In this study, we present a comparative analysis of gradient descent-based optimizers frequently used in Convolutional Neural Networks (CNNs), including SGD, mSGD, RMSprop, Adadelta, Nadam, Adamax, Adam, and the recent EVE optimizer. To explore the interaction between optimization strategies and activation functions, we systematically evaluate all combinations of these optimizers with four activation functions—ReLU, LeakyReLU, Tanh, and GELU—across three benchmark image classification datasets: CIFAR-10, Fashion-MNIST (F-MNIST), and Labeled Faces in the Wild (LFW). Each configuration was assessed using multiple evaluation metrics, including accuracy, precision, recall, F1-score, mean absolute error (MAE), and mean squared error (MSE). All experiments were performed using k-fold cross-validation to ensure statistical robustness. Additionally, two-way ANOVA was employed to validate the significance of differences across optimizer–activation combinations. This study aims to highlight the importance of jointly selecting optimizers and activation functions to enhance training dynamics and generalization in CNNs. We also consider the role of critical hyperparameters, such as learning rate and regularization methods, in influencing optimization stability. This work provides valuable insights into the optimizer–activation interplay and offers practical guidance for improving architectural and hyperparameter configurations in CNN-based deep learning models.

This research paper aims to examine the patterns of Arctic sea ice extent (ASIE) between 1979 and 2020 by using monthly data acquired from NSIDC. Our study employs seasonal time series models such as SARIMA, SARIMAX with global temperature anomalies as an exogenous variable, as well as the neural network (NNAR) models to forecast the future of ASIE in the next 100 years. To evaluate the accuracy of these models, we conduct fittings and analyze various prediction error metrics, MSE, MAE, and MAPE. Our findings reveal that the SARIMAX model outperforms others in out-of-sample forecasting (testing set), effectively integrating data components and accurately capturing both trends and seasonal variations, thus enhancing predictive accuracy for the ASIE relative to other models. According to the SARIMAX model, the overall coverage of sea ice is diminishing, and it is projected that the Arctic will experience its first ice-free month no later than September of 2076, with September of 2074 being a potential milestone. Additionally, the NNAR model suggests that ASIE will not disappear within the next 100 years.

With ongoing climate transformations, reliable Arctic sea ice forecasts are essential for understanding impacts on shipping, ecosystems, and climate teleconnections. This research examines physics-free neural architectures versus physics-informed statistical models for long-term Arctic projections by implementing Fourier Neural Operator (FNO) and Convolutional Neural Network (CNN) alongside a seasonal SARIMAX time series model incorporating physical predictors including temperature anomalies and ice thickness. We test whether neural models trained on historical ice data can match physics-informed SARIMAX reliability, and whether approaches exhibit systematic biases toward specific emission pathways. Using data from January 1979 to December 2024, we conducted forecasts through 2100, with SARIMAX driven by CMIP6 sea ice thickness under SSP2-4.5 and SSP5-8.5 scenarios. Results decisively reject the first hypothesis: both neural models projected ice free Arctic summer by September 2089 regardless of emission scenario, while SARIMAX maintained physically plausible seasonal coverage throughout the century under both pathways. Neural approaches demonstrated systematic bias toward extreme warming exceeding even high-emission projections, revealing fundamental limitations in physics-free deep learning for climate forecasting where physical constraints are paramount.

The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$ with $d=2,$ $1\leq \alpha\leq 2$ and $d\geq 3,$ $0< \alpha\le 2$. Our main result yields a sharper lower bound, in the sense of Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous result in [36]. Furthermore, we give a result improving the bounds for analogous elliptic operators in [19].

We introduce Neural Operator Algebra in Hilbert Space (NOAH), a new framework that learns an adaptive family of operators instead of a single fixed mapping. Its spectral realization, F-NOAH, adds three physics-constrained generators, for advection, diffusion, and reaction, to a Fourier Neural Operator (FNO) backbone, and it restores their defining constraints exactly by closed-form projection after every optimizer step. We test the module in the regime that matters most in practice, a small training set and a short training budget: 500 initial conditions and 50 epochs on the two-dimensional Navier–Stokes equations at viscosity ${10}^{-2}$, trained at $64\times 64$ and evaluated zero-shot up to $1024\times 1024$. The baseline is the Fourier Neural Operator as published. Over fourteen independent repetitions, F-NOAH attains the lower mean squared error in all 280 paired measurements, the pooled improvement is a factor of $2.50$, and every cell remains significant after Holm correction. In the best run, the error falls by $88.3\%$ at $64\times 64$ and by $87.2\%$ at $1024\times 1024$ at $T=10$. The NOAH module supplies a remarkable correction when data and computational resources are limited.