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Recently, the development of deep learning (DL), which has accomplished unbelievable success in many fields, especially in scientific computational fields. And almost all computational problems and physical phenomena can be described by partial differential equations (PDEs). In this work, we proposed two potential high-order geometric flows. Motivation by the physicalinformation neural networks (PINNs) and the traditional level set method (LSM), we have integrated deep neural networks (DNNs) and LSM to make the proposed method more robust and efficient. Also, to test the sensitivity of the system to different input data, we set up three sets of initial conditions to test the model. Furthermore, numerical experiments on different input data are implemented to demonstrate the effectiveness and superiority of the proposed models compared to the state-of-the-art approach.

Inverse problems (IPs) begin with measured data and try to estimate the model parameters. In science and engineering, many problems are seen as inverse problems. Partial differential equations (PDEs) or variational problems are also used to characterize similar issues (VPs). A VP is usually an energy functional that is solved by lowering the energy function. Since curvature-driven regularities have been proven to need considerable prior understanding of physics, they have gotten a lot of attention. Unfortunately, the curvaturedriven regularities correlate to the higher-order EulerLagrangian equations. Furthermore, they frequently have non-smooth and non-convex features, making numerical solutions a difficult challenge in a variety of applications. In this paper (AD), we introduced a method based on physics-constrained deep learning (PCL) and automatic differentiation to handle inverse issues from noisy data. In addition, to address this challenge, we combine standard variational approaches (VMs) with DL-based algorithms. The operator split technique may successfully break non-convex variational models into multiple simple sub-problems to solve. Each sub-problem corresponds to an Euler-Lagrangian PDE, which is effectively solved using deep neural networks (DNNs) via the AD process.

The coronavirus disease, COVID-19, has become a global challenging pandemic. Causing significant loss of life, property, and economics worldwide. The study of infectious disease models allows us to have a deeper understand of spreading trend of highly infectious diseases. Therefore, studying infectious disease models is imminent. In this work, we propose learning spacial domain infectious disease control model for vaccine distribution by physics-constraint machine learning. Usually, these dynamical systems are utilized in epidemiology models to predict the time evolution and development trend of highly infectious diseases such as COVID-19. We reformulate the SIR models and give corresponding policies via dynamical systems. More importantly, we obtain the approximating numerical solution of the systems of dynamical PDEs via the PINNs algorithm, within the acceptable range of approximation error. Additionally, we present several numerical solutions of the PDEs under a variety of scenarios.

Machine learning techniques have provided valuable insights into understanding the fundamental principles of physics. Recent studies have demonstrated the ability of deep neural networks (DNNs) to learn a wide range of differential equations (DEs) and classical Hamiltonian mechanics systems. This has led to the emergence of solving the inverse problem associated with partial differential equations using DNNs. In this paper, we propose a novel method for detecting quantum confinement by numerically solving the Schr¨odinger system on manifolds with nonlinear coupling. Our method exhibits remarkable performance in identifying various quantum phenomena. Moreover, our research introduces a promising approach for principled modeling and prediction in quantum processes. By combining deep learning with traditional models, our hybrid model leverages the strengths of both approaches, enhancing computational efficiency and enabling meaningful predictions for complex quantum physical processes. The integration of multiple deep learning principles with quantum physics enables effective handling of large datasets and addresses inverse challenges related to intricate dynamical systems and physical phenomena.