Scope-Specific Penalty Operator

• Scope-specific penalty operators are specialized constructs that apply targeted penalties based on structural, spatial, or statistical criteria.
• They are implemented in various frameworks including monotone inclusions, adaptive PDE solvers, sparse estimation, and deep operator networks, enhancing convergence and stability.
• Their design enables bias control, local active-set identification, and numerical stability through methods such as Fitzpatrick functions and spatially adaptive penalties.

A scope-specific penalty operator is a regularization or constraint-enforcing construct tailored to particular components, groups, regions, or modes within an optimization, operator-learning, or statistical inference framework. Rather than employing a global or uniform penalty, scope-specific operators modulate penalization based on structural, spatial, functional, or statistical criteria intrinsic to the problem. This principle appears across diverse contexts including monotone inclusion problems, adaptive numerical PDE solvers, Bayesian sparse estimation, and partition-of-unity neural architectures.

1. Fitzpatrick-Based Penalty Operators in Monotone Inclusion Problems

In the context of monotone inclusion formulations, scope-specific penalties are rigorously constructed using Fitzpatrick functions associated with maximally monotone operators. Given a constraint set $C = \{x \in H : B x = 0\}$ specified by a maximally monotone %%%%1%%%% on a Hilbert space, the penalty function is

$\Pen_C(x) := \varphi_B(x, 0) = \sup_{(y, v) \in \Gr B} \langle x - y, v \rangle$

with Fréchet subdifferential $\partial \Pen_C(x) = B(x)$. This operator penalizes deviations from $C$, confining iterates toward feasibility (Bot et al., 2013).

These penalty functions are utilized in:

• Forward-Backward Penalty Schemes: Iterative zeros of $A x + D x + N_C(x)$ are generated via

$$x_{n+1} = J_{\lambda_n A}\bigl(x_n - \lambda_n D x_n - \lambda_n \beta_n B x_n\bigr)$$

with $(\lambda_n)$ and $(\beta_n)$ controlling step and penalty intensity.

• Tseng-Type Schemes: Replacing cocoercivity by Lipschitz continuity, the procedure updates

$$\begin{aligned} p_n &= J_{\lambda_n A}[x_n - \lambda_n(D x_n + \beta_n B x_n)] \ x_{n+1} &= p_n + \lambda_n (D x_n - D p_n) + \lambda_n \beta_n (B x_n - B p_n) \end{aligned}$$

Convergence requires the summability condition involving $\varphi_B$ (the Fitzpatrick function), ensuring that penalization is precisely scope-specific to violations of $C$ (Bot et al., 2013).

2. Spatially Adaptive Penalty Operators in Numerical PDEs

The Adaptive Penalty Method (APM) defines spatially adaptive penalty operators for enforcing inequality constraints in variational formulations on Banach domains $\Omega$. Rather than a global penalty parameter, APM introduces $\sigma(x)$ as the solution of a mesh-local elliptic PDE: $$\sigma(x) - \epsilon \Delta \sigma(x) = \gamma\, |M(f - Au, u - g)|(x)$$ where $M$ encodes the complementarity residual, and $\epsilon, \gamma$ are tunable. $\sigma(x)$ thus adapts penalty strength in response to local constraint deviation.

The scope-specific penalty operator acts via

$$F_{\sigma}(u) = Au - f + [f - Au + c(u-g)]_{\sigma}$$

with the "smoothed ramp" $[\,\cdot\,]_\sigma$ regularizing complementarity. At each iteration, the associated Jacobian is approximated by

$$J(u; \sigma) = (I - \alpha_\sigma) A + \alpha_\sigma c$$

where $$\alpha_\sigma(x) = (1 + e^{-([f - Au + c(u-g)]/\sigma(x))})^{-1}$$ transitions from global penalization to block-local active-set identification. As the solution converges ($\sigma(x)\to 0$ on active regions), APM morphs into the primal-dual active set method (Boon et al., 2022).

3. Variable-Coefficient and Group-Specific Penalty Operators in Sparse Estimation

Sparse Bayesian Lasso with scope-specific variable-coefficient $\ell_1$ penalty introduces learnable penalty weights $\lambda_p$ (and in extension, $\lambda_{G}$ for groups/scopes). The penalized objective is

$$\min_{\beta,\theta,\lambda > 0} \quad L(\beta,\theta) + \tau \sum_p \lambda_p |\beta_p| - \sum_p \log \lambda_p - \sum_p \log p_\lambda(\lambda_p)$$

with hyperpriors $p_\lambda$ (Half-Cauchy, Gamma, etc.) on each $\lambda_p$ (Wycoff et al., 2022). The proximal operator for the scalar penalty $g(x,\lambda)=\lambda|x|$ solves

$$(x^*,\lambda^*) = \operatorname*{arg\,min}_{x,\lambda > 0} \left[ \lambda|x| + \frac{(x-x_0)^2}{2s_x} + \frac{(\lambda - \lambda_0)^2}{2s_\lambda} \right]$$

giving adaptive shrinkage and low bias on large coefficients. Scope-specificity is extended to block penalties for grouped coordinates: $$g(\beta, \{\lambda_s\}) = \sum_{s \in \mathrm{Scopes}} \lambda_s \|\beta_s\|_\alpha$$ enabling simultaneous learning of both penalized weights and coefficients per scope (Wycoff et al., 2022).

4. Partition Penalty Operators in Deep Operator Networks

The Partition Penalty (“$\mathrm{P}^2$”) operator enforces scope-specific regularization over trunk network modes in DeepONet-type architectures. For $p$ trunk outputs $\operatorname{tr}_j(x)$, the $\mathrm{P}^2$ penalty is defined via

• Signed Partition:

$\sum_{j=1}^p \operatorname{tr}_j(x) = 1$

• Magnitude Partition:

$\sum_{j=1}^p |\operatorname{tr}_j(x)| = 1$

deviations from which are penalized by

$$\mathcal{L}_{P^2}(\theta) = \frac{1}{N} \sum_{k=1}^N \left( \sum_{j=1}^p \operatorname{tr}_j(x_k) - 1 \right)^2$$

incorporated additively in the full loss: $$\mathcal{L}_{\mathrm{PIP}^2}(\theta) = w_{\mathrm{data}}\mathcal{L}_{\mathrm{data}} + w_{\mathrm{physics}}\mathcal{L}_{\mathrm{physics}} + w_{\mathrm{bc}}\mathcal{L}_{\mathrm{bc}} + \lambda_{P^2}\mathcal{L}_{P^2}$$ This partition penalty stabilizes mode outputs, prevents collapse, and yields marked improvements relative to baselines in empirical PDE solution accuracy (Mi et al., 17 Dec 2025).

5. Algorithmic Frameworks and Implementation

The following table organizes the principal scope-specific penalty operator frameworks.

Application Domain	Penalty Functional	Scope Definition
Monotone Inclusions	$\Pen_C(x) = \varphi_B(x, 0)$	Zeros of maximally monotone $B$
Adaptive PDE Methods	$[\,\cdot\,]_{\sigma(x)}$ via elliptic PDE	Spatial mesh points $\Omega$
Sparse Estimation	$\sum_s \lambda_s \|\beta_s\|_\alpha$	Coordinate/group/fused blocks
Deep Operator Networks	$\mathcal{L}_{P^2}$ partition penalty	Trunk mode outputs across domain

Each framework employs scope-specific penalization either through analytical objects (Fitzpatrick functions), spatially adaptive PDE solves, hierarchical regularization coefficients, or structural architecture components. Training pipelines routinely incorporate these penalties into gradient-based minimization, proximal iterations, or explicitly in physics-informed deep learning models, as in the presented Python pseudocode (Mi et al., 17 Dec 2025).

6. Convergence, Properties, and Empirical Impact

Scope-specific penalty operators contribute to improved feasibility, expressiveness, sparsity control, and numerical stability:

• Convergence: Fitzpatrick-based penalties guarantee weak ergodic (and under further monotonicity, strong) convergence in monotone inclusion problems (Bot et al., 2013). Adaptive PDE penalties enforce exact constraints in the vanishing limit, providing locally superlinear Newton-type convergence (Boon et al., 2022).
• Bias control and selection consistency: Variable-coefficient penalties in Bayesian Lasso enjoy bias reduction for large signals and oracle-consistent likelihood penalties (Wycoff et al., 2022).
• Architectural stability: Partition penalties in PIP$^2$ Net regularize trunk mode outputs, mitigating instability and mode collapse. Empirical $L^2$ errors across nonlinear PDE benchmarks consistently favor partition-penalized models, with error reductions by up to two orders of magnitude (Mi et al., 17 Dec 2025).

A plausible implication is that scope-specific penalty operators, when properly constructed and tuned to the problem structure, yield both theoretical guarantees and practical improvements that are unattainable via global penalization.

7. Extension, Tuning, and Limit Behavior

Extensions of scope-specific penalty operators involve learning penalties at increasing granularity: per group, region, modality, or architecture subcomponent. Tuning of associated hyperparameters $(\lambda_{P^2}, \gamma, s_x s_\lambda)$ is guided by cross-validation and monitoring error/improvement criteria. In adaptive frameworks, scope-specific penalties can transition between regimes—e.g., from smooth penalization to active-set enforcement as residuals vanish (Boon et al., 2022).

In summary, scope-specific penalty operators are foundational regularization primitives tailored to problem-intrinsic decompositions, affording improved enforcement of constraints, adaptivity, and interpretability across monotone inclusion, PDE optics, sparsity modeling, and deep operator learning.