconstruction from sparse measurements orders of magnitude faster than RANS or LES simulations, consistently outperforming state-of-the-art methods. The framework also extends, without algorithmic modification, to experimental passive scalar concentration dataset, a direct proxy for pollutant dispersion, measured in a 1:2400 scale water-flume model of the Beijing Haidian neighbourhood under realistic urban flow conditions. Shapley-guided sensor placement achieves up to 57% lower reconstruction error than randomly placed sensors at extreme sparsity, identifying compact and physically interpretable configurations. These results establish Diff--SPORT as a modular foundation offering a zero-shot alternative to retraining-intensive downstream strategies, supporting scalable urban flow monitoring for air quality management and resilient city design.

Abhijeet Vishwasrao, Sai Bharath Chandra Gutha, Andres Cremades +8

Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA · FLOW, Engineering Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden · Division of Robotics, Perception and Learning, School of EECS, KTH Royal Institute of Technology, Stockholm, Sweden +7

Diffusion-Based Inverse Problems Inverse Problems Computational Fluid Dynamics