In this work, we establish the initial boundary value problem of a class of Petrovsky equation of Hartree-type. We prove the global existence of weak solutions by applying potential well theory. Furthermore, we investigate the blow-up phenomena of solutions under nonnegative or negative initial energy conditions.

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.

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.

Regularization plans, acceleration mechanisms, and optimization strategies all have a significant impact on the convergence behavior of neural network training algorithms. A thorough analysis of a batch gradient method (BGM) improved with momentum and penalty terms for training wavelet feedforward neural networks (WFNNs) is presented in this research. In nonlinear function approximation challenges, the suggested approach seeks to enhance convergence speed, stability, and generalization performance. To prove the monotonicity, weak convergence, and strong convergence of the training process under the combined influence of momentum regularization and penalty, a rigorous mathematical framework is created. While the momentum term speeds up convergence by reducing oscillations in the gradient trajectory, the penalty term suppresses weight overgrowth and overfitting. The algorithm is applied to the Peaks function, Sinc function, and 5-bit parity problem to support these theoretical findings. The findings show that in terms of quicker convergence, lower mean squared error (MSE), and smaller gradient norms, the suggested BGM with penalty and momentum performs better than traditional BGM with L₂ regularization. Smoother optimization dynamics and improved generalization are revealed by the training convergence and gradient norm evolution curves. The suggested method offers a cohesive viewpoint on integrating momentum acceleration, regularization, and penalty into gradient-based optimization for WFNNs. It is a promising technique for function approximation, pattern recognition, and control system applications since it provides both theoretical convergence guarantees and real-world increases in learning efficiency.

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.