Data-Driven Inverse Problems

Updated 30 June 2025

Data-Driven Paradigm for Inverse Problems is a shift from traditional analytic models to data-guided techniques that extract solution behavior from observed data.
These methods integrate adaptive model reduction, learned priors, and plug-and-play regularization to enhance reconstruction accuracy and computational efficiency.
They are widely applied in imaging, geosciences, and control, offering robust and scalable solutions for complex, high-dimensional, and ill-posed inverse challenges.

Data-driven approaches for inverse problems encompass a broad set of methodologies in which the solution—and, often, the regularization, reconstruction strategy, or the governing model itself—is derived or guided by observed data rather than analytic modeling alone. This paradigm has emerged in response to the limitations of classical methods, particularly in situations involving complex, high-dimensional, or poorly understood underlying processes. By leveraging large datasets, modern machine learning models, and computational advances, data-driven approaches have reshaped the landscape of inverse problems in fields such as imaging, geosciences, control, and design.

1. Key Concepts and Historical Context

Inverse problems are mathematical frameworks for inferring unknown parameters, fields, or signals from indirect or noisy observations, typically modeled as: $y = A(x) + n$ where $y$ denotes measurements, $x$ the object of interest, $A$ a (possibly nonlinear or ill-posed) forward operator, and $n$ noise. Classical solution frameworks employ analytic regularization—such as Tikhonov, total variation, or sparsity-invoking techniques—aimed at constraining ill-posedness by integrating structural priors about $x$ .

The data-driven paradigm, by contrast, utilizes observed data—either to inform or to replace explicit prior modeling. Its rise has followed increased data availability and the success of over-parameterized models in areas like deep learning. A distinguishing feature is the adaptation of solution methods to empirical data distributions, which aligns the inversion process with the statistics and complexity of real-world targets.

2. Data-Driven Model Reduction and Surrogates

Early and foundational work in data-driven inverse problems demonstrated that computational cost for large-scale, PDE-governed inference (particularly Bayesian approaches) could be substantially reduced using adaptive, problem-specific model reduction. Rather than constructing reduced-order models (ROMs) using prior-based sampling, data-driven ROMs focus basis construction on the most probable regions of parameter space determined by the posterior distribution (1403.4290). For example, when inferring permeability in subsurface flow, snapshots for basis enrichment are adaptively selected using the error of the ROM as the Markov chain explores the posterior. This gives rise to a posterior-oriented ROM—achieving high accuracy in regions of actual inferential relevance—and leads to dramatic efficiency gains when coupled with techniques like delayed acceptance MCMC.

This "data-driven model reduction" framework is extensible and underpins more recent surrogate modeling approaches, including autoencoder-based dimension reduction of solution spaces (1912.10840, 2501.14636). Here, separate (possibly nonlinear) autoencoders compress the forward and inverse solution spaces, and mappings in latent coordinates serve as surrogates for both forward modeling and inversion. Such surrogates enable practical inversion in problems where only unpaired data (input or solution, but not both) are available, and the mapping between latent spaces can often be approached using Bayes risk minimization.

3. Learning Priors, Regularizers, and Physical Models

Modern data-driven paradigms have expanded from ROMs to learning explicit priors and regularization functionals from data. Rather than defining regularization via handcrafted norms or constraints, approaches such as adversarial regularization learn, via discriminative training, functionals that distinguish desirable (real) from undesirable (artifact-laden or noisy) solutions (2309.09250, 2506.11732). These learned regularizers can be enforced as convex (or weakly-convex) neural networks, enabling unique and stable solutions with theoretical convergence guarantees—provided suitable structural assumptions (e.g., convexity, Lipschitz continuity) are maintained.

Plug-and-play (PnP) methods represent another influential line, inserting (possibly learned) denoisers into iterative optimization frameworks as a substitute for explicit regularizers. While early PnP work relied on pre-existing denoisers, current developments analyze and train neural denoisers such that the overall inversion inherits provable regularization properties, including convergence and spectral filtering interpretation for the special case of linear denoisers (2506.11732).

In quantitative imaging—such as qMRI—data-driven techniques now extend to learning the parameter-to-image operator using black-box surrogates (e.g., neural networks) for complex or partially understood physical processes (2404.07886). This allows for both correction of model error and acceleration of inversion.

4. Learning from Imperfect, Scarce, or Misspecified Data

A recurring challenge is adapting data-driven priors and regularizers when available datasets are scarce, noisy, or statistically mismatched from the true target distributions. Several approaches have emerged:

Distributionally Robust Optimization (DRO): Frameworks that minimize the worst-case risk over probability distributions within a Wasserstein ball of the empirical distribution—thus regularizing against noise, measurement error, model uncertainty, and bounded rationality (1512.05489). Key theoretical results guarantee out-of-sample performance, tractability, and robustness.
Iterative Prior Correction and Empirical Bayes: When training data for learned priors are drawn from simulations or corrupted distributions (leading to misspecification), iterative empirical Bayes or sample-based retraining can progressively update the prior using posterior samples from observed data, correcting for distributional shift and promoting unbiased inference (2407.17667). This strategy is particularly crucial in observational sciences such as astronomy and remote sensing, where the true data-generating process is only partially accessible.
Self-Supervised and Semi-Supervised Structures: Paired autoencoder architectures and semi-supervised learning frameworks exploit large amounts of unpaired training data—decoupling the learning of representations in observation and solution spaces, and using small paired datasets to learn the mapping between them (2501.14636). Empirical Bayes interpretations and metrics for assessing solution trustworthiness complement scarcity-mitigation strategies.

5. Stochastic, Non-Smooth, and Non-Convex Learning Schemes

Inverse problems—especially when scaled up—often involve non-smooth and non-convex objectives (e.g., due to total variation, neural network regularizers, or implicit physical constraints). Data-driven settings further complicate gradients, as surrogate models may be black-box or learned. Several methodologies address these challenges:

Stochastic (and Data-Driven) Landweber Variants: Iteratively regularized Landweber and Bouligand Landweber methods have been extended to operate with data-driven, black-box regularization operators constructed from training data, Bouligand subdifferentials to address non-smoothness, and stochastic updates for scalability in systems with many components or training pairs (1812.00272, 2402.04772).
Stochastic Bilevel Optimization: Parameter learning for regularization or experimental design can be treated as a bilevel problem, where the lower (inner) problem is a variational inverse reconstruction, and the upper (outer) problem learns the best (hyper-)parameters via empirical risk minimization across observed data sets. Derivative-free stochastic algorithms (using smoothing and finite-differences) overcome the lack of analytic hypergradients in non-smooth or black-box lower-level solvers, with proven complexity and convergence rates (2411.18100, 2311.15845, 2007.02677).

6. Theory: Well-Posedness, Accuracy, and Generalization

Despite rapid progress in empirical methods, there is substantial focus on establishing theoretical properties:

Convergence and Well-Posedness: For approaches using convex (or weakly-convex) learned regularizers, existence, uniqueness, and stability of reconstructions follow classical regularization theory (e.g., Morozov, Tikhonov), extended to accommodate non-convex and even noise-adaptive network-based penalties (2310.14290, 2309.09250). Key results establish that, under proper architectural and training restrictions, learned regularizers enable solutions that converge with diminishing noise, with controllable bias.
Out-of-Sample Guarantees: Distributionally robust frameworks provide rigorous out-of-sample risk certificates against noise and bounded rationality (1512.05489). For hyperparameter learning (e.g., regularization parameter selection) by ERM, sharp minimax optimal rates are achieved without explicit knowledge of noise or regularity (2311.15845).
Generalization and Adaptivity: Many data-driven approaches are analyzed through a statistical learning lens, emphasizing adaptation to unknown noise, smoothness, or distributional shift. Empirical and theoretical results confirm scalability and generalization to unseen data, across a range of inverse problem types (2007.02677, 2411.18100, 2501.14636, 2506.11732).

7. Applications, Implications, and Open Challenges

Data-driven approaches have transformed inverse problem methodology in disciplines ranging from subsurface exploration and computational engineering to imaging (MRI, CT, PET), astrophysics, and environmental modeling. Notable practical outcomes include:

Drastic computational savings (orders of magnitude) in high-dimensional Bayesian or optimization-based inference (1403.4290, 2110.07676, 2104.13070).
Statistically principled, robust, and distribution-aware inversion in settings with imperfect or misspecified priors (2407.17667, 1512.05489).
Enhanced reconstruction quality through learned priors, regularizers, and denoisers—frequently surpassing handcrafted methods (e.g., total variation, sparsity) in complex, realistic scenarios (2309.09250, 2506.11732).
Uncertainty quantification and Bayesian posterior sampling in computationally demanding PDE systems (2104.13070).

However, open challenges remain. These include general theorems for convergence and stability with arbitrary learned (and often non-convex) priors; strategies to ensure interpretability, reliability, and explainability in black-box neural inversion; scalable and trustworthy approaches in high-dimensional or computationally intensive contexts; systematic reassurance of generalization across domains with limited or mismatched data; and integration of uncertainty quantification within data-driven paradigms.

Methodology	Principle	Theoretical Features
Data-driven model reduction (ROMs)	Adaptive, posterior-focused	Error bounds, efficiency proven
Learned priors/regularizers	Adversarial, empirical	Convexity (if enforced), rates
Plug-and-play denoising	Iterative, modular	Convergence for certain classes
Bilevel hyperparameter learning	Empirical/statistical risk	Minimax rates, ERM theory
Stochastic, derivative-free methods	Zeroth-order, scalable	Complexity analysis

Data-driven paradigms for inverse problems continue to evolve, unifying classical mathematical rigor with machine learning and statistical principles. The literature demonstrates both profound practical gains and a vibrant ongoing exploration of interpretability, reliability, and statistical soundness in the analysis and solution of inverse problems.