Multi-stage Neural Networks: Function Approximator of Machine Precision

Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below $O(10^{-5})$ even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision $O(10^{-16})$ of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone.Comment: 38 pages, 17 page

The First Protocol Of Reaching Consensus Under Unreliable Mobile Edge Computing Paradigm

Mobile Edge Computing (MEC) is an emerging technology that enables computing directly at the edge of the cloud computing network. Therefore, it is important that MEC is applied with reliable transmission. The problem of reaching consensus in the distributed system is one of the most important issues in designing a reliable transmission network. However, all previous protocols for the consensus problem are not suitable for an MEC paradigm. It is the first time an optimal protocol of reaching consensus is pro- posed for MEC paradigm. The protocol makes all fault-free nodes communicate with each other and collect the exchanged messages to decide a common value. Based on the common value, the protocol ensures all fault-free nodes reach consensus without the influence of unreliable transmission. Finally, we proved theoretically that the proposed protocol can tolerate the maximum number of faulty components and using only two rounds of message exchanges