DiffGrad for Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) are regarded as state-of-the-art tools for addressing highly nonlinear problems based on partial differential equations. Despite their broad range of applications, PINNs encounter several performance challenges, including issues related to efficiency, minimization of computational cost, and enhancement of accuracy. Burgers' equation, a fundamental equation in fluid dynamics that is extensively used in PINNs, provides flexible results with the Adam optimizer that does not account for past gradients. This paper introduces a novel strategy for solving Burgers' equation by incorporating DiffGrad with PINNs, a method that leverages the difference between current and immediately preceding gradients to enhance performance. A comprehensive computational analysis is conducted using optimizers such as Adam, Adamax, RMSprop, and DiffGrad to evaluate and compare their effectiveness. Our approach includes visualizing the solutions over space at various time intervals to demonstrate the accuracy of the network. The results show that DiffGrad not only improves the accuracy of the solution but also reduces training time compared to the other optimizers.

SwishReLU: A Unified Approach to Activation Functions for Enhanced Deep Neural Networks Performance

ReLU, a commonly used activation function in deep neural networks, is prone to the issue of "Dying ReLU". Several enhanced versions, such as ELU, SeLU, and Swish, have been introduced and are considered to be less commonly utilized. However, replacing ReLU can be somewhat challenging due to its inconsistent advantages. While Swish offers a smoother transition similar to ReLU, its utilization generally incurs a greater computational burden compared to ReLU. This paper proposes SwishReLU, a novel activation function combining elements of ReLU and Swish. Our findings reveal that SwishReLU outperforms ReLU in performance with a lower computational cost than Swish. This paper undertakes an examination and comparison of different types of ReLU variants with SwishReLU. Specifically, we compare ELU and SeLU along with Tanh on three datasets: CIFAR-10, CIFAR-100 and MNIST. Notably, applying SwishReLU in the VGG16 model described in Algorithm 2 yields a 6% accuracy improvement on the CIFAR-10 dataset.