Although wavelet neural networks (WNNs) combine the expressive capability of neural models with multiscale localization, there are currently few theoretical guarantees for their training. We investigate the weight decay (${\mathrm{L}}_2$ regularization) optimization dynamics of gradient descent (GD) for WNNs. Using explicit rates controlled by the spectrum of the regularized Gram matrix, we first demonstrate global linear convergence to the unique ridge solution for the feature regime when wavelet atoms are fixed and only the linear head is trained. Second, for fully trainable WNNs, we demonstrate linear rates in regions satisfying a Polyak–Łojasiewicz (PL) inequality and establish convergence of GD to stationary locations under standard smoothness and boundedness of wavelet parameters; weight decay enlarges these regions by suppressing flat directions. Third, we characterize the implicit bias in the over-parameterized neural tangent kernel (NTK) regime: GD converges to the minimum reproducing kernel Hilbert space (RKHS) norm interpolant associated with the WNN kernel with ${\mathrm{L}}_2$. In addition to an assessment process on synthetic regression, denoising, and ablations across λ and stepsize, we supplement the theory with useful recommendations on initialization, stepsize schedules, and regularization scales. Together, our findings give a principled prescription for dependable training that has broad applicability to signal processing applications and shed light on when and why ${\mathrm{L}}_2$-regularized GD is stable and quick for WNNs.

The stability of the bcc FM of Fe, the magnetic moments, electronic structure, and optical properties have calculated for the cases of bcc and fcc Fe, for non-magnetic (NM), ferromagnetic (FM), and antiferromagnetic (AFM) states, using the full-potential linearized augmented plane wave method (FP-LAPW) with the generalized gradient approximation (GGA) and GGA+U within the framework of density functional theory (DFT), as implemented in the WIEN2k code. The magnetic moments increase when significant correlations are taken into account. The Density of State (DOS) changes significantly with the FM state but not much with the NM or AFM states, which is consistent with Iota's findings. The optical properties results obtained for the interband dielectric function ε(w), variation in the energy-loss spectrum L(w), absorption coefficient α(w), refractive index n(w), reflectivity R(w), and optical absorption σ(w), are comparable with other studies. Our findings are in good agreement with earlier experimental and theoretical Findings.

This study employs an in-situ growth strategy utilizing a catalyst composed of NiMn2O4/Tamarind Seed-derived carbon (TSAC), applying a hydrothermal synthesis method to affix flower-like nanostructures of NiMn2O4 onto Tamarind Seed-derived activated carbon. The resulting hybrid composite catalyst, NiMn2O4/Tamarind Seed-derived carbon exhibits exceptional electrochemical performance in the oxygen evolution reaction (OER). Significantly enhanced OER catalytic activity is observed in the NiMn2O4/TSAC composite compared to the original NiMn2O4 sample, attributing this improvement to the hierarchical structure of the NiMn2O4 /TSAC system. The NiMn2O4 nanoflowers within the system contribute to an increased abundance of active OER sites, with TSAC facilitating rapid charge transfer. Validation of these attributes is provided through the utilization of tip Spar structure compounds for OER electrocatalysis. Furthermore, the NiMn2O4/TSAC electrode exhibits an impressive specific capacity of 207 mAh g-1 under a current density of 500 mA g-1, emphasizing its remarkable performance and potential for high-performance energy storage applications. The closely packed arrangement of anions, alongside Ni2+, Mn2+, Ni3+, and Mn3+ cations, forms an intricate composition, activating octahedral active sites even under extremely low activation conditions, resulting in enhanced electrochemical performance.

A concentration-dependent investigation of Fe-doped anatase TiO₂ was performed using the full-potential linearized augmented plane wave method implemented in WIEN2k, combined with GGA, GGA+U, and physics-informed neural network (PINN) modeling. Fe concentrations of 3.125%, 6.25%, 12.5%, and 25% were examined using 2 × 2 × 2 anatase TiO₂ supercells. Fe incorporation systematically expands the anatase lattice, with the equilibrium volume increasing from 17.14 ų/f.u. for pristine TiO₂ to 18.49 ų/f.u. within GGA and up to 18.91 ų/f.u. using Hubbard-corrected treatments at 25% Fe. The bulk modulus decreases from 179.2 GPa to 160.5 GPa in GGA, indicating concentration-driven lattice softening, while the formation energy becomes more negative, reaching −10.91 eV/f.u. for GGA+U_Ti+U_Fe at 25% Fe. Electronically, Fe doping introduces localized Fe-3d impurity states within the forbidden gap, narrowing the band gap from 2.18 eV for pristine TiO₂ to 1.23 eV in GGA and to 1.59 eV in GGA+U_Ti+U_Fe at 25% Fe. The ferromagnetic magnetic moment increases with Fe concentration, reaching 4.41 μB/f.u. for the combined Hubbard correction. Optical calculations reveal enhanced dielectric, conductivity, absorption, reflectivity, and loss responses in the visible-to-near-UV range due to Fe-3d/O-2p/Ti-3d hybridization. The PINN model closely reproduces the FP-LAPW/GGA+U trends, including band gaps from 2.25 eV at 3.125% Fe to 1.48 eV at 25% Fe, demonstrating its usefulness as an efficient surrogate for correlated-property prediction. These results establish Fe doping as an effective route for multifunctional tuning of anatase TiO₂ for photocatalytic, photovoltaic, optoelectronic, and spintronic applications.

The convergence characteristics of an online gradient learning technique for training Pi–Sigma higher-order neural networks under a smoothed L1 regularization framework with adaptive momentum are examined in this research. Although Pi-Sigma networks can effectively represent high-order nonlinear interactions, nonconvexity, parameter coupling, and noise sensitivity in online contexts make training them difficult. In order to overcome these problems, we avoid the nondifferentiability of the traditional L1 norm while promoting sparsity by using a differentiable approximation of the L1 penalty. Furthermore, an adaptive momentum term is added to speed up convergence and stabilize weight updates. We construct important lemmas that describe the behavior of the smoothed regularizer and momentum dynamics, make modest assumptions on the activation functions, data sequence, and learning parameters, and create a unified mathematical model for the suggested learning rule. These findings allow us to demonstrate that the online approach guarantees the weight sequence's boundedness, the gradients' convergence to zero, and the corresponding energy function's monotonic reduction. In comparison to existing models, numerical studies show that the suggested approach produces stable convergence, enhanced sparsity, and decreased gradient oscillation. Empirical plots of loss evolution, gradient norms, and weight norms are used to validate the theoretical results. The robustness of the suggested learning framework and its applicability for nonlinear function approximation and online learning tasks in higher-order neural networks are highlighted by the agreement between theoretical results and actual data.

Although wavelet neural networks (WNNs) combine the expressive capability of neural models with multiscale localization, there are currently few theoretical guarantees for their training. We investigate the weight decay (L₂ regularization) optimization dynamics of gradient descent (GD) for WNNs. Using explicit rates controlled by the spectrum of the regularized Gram matrix, we first demonstrate global linear convergence to the unique ridge solution for the feature regime when wavelet atoms are fixed and only the linear head is trained. Second, for fully trainable WNNs, we demonstrate linear rates in regions satisfying a Polyak–Łojasiewicz inequality and establish convergence of GD to stationary locations under standard smoothness and boundedness of wavelet parameters; weight decay enlarges these regions by suppressing flat directions. Third, we characterize the implicit bias in the over-parameterized (NTK) regime: GD converges to the minimum-RKHS-norm interpolant associated with the WNN kernel with L₂. In addition to an assessment process on synthetic regression, denoising, and ablations across λ and stepsize, we supplement the theory with useful recommendations on initialization, stepsize schedules, and regularization scales. Together, our findings give a principled prescription for dependable training that has broad applicability to signal processing applications and shed light on when and why L₂-regularized GD is stable and quick for WNNs.