Physics-informed radial basis network (PIRBN): A local approximating neural network for solving nonlinear PDEs

Our recent intensive study has found that physics-informed neural networks (PINN) tend to be local approximators after training. This observation leads to this novel physics-informed radial basis network (PIRBN), which can maintain the local property throughout the entire training process. Compared to deep neural networks, a PIRBN comprises of only one hidden layer and a radial basis "activation" function. Under appropriate conditions, we demonstrated that the training of PIRBNs using gradient descendent methods can converge to Gaussian processes. Besides, we studied the training dynamics of PIRBN via the neural tangent kernel (NTK) theory. In addition, comprehensive investigations regarding the initialisation strategies of PIRBN were conducted. Based on numerical examples, PIRBN has been demonstrated to be more effective and efficient than PINN in solving PDEs with high-frequency features and ill-posed computational domains. Moreover, the existing PINN numerical techniques, such as adaptive learning, decomposition and different types of loss functions, are applicable to PIRBN. The programs that can regenerate all numerical results can be found at https://github.com/JinshuaiBai/PIRBN.

Physics-guided deep learning for data scarcity

Data are the core of deep learning (DL), and the quality of data significantly affects the performance of DL models. However, high-quality and well-annotated databases are hard or even impossible to acquire for use in many applications, such as structural risk estimation and medical diagnosis, which is an essential barrier that blocks the applications of DL in real life. Physics-guided deep learning (PGDL) is a novel type of DL that can integrate physics laws to train neural networks. It can be used for any systems that are controlled or governed by physics laws, such as mechanics, finance and medical applications. It has been shown that, with the additional information provided by physics laws, PGDL achieves great accuracy and generalisation when facing data scarcity. In this review, the details of PGDL are elucidated, and a structured overview of PGDL with respect to data scarcity in various applications is presented, including physics, engineering and medical applications. Moreover, the limitations and opportunities for current PGDL in terms of data scarcity are identified, and the future outlook for PGDL is discussed in depth.