Modelling and predicting the behaviour of infectious diseases is essential for early warning and evaluating the most effective interventions to prevent significant harm. Compartmental models produce a system of ordinary differential equations (ODEs) that are renowned for simulating the transmission dynamics of infectious diseases. However, the parameters in compartmental models are often unknown, and they can even change over time in the real world, making them difficult to determine. This study proposes an advanced artificial intelligence approach based on physics-informed neural networks (PINNs) to estimate time-varying parameters from given data for the compartmental model. Our proposed PINNs method captures the complex dynamics of COVID-19 by integrating a modified Susceptible-Exposed-Infectious-Recovered-Death (SEIRD) compartmental model with deep neural networks. Specifically, we modelled the system of ODEs as one network and the time-varying parameters as another network to address significant unknown parameters and limited data. Such structure of the PINNs method is in line with the prior epidemiological correlations and comprises the mismatch between available data and network output and the residual of ODEs. The experimental findings on real-world reported data data have demonstrated that our method robustly and accurately learns the dynamics and forecasts future states. Moreover, as more data becomes available, our proposed PINNs method can be successfully extended to other regions and infectious diseases.

Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential equations (PDEs) with multiple scales by utilizing Fourier-based mixed physics informed neural networks (dubbed FMPINN), its solver is configured as a multi-scale deep neural network. In contrast to the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix caused by the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, to validate the efficiency and robustness of the proposed FMPINN algorithm, we present several numerical examples of multi-scale problems in various dimensional Euclidean spaces. These examples cover low-frequency and high-frequency oscillation cases, demonstrating the effectiveness of our approach. All code and data accompanying this manuscript will be publicly available at https://github.com/Blue-Giant/FMPINN.

Learning with noise is a practically challenging problem in deep face recognition. Despite the success of large margin softmax loss functions, these methods are designed for clean face databases. Considering the inevitable noise in the large scale databases, we first analyze the performance of noise in the training databases. For noise-robust deep face recognition, we propose a dynamic training data dropout (DTDD) method to dynamically filter the noise in the training database and gradually form a stable refined database for model learning. Specifically, we leverage the information provided by the model predictions of accumulated training epochs, which can distinguish regular samples and noise effectively and accurately. The proposed DTDD method is easy and stable for implementation, and can be combined with existing state-of-the-art loss functions and network architectures. Extensive experiments on CASIA-WebFace, VGGFace2, and MS-Celeb-1 M databases empirically demonstrate that our proposed method can robustly train deep face recognition models in the presence of label noise and low quality images.