Tikhonov Regularization with Neural Operators

• Tikhonov regularization with neural operators is a framework that integrates variational regularization with neural models to stabilize ill-posed inverse problems.
• Different methods learn distinct components—such as regularized inverse maps, surrogate forward operators, or Newton factors—balancing data noise, bias, and approximation errors.
• Applications range from DaROL for stable label learning to adaptive-punishment schemes in imaging, offering provable convergence and efficient reconstructions.

Searching arXiv for papers on Tikhonov regularization and neural operators / related operator-learning formulations. “Tikhonov regularization with neural operators” denotes a family of methods that combine variational regularization with neural or operator-learning components, but the phrase is not uniform across the literature. In one line of work, Tikhonov regularization is applied first to define a stable inverse map, and a neural network is then trained to learn that regularized map (Chen et al., 2023). In another, the exact forward operator in a Tikhonov functional is replaced by a learned neural-operator surrogate, so the regularization theory must balance data noise, regularization bias, and surrogate approximation error (Scherzer et al., 26 Aug 2025). A third line embeds Tikhonov terms into continuous-time or unrolled neural dynamics for solving operator equations, sometimes in a broad operator-theoretic sense rather than in the modern DeepONet/FNO sense (Anh et al., 2024). The resulting field is therefore best understood through the role played by the regularized operator, the learned object, and the convergence target.

1. Conceptual scope and taxonomy

The literature uses the same words for substantially different constructions. Some papers study neural operators in the modern sense of learned maps between function spaces; others study neural-network-assisted variational regularization; still others use “neural network” to describe projected dynamical systems whose vector fields are built from operators in Hilbert space rather than learned from data (Anh et al., 2024, Scherzer et al., 26 Aug 2025).

Setting	Tikhonov object	Neural component
DaROL	Offline regularized inverse map $f_{\mathrm{reg}}^{-1}$	ReLU network learns $m \mapsto \hat a$
Surrogate forward model	$T_N^{\alpha,\delta}[x]$ with $F_N$ replacing $F$	DeepONet-type neural operator approximates $F$
Projected neural dynamics	$F_\alpha = F+\alpha I$	Continuous-time projected dynamical system
Learned-center Tikhonov	$\lambda\|x-f_\theta(A^\dagger y)\|_2^2$	CNN generates prior image
Parameter-map learning	Sample-wise scalar $\alpha^n$	MLP predicts regularization map
Regularized Newton–Kantorovich	$J^*J+\lambda I$ in each linearized step	Neural surrogate predicts Cholesky factors

A central distinction concerns what is being learned. In DaROL, the learned object is the regularized inverse map itself (Chen et al., 2023). In surrogate-forward approaches, the neural operator approximates the forward map appearing inside the Tikhonov functional (Scherzer et al., 26 Aug 2025). In CHONKNORIS, the learned object is neither the full forward map nor the full inverse map, but the Cholesky factors of the inverse Tikhonov-regularized Gauss–Newton operator (Bacho et al., 25 Nov 2025). In medical-imaging and adaptive-penalty formulations, the neural network may instead generate a prior center, a graph geometry, or a sample-dependent regularization parameter, while the final reconstruction still comes from a classical optimization problem (Kofler et al., 2019, Bianchi et al., 2023, Pourahmadian et al., 8 Jun 2025).

This terminological dispersion matters because stability, consistency, and approximation theory are attached to different mathematical objects in each case. A statement about learning a regularized inverse map does not immediately transfer to replacing the forward operator by a learned surrogate, and neither is equivalent to learning a damping rule or a prior image.

2. Learning the regularized inverse map

A direct synthesis of Tikhonov regularization and operator learning is the “data-regularized operator learning” framework DaROL. Its central move is to regularize the inverse problem first and to train on the resulting regularized labels rather than on raw forward-generated pairs. The regularized inverse is defined by

$m \mapsto \hat a$0

and the training set is

$m \mapsto \hat a$1

so supervised learning targets a stable inverse map rather than an ill-posed one (Chen et al., 2023).

In the paper’s main Tikhonov example, the forward map is linear and the penalty is $m \mapsto \hat a$2, giving the LASSO problem

$m \mapsto \hat a$3

Under a nondegeneracy condition, the regularized map $m \mapsto \hat a$4 is globally Lipschitz, with

$m \mapsto \hat a$5

This Lipschitz property is then used as the regularity assumption needed for network approximation and generalization theory (Chen et al., 2023).

The learned model is a vector-valued ReLU network $m \mapsto \hat a$6, interpreted as an operator between discretized parameter and measurement spaces. The approximation theorem states that if $m \mapsto \hat a$7 is $m \mapsto \hat a$8-Lipschitz, then there exists a ReLU neural network operator of width

$m \mapsto \hat a$9

and depth

$T_N^{\alpha,\delta}[x]$0

such that

$T_N^{\alpha,\delta}[x]$1

The same paper derives a Rademacher-complexity generalization bound and a combined learning-error bound whose structure is the sum of a polynomially decaying approximation term in network size and a $T_N^{\alpha,\delta}[x]$2-type generalization term (Chen et al., 2023).

This formulation is significant because Tikhonov regularization is not a penalty on network weights and not part of end-to-end training. It is an offline inverse solver that defines the supervised target. A plausible implication is that the well-posedness of the learned problem is inherited from the regularized inverse map rather than from implicit architectural bias.

3. Learned operators inside classical Tikhonov solvers

A different use of neural operators appears when the exact forward operator $T_N^{\alpha,\delta}[x]$3 is unavailable and is replaced inside the Tikhonov functional by a learned surrogate $T_N^{\alpha,\delta}[x]$4. The baseline exact functional is

$T_N^{\alpha,\delta}[x]$5

while the surrogate functional is

$T_N^{\alpha,\delta}[x]$6

The approximation error is measured locally uniformly near the solution by

$T_N^{\alpha,\delta}[x]$7

The regularization theory then balances three quantities: data noise $T_N^{\alpha,\delta}[x]$8, regularization parameter $T_N^{\alpha,\delta}[x]$9, and surrogate error $F_N$0 (Scherzer et al., 26 Aug 2025).

The key parameter rule is

$F_N$1

and under nonlinear source conditions the reconstruction error satisfies

$F_N$2

For $F_N$3-domain problems, the paper introduces mollification $F_N$4 because point evaluations are not defined, leading to

$F_N$5

and an additional mollification term in the rate (Scherzer et al., 26 Aug 2025).

The neural-operator class analyzed there is DeepONet-type: $F_N$6 with branch-net sampling at sensor points $F_N$7 and trunk-net dependence on the output variable $F_N$8. A major technical contribution is the extension of approximation theory from compact subsets of continuous functions to Sobolev and Lebesgue settings, which is the topology in which inverse-problem stability is usually formulated (Scherzer et al., 26 Aug 2025).

A second, more algorithmic, line appears in CHONKNORIS. Here the nonlinear problem is written as $F_N$9, and each Newton–Kantorovich step is regularized by solving

$F$0

which yields the Hilbert-space Levenberg–Marquardt step

$F$1

The learned object is a triangular factor $F$2 such that

$F$3

and the iteration becomes

$F$4

The forcing term

$F$5

splits the inexactness into a Tikhonov bias term and a learned-operator approximation term. If $F$6, the inexact Newton iteration remains contractive; if $F$7, the rate becomes superlinear; if $F$8, the rate becomes quadratic (Bacho et al., 25 Nov 2025).

This makes the role of Tikhonov regularization unusually explicit. The neural operator is not trained to approximate the full nonlinear solution map. It is trained to emulate the regularized linear algebra inside each Newton step. The paper reports median relative $F$9 errors near machine precision on several forward and inverse PDE problems, including $F$0 for a nonlinear elliptic equation and $F$1 for Burgers’ equation (Bacho et al., 25 Nov 2025).

4. Operator-theoretic neural dynamics in Hilbert spaces

The phrase also covers an operator-theoretic tradition in which the “neural network” is a continuous-time dynamical system designed to solve a regularized operator equation rather than a trained model. A representative example studies the general variational inequality

$F$2

posed in a real Hilbert space $F$3 with nonempty closed convex $F$4. The pair $F$5 is called monotone if

$F$6

Tikhonov regularization is introduced by modifying $F$7 to

$F$8

If $F$9 is $F_\alpha = F+\alpha I$0-strongly monotone and $F_\alpha = F+\alpha I$1 is monotone, then

$F_\alpha = F+\alpha I$2

so $F_\alpha = F+\alpha I$3 becomes $F_\alpha = F+\alpha I$4-strongly monotone and the regularized GVI has a unique solution $F_\alpha = F+\alpha I$5 (Anh et al., 2024).

The regularization is not merely a device for proving existence. The paper establishes a selection principle: $F_\alpha = F+\alpha I$6 where $F_\alpha = F+\alpha I$7 is the solution selected by a bilevel variational inequality on the convex image set $F_\alpha = F+\alpha I$8. This is not, in general, a minimum-norm selection; it is the unique solution induced by the auxiliary bilevel problem (Anh et al., 2024).

The associated “neural network” is the projected dynamical system

$F_\alpha = F+\alpha I$9

whose equilibria coincide with regularized GVI solutions. The paper proves existence and uniqueness of a global strong solution for positive continuous $\lambda\|x-f_\theta(A^\dagger y)\|_2^2$0 and $\lambda\|x-f_\theta(A^\dagger y)\|_2^2$1, using the Cauchy–Lipschitz–Picard theorem. Under additional assumptions, including $\lambda\|x-f_\theta(A^\dagger y)\|_2^2$2 linear and self-adjoint and

$\lambda\|x-f_\theta(A^\dagger y)\|_2^2$3

the solution converges strongly: $\lambda\|x-f_\theta(A^\dagger y)\|_2^2$4 The explicit Euler discretization

$\lambda\|x-f_\theta(A^\dagger y)\|_2^2$5

produces a strongly convergent iterative regularization scheme with $\lambda\|x-f_\theta(A^\dagger y)\|_2^2$6 under the corresponding step-size and parameter conditions (Anh et al., 2024).

This body of work is relevant to the topic only in a broad operator-theoretic sense. The paper explicitly emphasizes that it is not a trained neural-operator architecture of the DeepONet, Fourier Neural Operator, or operator-transformer kind. There is no training data and no learned operator between function spaces. The “neural” element is the continuous-time projected dynamics.

5. Learned priors, adaptive penalties, and unrolled Tikhonov networks

A large applied literature keeps the final reconstruction variational while using neural networks to supply priors, parameter maps, or adaptive penalty operators. In large-scale medical imaging, one approach computes an initial reconstruction $\lambda\|x-f_\theta(A^\dagger y)\|_2^2$7, applies a trained CNN $\lambda\|x-f_\theta(A^\dagger y)\|_2^2$8, and then solves

$\lambda\|x-f_\theta(A^\dagger y)\|_2^2$9

The network therefore produces the center of a quadratic Tikhonov penalty, while data consistency is enforced by a separate optimization. For undersampled 2D radial cine MRI the final reconstruction achieved PSNR $\alpha^n$0, NRMSE $\alpha^n$1, SSIM $\alpha^n$2, and HPSI $\alpha^n$3; for 3D cone-beam low-dose CT it outperformed TV and dictionary-based regularization on all reported measures (Kofler et al., 2019).

Another direction learns not the reconstruction itself but the regularization parameter field. In inverse scattering, the paper considers a large family of Tikhonov problems

$\alpha^n$4

with one scalar $\alpha^n$5 per right-hand side $\alpha^n$6. An MLP takes the projected right-hand side $\alpha^n$7 and predicts $\alpha^n$8, thereby learning a regularization map. Training is discrepancy-informed: low-resolution Morozov labels provide weak supervision in stage 1, and stage 2 minimizes the Tikhonov imaging loss directly. The paper reports, for example, that at $\alpha^n$9 operator noise and time step $J^*J+\lambda I$0, mean contrast improves from $J^*J+\lambda I$1 for Morozov to $J^*J+\lambda I$2 after stage 2, while max contrast improves from $J^*J+\lambda I$3 to $J^*J+\lambda I$4 (Pourahmadian et al., 8 Jun 2025).

A closely related adaptive-penalty idea appears in graphLa+$J^*J+\lambda I$5. Starting from a preliminary reconstructor $J^*J+\lambda I$6, the method builds a graph Laplacian $J^*J+\lambda I$7 and solves

$J^*J+\lambda I$8

The graph depends on data only through the preliminary reconstruction, which may come from FBP, standard Tikhonov, TV, or a trained ResU-net. The paper proves that this data-dependent variational method is a regularization method under weak assumptions on $J^*J+\lambda I$9, including the case where $m \mapsto \hat a$00 is a fixed Lipschitz neural network. On sparse-view CT, graphLa+Net outperformed the standalone network and the other graphLa+$m \mapsto \hat a$01 variants across reported PSNR and SSIM trends (Bianchi et al., 2023).

There are also formulations in which the Tikhonov penalty is placed directly on neural-network hypothesis classes. For shallow RePU networks

$m \mapsto \hat a$02

one paper studies

$m \mapsto \hat a$03

with $m \mapsto \hat a$04 taken as the extended Barron norm, the variation norm, or the Radon-BV seminorm. The resulting regularization theory yields explicit $m \mapsto \hat a$05-error bounds for function and derivative recovery, with dimension dependence tracked through the relevant network norm embeddings (Li et al., 2024).

A further variant unfolds a forward-backward algorithm for

$m \mapsto \hat a$06

into a network layer

$m \mapsto \hat a$07

Here Tikhonov regularization is embedded directly in the linear part of every layer, and the paper derives robustness bounds not only with respect to the input state but also with respect to the bias $m \mapsto \hat a$08, which encodes the observed data (Chouzenoux et al., 2021).

6. Theoretical themes, misconceptions, and limits

A persistent misconception is that all of these methods are “neural operators” in the same sense. Several cited papers explicitly deny that interpretation. The regularized Hilbert-space dynamics for variational inequalities are not data-driven operator learning (Anh et al., 2024). DaROL uses standard ReLU networks on discretized spaces rather than architecture-specific neural operators (Chen et al., 2023). The CNN-prior method for MRI and CT is a learned image-restoration operator inside a Tikhonov functional, not a DeepONet or Fourier Neural Operator (Kofler et al., 2019). Even the regularization-map learner for inverse scattering uses an MLP that predicts scalar parameters, although the paper places it conceptually near neural-operator logic (Pourahmadian et al., 8 Jun 2025).

A second recurrent theme is that Tikhonov regularization changes what is learnable by changing the target map. In DaROL, the regularized inverse $m \mapsto \hat a$09 becomes Lipschitz and thus amenable to approximation theory (Chen et al., 2023). In surrogate-forward formulations, the approximation error of the learned operator enters exactly like a discretization error, so the regularization parameter must be chosen against $m \mapsto \hat a$10 as well as $m \mapsto \hat a$11 (Scherzer et al., 26 Aug 2025). In CHONKNORIS, learning succeeds because the learned object is a regularized Newton linear solve, not the full nonlinear solution operator (Bacho et al., 25 Nov 2025).

A third theme is the contrast between infinite-dimensional theory and discretized practice. Some papers work explicitly in Hilbert or Sobolev spaces and formulate local uniform operator approximation there (Scherzer et al., 26 Aug 2025, Anh et al., 2024). Others pass to fixed discretizations before learning and analyze operator learning only on compact subsets of Euclidean space (Chen et al., 2023). A cautionary result for learned variational penalties shows that if a learned $m \mapsto \hat a$12-type transform $m \mapsto \hat a$13 fails to control all directions in infinite dimensions, reconstruction error can become unbounded, whereas in finite dimensions the minimizer map remains Hausdorff-Lipschitz continuous (Behrens et al., 1 Apr 2026). This suggests that finite-dimensional empirical robustness does not by itself resolve the function-space regularization question.

Finally, the literature repeatedly returns to operator-adapted priors. A non-neural source-identification paper shows that replacing the standard penalty $m \mapsto \hat a$14 by $m \mapsto \hat a$15, with $m \mapsto \hat a$16 constructed from the geometry of $m \mapsto \hat a$17, removes the boundary bias induced by classical pseudoinverse selection (Elvetun et al., 2020). A plausible implication is that learned neural-operator priors may need to emulate not merely smoothness or sparsity, but the nullspace geometry and identifiability structure of the forward problem.

Taken together, these works show that “Tikhonov regularization with neural operators” is not a single method but a structured research area organized by where the regularization acts and what the neural component learns. The most stable formulations preserve an explicit variational or iterative regularization backbone and use neural models to learn a regularized inverse map, a surrogate forward operator, a regularized Newton factorization, a prior center, a graph geometry, or a parameter-selection rule. The central mathematical issue across these variants is the same: the learned component must be integrated in a way that preserves, or at least quantifiably perturbs, the stability mechanism supplied by Tikhonov regularization.