In this study, we construct an error estimate for a fully discrete finite element scheme that satisfies the criteria of unconditional energy stability, as suggested in ^[1]. Our theoretical findings, in more detail, demonstrate that this system has second-order accuracy in both space and time. Additionally, we offer a powerful space and time adaptable approach for solving the Cahn-Hilliard problem numerically based on the posterior error estimation. The major goal of this technique is to successfully lower the calculated cost by controlling the mesh size using a Superconvergent Cluster Recovery (SCR) approach in accordance with the error estimation. To demonstrate the effectiveness and stability of the suggested SCR-based algorithm, numerical results are provided.

In this paper, an accurate fractional physical information neural network with an adaptive learning rate (adaptive-fPINN-PQI) was first proposed for solving fractional partial differential equations. First, piecewise quadratic interpolation (PQI) in the sense of the Hadamard finite-part integral was introduced in the neural network to discretize the time-fractional derivative in the Caputo sense. Second, the adaptive learning rate residual network was constructed to keep the network from being stuck in the locally optimal solution, which automatically adjusts the weights of different loss terms, significantly balancing their gradients. Additionally, different from the traditional physical information neural networks, this neural network employs a new composite activation function based on the principle of Fourier transform instead of a single activation function, which significantly enhances the network's accuracy. Finally, numerous time-fractional diffusion and time-fractional phase-field equations were solved using the proposed adaptive-fPINN-PQI to demonstrate its high precision and efficiency.