End-to-End Inverse Operator Learning

• The paper introduces a direct approximation of inverse operators, bypassing iterative forward simulations for efficient reconstruction.
• End-to-end inverse operator learning maps observation data to latent parameters or distributions, addressing ill-posedness with tailored regularization.
• Key architectures like SINO and DeepONet unroll iterative updates and operator-to-function mappings, demonstrating improved speed and accuracy in PDE inverse problems.

to=browser.search 天天中彩票有人말? to=browser.search 万亚json {"query":"site:arxiv.org end-to-end inverse operator learning arXiv SINO neural inverse operators", "max_results": 10} to=browser.search 彩神争霸可以json {"query":"site:arxiv.org \"Starter-Iterator Neural Operator\" arXiv (Qin et al., 16 Jun 2026)", "max_results": 5} to=browser.open qq天天中彩票json {"url":"https://arxiv.org/abs/([2606.18305](/papers/2606.18305))"} End-to-end inverse operator learning denotes the direct approximation of an inverse solution operator from observations to latent quantities of interest—functions, coefficient fields, trajectories, or posterior distributions—without requiring explicit test-time inversion of the forward map. In the canonical formulation, the forward model is written as $y=\mathcal G(u)+\eta$, while the learned object is an inverse map or posterior map from data space to parameter space, rather than a surrogate for $\mathcal G$ itself. This distinguishes the paradigm both from standard forward operator learning and from classical variational or Bayesian inversion, which repeatedly invoke the forward operator during inference. Recent work has developed deterministic, probabilistic, and solver-unrolled variants of this idea, including operator-to-function architectures for PDE inverse problems, measure-aware inverse maps, latent diffusion posteriors, and unified forward/inverse neural operators such as SINO (Nelsen et al., 27 Aug 2025, Molinaro et al., 2023, Qin et al., 16 Jun 2026).

1. Conceptual scope and distinguishing features

The central object in end-to-end inverse operator learning is a reconstruction operator $R:Y\to X$ or, more generally, an inverse solution map $\Psi^\star:Y\to U$ learned from paired examples $(y_n,u_n)$. In the deterministic supervised setting, the training problem takes the form

$$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$

This reverses the direction of standard operator learning, where one learns forward maps from parameters or source terms to solutions or observables. The reversal is not merely notational: inverse maps may be non-unique, unstable, or undefined outside the range of the forward operator, so the learned model must typically act as a regularized extension from noisy off-range data to admissible reconstructions (Nelsen et al., 27 Aug 2025).

A recurrent misconception is that end-to-end inverse operator learning is just “forward operator learning with inputs and outputs swapped.” The literature explicitly argues otherwise. Neural Inverse Operators formulate many PDE inverse problems as operator-to-function maps, not function-to-function maps, because the observation itself is an operator such as a Dirichlet-to-Neumann map or a set of boundary measurements (Molinaro et al., 2023). The probabilistic literature further emphasizes that many inverse problems are more naturally data-to-distribution maps than data-to-point maps, since multiple latent states can be consistent with the same observation (Thorpe et al., 19 Dec 2025).

A second misconception is that the paradigm is intrinsically one-shot regression. Several influential formulations are instead iterative or unrolled. Learned iterative networks define a learned reconstruction operator through a truncated sequence of learned updates, and solver-emulation methods such as CHONKNORIS learn a local inverse linearized operator inside a Newton--Kantorovich loop rather than regressing the full inverse map in one pass (Hauptmann et al., 9 Dec 2025, Bacho et al., 25 Nov 2025). This suggests that “end-to-end” refers to the training and inference interface—data in, reconstruction out—rather than to a single architectural style.

2. Mathematical formulations of inverse operators

A common starting point is the forward inverse-problem model

$y=\mathcal G(u)+\eta,$

with $u$ the latent parameter or field and $y$ the observation. In classical inverse theory, one reconstructs $u$ by optimization or posterior computation using explicit knowledge of $\mathcal G$0. End-to-end inverse operator learning instead approximates the inverse solver directly, either as a deterministic map $\mathcal G$1 or as a probabilistic map $\mathcal G$2 or $\mathcal G$3 (Nelsen et al., 27 Aug 2025).

In PDE inverse problems, the operator-theoretic formulation is often more specific. NIO writes the PDE as

$\mathcal G$4

with unknown coefficient $\mathcal G$5, and defines the forward coefficient-to-operator map

$\mathcal G$6

The inverse problem is then

$\mathcal G$7

which formalizes inverse EIT, inverse scattering, optical tomography, and seismic imaging as operator-to-function learning (Molinaro et al., 2023).

SINO proposes a different unifying abstraction. It begins from

$\mathcal G$8

where $\mathcal G$9 is the unknown latent solution and $R:Y\to X$0 is the observed data or forcing/measurement term. For time-dependent PDEs, backward Euler and Picard-type linearization reduce each time step to a stationary operator equation. In the inverse setting, $R:Y\to X$1 is interpreted as the forward measurement map “projecting the latent field from the parameter space to the data space,” so the same operator representation used for forward simulation extends to inverse reconstruction (Qin et al., 16 Jun 2026).

Probabilistic inverse learning replaces a single inverse map by a conditional distribution. In high-energy unfolding, the detector response defines a forward relation

$R:Y\to X$2

while the desired pseudo-inverse is

$R:Y\to X$3

Here the learned inverse operator is the posterior sampler $R:Y\to X$4, not a deterministic regression function (Shmakov et al., 2023). B2B$R:Y\to X$5 makes the same point in function-space language by defining

$R:Y\to X$6

so that inverse operator learning maps an observed function to a distribution over plausible input functions (Thorpe et al., 19 Dec 2025).

3. Architectural patterns

A major design axis is the structure of the input data. NIO addresses operator-valued observations by composing DeepONets and FNOs. The DeepONet block handles the boundary-to-interior transfer, while the FNO block provides nonlinear mode mixing that DeepONet alone lacks. The architecture is permutation-invariant with respect to the set of measurements, and randomized batching makes it robust to varying numbers of boundary samples (Molinaro et al., 2023).

SINO introduces a unified starter-iterator decomposition. The global learned operator is

$R:Y\to X$7

with a frequency-domain starter

$R:Y\to X$8

and a learned fixed-point-like iterator

$R:Y\to X$9

In inverse tasks, the starter recovers dominant low-frequency structure while the iterator refines missing details through residual corrections. The explicit separation of initialization and refinement is the paper’s main architectural distinction from monolithic FNO-style mappings, DeepONet-style basis models, MgNO, and HINTs (Qin et al., 16 Jun 2026).

A broader unrolling perspective appears in learned iterative networks. There, a learned reconstruction operator is defined by

$\Psi^\star:Y\to U$0

with learned gradient, proximal, variational, primal-dual, Newton, Gauss--Newton, or quasi-Newton update rules. The chapter explicitly separates “what to compute,” namely the target inverse operator, from “how to compute,” namely the unrolled architecture used to approximate it (Hauptmann et al., 9 Dec 2025).

Operator-valued inputs motivate still more specialized designs. ORNNs reinsert the operator input multiplicatively at every layer,

$\Psi^\star:Y\to U$1

rather than vectorizing $\Psi^\star:Y\to U$2 once at the input. This architecture is presented as a learned analogue of boundary-control reconstruction schemes for inverse boundary value problems (Hoop et al., 2019).

4. Probabilistic, latent, and generative inverse operators

The probabilistic strand of end-to-end inverse operator learning is centered on learning data-to-posterior maps. VLD for high-energy unfolding combines a detector encoder, a parton VAE, and a latent diffusion model trained jointly under a single ELBO-like objective. The detector encoder is permutation-invariant and based on SPANet; the diffusion denoises in latent space rather than in the original truth space; and the model adds the consistency loss

$\Psi^\star:Y\to U$3

with $\Psi^\star:Y\to U$4 to enforce relativistic structure. The result is a sampleable inverse operator $\Psi^\star:Y\to U$5 that reconstructs global distributions of theoretical kinematic quantities while adhering to known physics constraints (Shmakov et al., 2023).

B2B$\Psi^\star:Y\to U$6 separates representation learning from inversion. Functions are expanded in learned bases,

$\Psi^\star:Y\to U$7

and the inverse model acts only on coefficients, $\Psi^\star:Y\to U$8. This enables deterministic, invertible, and probabilistic inverse models within one framework, including linear regressors, MLPs, INNs, cVAEs, MDNs, RealNVP, and probabilistic cINNs. Evaluation is performed primarily by forward re-simulation,

$\Psi^\star:Y\to U$9

rather than by direct input-space error, because the inverse may be non-unique (Thorpe et al., 19 Dec 2025).

OL-GAN integrates operator learning with Bayesian inversion by learning the joint distribution of parameters and responses,

$(y_n,u_n)$0

via a coordinate-aware generator $(y_n,u_n)$1. After training, posterior inference is transferred to latent space: $(y_n,u_n)$2 Because the generator takes coordinates explicitly, inference can be carried out on arbitrary sensor sets and misaligned grids without retraining, which the paper terms resolution-independent inversion (Jiang et al., 2023).

These developments align with the measure-centric view emphasized in the survey literature: inverse operator learning may target a point estimate, a posterior distribution, or even a transport map between measures. A plausible implication is that the choice between deterministic and probabilistic inverse operators should be governed by the degree of ill-posedness rather than by architectural convenience alone (Nelsen et al., 27 Aug 2025).

5. Theory, stability, and regularization

Theoretical analysis focuses on approximation, contraction, generalization, and noise robustness. SINO proves a universal approximation theorem: $(y_n,u_n)$3 provided the preconditioned residual operator satisfies

$(y_n,u_n)$4

Its proof decomposes the total approximation error into finite iterative convergence error, iterator approximation error, and starter approximation error, thereby making the initialization-refinement split part of the theory rather than only an implementation detail (Qin et al., 16 Jun 2026).

CHONKNORIS develops a stronger solver-theoretic result. Instead of regressing the full inverse operator, it learns the inverse of the regularized linearized operator

$(y_n,u_n)$5

parameterized through a Cholesky factor $(y_n,u_n)$6. Under an inexact Newton--Kantorovich condition

$(y_n,u_n)$7

the iteration is contractive and converges to a unique zero in a neighborhood, with linear, superlinear, or quadratic behavior depending on the accuracy regime. The paper explicitly argues that machine precision arises because the learned model need only preserve contraction of the solver, not approximate the full nonlinear solution map globally (Bacho et al., 25 Nov 2025).

The survey literature isolates noise as the distinctive difficulty of end-to-end inversion. Since the true inverse is defined only on $(y_n,u_n)$8, noisy data typically lie outside its domain. The learned model must therefore approximate an extension $(y_n,u_n)$9, and robustness depends either on the continuity of the network itself or on the regularity of that extension (Nelsen et al., 27 Aug 2025). UAR makes this explicit by combining a measurement-space fidelity term with a Wasserstein-1 distributional term,

$$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$0

and proves equivalence to an adversarial min-max formulation together with stability under perturbations of the noisy measurement distribution (Mukherjee et al., 2021).

ORNNs contribute a complementary perspective: architecture-specific regularization. Their Schatten-$$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$1-type sparsity regularizer induces low-rank structure in the learned matrices, improves covering-number estimates, and yields explicit generalization bounds on operator-valued inputs (Hoop et al., 2019). This suggests that structure-aware regularization remains central even when the inverse map is learned end-to-end.

6. Empirical domains and representative results

The empirical range of end-to-end inverse operator learning is broad. In PDE inverse problems, NIO reports strong results on Calderón/EIT, inverse wave scattering, radiative transport, and seismic imaging. On the main benchmark table, NIO achieves $$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$2 $$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$3 and $$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$4 $$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$5 on trigonometric Calderón, $$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$6 and $$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$7 on inverse wave scattering, and $$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$8 and $$\min_{\Psi\in H}\frac1N\sum_{n=1}^N \ell\bigl(u_n,\Psi(y_n)\bigr), \qquad y_n=\mathcal G(u_n)+\eta_n.$$9 on radiative transport, consistently outperforming FCNN and DeepONet baselines. Against classical methods, the heart–lungs EIT experiment reports roughly $y=\mathcal G(u)+\eta,$0 hours per sample for D-bar versus $y=\mathcal G(u)+\eta,$1 seconds for NIO on CPU, while inverse wave scattering compares less than $y=\mathcal G(u)+\eta,$2 second for NIO to $y=\mathcal G(u)+\eta,$3 hours for PDE-constrained optimization (Molinaro et al., 2023).

SINO provides one of the clearest demonstrations of unified end-to-end forward and inverse operator learning. In super-resolution microscopy on the BioSR dataset, containing more than 2,200 paired low- and high-resolution images across CCPs, ER, MTs, and F-actin, SINO is compared with RCAN and DFCAN using MSE, PSNR, and SSIM. The reported SINO metrics are $y=\mathcal G(u)+\eta,$4, $y=\mathcal G(u)+\eta,$5, and $y=\mathcal G(u)+\eta,$6 for CCPs; $y=\mathcal G(u)+\eta,$7, $y=\mathcal G(u)+\eta,$8, and $y=\mathcal G(u)+\eta,$9 for ER; $u$0, $u$1, and $u$2 for MTs; and $u$3, $u$4, and $u$5 for F-actin. The paper states that its MSE is about two orders of magnitude lower than DFCAN and one order of magnitude lower than RCAN in these inverse tasks (Qin et al., 16 Jun 2026).

In probabilistic inverse learning, VLD is evaluated on $u$6 training events and $u$7 test events for semi-leptonic $u$8 unfolding. For the total distance over all 55 components, VLD reports Wasserstein $u$9, Energy $y$0, K-S $y$1, $y$2 $y$3, $y$4 $y$5, and $y$6 $y$7. The paper states that its unified latent diffusion approach achieves a distribution-free distance to truth over 20 times smaller than the non-latent state-of-the-art baseline and about 3 times smaller than traditional latent diffusion models (Shmakov et al., 2023).

Other domains show the same pattern of amortized inversion. UAR on low-dose CT reports $y$8 dB PSNR and $y$9 SSIM for the end-to-end model, improving to $u$0 dB and $u$1 SSIM with variational refinement, while retaining runtime about $u$2 smaller than AR (Mukherjee et al., 2021). INO for ODE parameter recovery reports about $u$3 s inference time and a $u$4 speedup over iterative gradient descent on stiff atmospheric chemistry, while improving mean parameter error from $u$5 to $u$6 on POLLU (Liu et al., 12 Mar 2026). Beyond scientific inversion, IDOL learns an inverse mapping from adjacent latent future BEV states to planning-relevant motion deltas and reports ablations of PDMS $u$7 without IDM, $u$8 with IDM, and $u$9 with IDM plus 2-stage closed-loop refinement (Zhang et al., 29 May 2026).

7. Limitations, failure modes, and open directions

The literature is explicit that end-to-end inverse operator learning does not eliminate the classical pathologies of inverse problems. Noise remains fundamental because the learner must act on data that typically fall outside the exact forward range; implicit regularization is poorly understood; and general nonlinear theory remains case-by-case rather than comprehensive (Nelsen et al., 27 Aug 2025). This suggests that end-to-end methods should be interpreted as learned regularized inverses, not exact inverses.

Method-specific limitations are equally prominent. SINO reports strong dependence on data fidelity, system conditioning, and the model-physics gap; the iterator’s efficiency declines in more ill-conditioned scenarios; and very deep multiscale architectures can become overparameterized with diminishing returns (Qin et al., 16 Jun 2026). NIO assumes access to many simulated training pairs and is demonstrated mainly in low to moderate dimensions, while its original formulation does not provide theoretical approximation or generalization guarantees (Molinaro et al., 2023). VLD depends on the simulator prior used to generate training data and is demonstrated on a single topology, so prior mismatch and posterior calibration remain open concerns (Shmakov et al., 2023). B2B$\mathcal G$00 notes that basis smoothness can miss fine-scale details in wave scattering and that some FWI errors are limited by the learned forward surrogate rather than only by the inverse map (Thorpe et al., 19 Dec 2025).

A broader tension runs through the field. Direct inverse regressors provide amortized speed, but solver-unrolled methods such as learned iterative networks and CHONKNORIS retain more explicit ties to optimization, regularization, and convergence theory (Hauptmann et al., 9 Dec 2025, Bacho et al., 25 Nov 2025). Probabilistic models better match non-uniqueness, but they introduce questions of posterior faithfulness, sampling cost, and evaluation. Representation-decoupled methods such as B2B$\mathcal G$01, coordinate-aware generators such as OL-GAN, and unified forward/inverse architectures such as SINO all indicate that future progress is likely to depend less on a single canonical architecture than on tighter alignment between inverse-problem structure, data type, and the chosen notion of solution—point estimate, conditional distribution, or iterative solver state (Jiang et al., 2023, Qin et al., 16 Jun 2026).