Paulo Akira F. Enabe

Paulo Akira F. Enabe, Rodrigo Provasi

Low-order virtual element methods (VEM) compute a consistent finite-strain contribution through polynomial projections and rely on stabilization to control the unresolved modes in the projector kernel. In current hyperelastic VEM practice, stabilization is often defined by integrating a nonlinear surrogate energy over an auxiliary sub-triangulation and scaled through modified Lamé parameters and incompressibility factors; this can introduce sensitivity to the arbitrary internal tessellation, complicate consistent Newton linearization, and, most critically, inject bulk-dependent proxies into shear-type kernel penalties, artificially stiffening isochoric missing modes in the nearly incompressible regime. This work develops a submesh-free, kernel-only stabilization that decouples deviatoric and volumetric channels and is explicitly designed to scale like the current Newton tangent energy on the kernel: the deviatoric term is scaled solely by a shear measure and enhanced by bounded geometry-driven directional weights, while the volumetric term is scaled by an independent bulk measure and can be capped or suppressed as $ν\to 1/2$. A spectral framework is established in which the canonical VEM stability requirement on the kernel is characterized by generalized Rayleigh quotients and eigenvalue bounds, and it is shown under standard polygon regularity assumptions that the deviatoric stabilization is uniformly equivalent to $μ_E|\cdot|_{1,E}^2$ on the kernel with constants independent of mesh size and Poisson ratio. Element-level diagnostics confirm that classical surrogate-based stabilizations assign bulk-driven energy to isochoric kernel modes as $ν\to 1/2$, whereas the proposed decoupled stabilization remains shear-scaled; kernel spectra and Cook's membrane simulations in the nearly incompressible regime further support improved robustness across polygonal mesh families.

Paulo Akira F. Enabe

A unified framework for learning under covariate shift is presented, in which a constrained density-ratio network approximates the Radon-Nikodym derivative $r^\star = dP/dQ$ from source $Q$ to target $P$, supports an importance-weighted empirical risk, and feeds an anytime PAC-Bayes generalization certificate. A change-of-measure identity decomposes the gap between target risk and importance-weighted source risk into a ratio-bias term, controlled by the $L^2(Q)$ closeness of the learned ratio to $r^\star$, and a generalization-gap term, controlled by the variability of the weighted loss. Three structural identities of a Radon-Nikodym derivative, normalization, moment matching, and a second-moment penalty controlling the effective sample size, are imposed as hard integral constraints through an augmented-Lagrangian scheme. PAC-Bayes is then instantiated on the weighted risk in a fixed-time regime that yields Bernoulli-KL bounds, a KL-regularized objective whose minimizer is the network-weighted Gibbs posterior, and a stability statement on $L^2(Q)$ perturbations of the learned ratio, and in an anytime regime that builds a time-uniform certificate by geometric peeling across epochs. A pre-registered two-campaign protocol combining a patch test against analytic ground truth with a real-data deployment under intrinsic distribution shift validates the framework. The network produces a calibrated covariate ratio on real data, reduces the target $0/1$ loss relative to unweighted empirical risk minimization and to classical direct ratio-estimation baselines, and attains the anytime certificate as the construction promises. A single pre-registered failure of the fixed-time coverage claim is recorded, with per-split coverage aligning one-to-one with the magnitude of the label shift, confirming that the covariate-only assumption is operationally tight rather than a defect of the certificate.

Paulo Akira F. Enabe

Large language model agents that use external tools are often implemented through reactive execution, in which reasoning is repeatedly recomputed after each observation, increasing latency and sensitivity to error propagation. This work introduces Profile--Then--Reason (PTR), a bounded execution framework for structured tool-augmented reasoning, in which a language model first synthesizes an explicit workflow, deterministic or guarded operators execute that workflow, a verifier evaluates the resulting trace, and repair is invoked only when the original workflow is no longer reliable. A mathematical formulation is developed in which the full pipeline is expressed as a composition of profile, routing, execution, verification, repair, and reasoning operators; under bounded repair, the number of language-model calls is restricted to two in the nominal case and three in the worst case. Experiments against a ReAct baseline on six benchmarks and four language models show that PTR achieves the pairwise exact-match advantage in 16 of 24 configurations. The results indicate that PTR is particularly effective on retrieval-centered and decomposition-heavy tasks, whereas reactive execution remains preferable when success depends on substantial online adaptation.

Paulo Akira F. Enabe, Rodrigo Provasi

A hybrid framework integrating the Virtual Element Method (VEM) with deep learning is presented as an initial step toward developing efficient and flexible numerical models for one-dimensional Euler-Bernoulli beams. The primary aim is to explore a data-driven surrogate model capable of predicting displacement fields across varying material and geometric parameters while maintaining computational efficiency. Building upon VEM's ability to handle higher-order polynomials and non-conforming discretizations, the method offers a robust numerical foundation for structural mechanics. A neural network architecture is introduced to separately process nodal and material-specific data, effectively capturing complex interactions with minimal reliance on large datasets. To address challenges in training, the model incorporates Sobolev training and GradNorm techniques, ensuring balanced loss contributions and enhanced generalization. While this framework is in its early stages, it demonstrates the potential for further refinement and development into a scalable alternative to traditional methods. The proposed approach lays the groundwork for advancing numerical and data-driven techniques in beam modeling, offering a foundation for future research in structural mechanics.