Since the PDE in Eq. 5 in the main manuscript is equivalent to the E-L equation of the quadratic

Recall C(xt|(1 t)xt 1+ t/K)fordiffxt, which (1 t)+ t/K if xt = xt 1 and t/K otherwise.

The key task of machine learning is to minimize the loss function that measures the model fit to the training data. The numerical methods to do this efficiently depend on the properties of the loss function. The most decisive among these properties is the convexity or non-convexity of the loss function. The fact that the loss function can have, and frequently has, non-convex regions has led to a widespread commitment to non-convex methods such as Adam. However, a local minimum implies that, in some environment around it, the function is convex. In this environment, second-order minimizing methods such as the Conjugate Gradient (CG) give a guaranteed superlinear convergence. We propose a novel framework grounded in the hypothesis that loss functions in real-world tasks swap from initial non-convexity to convexity towards the optimum. This is a property we leverage to design an innovative two-phase optimization algorithm. The presented algorithm detects the swap point by observing the gradient norm dependence on the loss. In these regions, non-convex (Adam) and convex (CG) algorithms are used, respectively. Computing experiments confirm the hypothesis that this simple convexity structure is frequent enough to be practically exploited to substantially improve convergence and accuracy.

Section 3.1, we flatten and process the image as a sequence of the length of 784 pixel-by-pixel. The baseline LSTM models consist of one LSTM cell with 128 and 256 hidden units. Orthogonal initialization is used for input-to-hidden weights, while hidden-to-hidden weights are initialized to identity matrices. The gradient norms are clipped to 1 during training. The log-magnitude of these sequences is fed into the models as the input data.

Training deep neural networks remains computationally intensive due to the itera2 tive nature of gradient-based optimization. We propose Gradient Flow Matching (GFM), a continuous-time modeling framework that treats neural network training as a dynamical system governed by learned optimizer-aware vector fields. By leveraging conditional flow matching, GFM captures the underlying update rules of optimizers such as SGD, Adam, and RMSprop, enabling smooth extrapolation of weight trajectories toward convergence. Unlike black-box sequence models, GFM incorporates structural knowledge of gradient-based updates into the learning objective, facilitating accurate forecasting of final weights from partial training sequences. Empirically, GFM achieves forecasting accuracy that is competitive with Transformer-based models and significantly outperforms LSTM and other classical baselines. Furthermore, GFM generalizes across neural architectures and initializations, providing a unified framework for studying optimization dynamics and accelerating convergence prediction.

Graph neural networks (GNNs) have become essential tools for analyzing non-Euclidean data across various domains. During training stage, sampling plays an important role in reducing latency by limiting the number of nodes processed, particularly in large-scale applications. However, as the demand for better prediction performance grows, existing sampling algorithms become increasingly complex, leading to significant overhead. To mitigate this, we propose YOSO (You-Only-Sample-Once), an algorithm designed to achieve efficient training while preserving prediction accuracy. YOSO introduces a compressed sensing (CS)-based sampling and reconstruction framework, where nodes are sampled once at input layer, followed by a lossless reconstruction at the output layer per epoch. By integrating the reconstruction process with the loss function of specific learning tasks, YOSO not only avoids costly computations in traditional compressed sensing (CS) methods, such as orthonormal basis calculations, but also ensures high-probability accuracy retention which equivalent to full node participation. Experimental results on node classification and link prediction demonstrate the effectiveness and efficiency of YOSO, reducing GNN training by an average of 75\% compared to state-of-the-art methods, while maintaining accuracy on par with top-performing baselines.

Convolutional Neural Networks (CNNs) have been heavily used in Deep Learning due to their success in various tasks. Nonetheless, it has been observed that CNNs suffer from redundancy in feature maps, leading to inefficient capacity utilization. Efforts to mitigate and solve this problem led to the emergence of multiple methods, amongst which is kernel orthogonality through variant means. In this work, we challenge the common belief that kernel orthogonality leads to a decrease in feature map redundancy, which is, supposedly, the ultimate objective behind kernel orthogonality. We prove, theoretically and empirically, that kernel orthogonality has an unpredictable effect on feature map similarity and does not necessarily decrease it. Based on our theoretical result, we propose an effective method to reduce feature map similarity independently of the input of the CNN. This is done by minimizing a novel loss function we call Convolutional Similarity. Empirical results show that minimizing the Convolutional Similarity increases the performance of classification models and can accelerate their convergence. Furthermore, using our proposed method pushes towards a more efficient use of the capacity of models, allowing the use of significantly smaller models to achieve the same levels of performance.