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Our recent study has found that physics-informed neural networks (PINN) tend to be local approximators after training. This observation led to the development of a novel physics-informed radial basis network (PIRBN), which is capable of maintaining the local approximating property throughout the entire training process. Unlike deep neural networks, a PIRBN comprises 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. Numerical examples demonstrated that PIRBN is more effective than PINN in solving nonlinear partial differential equations 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 are available at https://github.com/JinshuaiBai/PIRBN. • We found that PINNs tend to be local approximators during training process. • We proposed the physics-informed radial basis network (PIRBN). • We proved that training a PIRBN can converge to Gaussian Processes. • We proved that PIRBNs can maintain the local approximating property during training. • PIRBN is effective for problems with high-frequency features and ill-posed domains. [ABSTRACT FROM AUTHOR]

Neural Particle Method (NPM) is a newly proposed Physics-Informed Neural Network (PINN) based, truly meshfree method for hydrodynamics modeling. In the NPM, PINN is applied to globally approximate field variables, and the high-order Implicit Runge–Kutta (IRK) method is used to treat the time integration. The NPM can easily achieve incompressibility and deal with the free-surface problem. However, the NPM is currently limited to inviscid hydrodynamics problems and is computationally expensive. In this work, we developed a general NPM (gNPM) for viscous hydrodynamics modeling. In the gNPM, a single pressure is output as the predicted pressure field rather than the multiple pressures in the original NPM. Thus, the size of the neural network is greatly reduced, making the gNPM computationally more efficient. Besides, the spatial derivatives in the governing equations are calculated with respect to the current spatial coordinates rather than the predicted space. In this manner, the gNPM is more straightforward to be implemented. Furthermore, by considering the viscous term in the conservation of momentum, the gNPM can be applied for viscous hydrodynamics modeling. The effectiveness and robustness of the gNPM have been demonstrated through several hydrodynamics benchmark cases with different boundary conditions. We highlight that the proposed gNPM is able to cope with highly uneven particle distributions, while the traditional meshfree method can produce severe failure. • A general Neural Particle Method (gNPM) is proposed for hydrodynamics modeling. • The gNPM is computationally more efficient and easier to be implemented than the NPM. • The effectiveness of the gNPM has been demonstrated through benchmark cases. • The gNPM is more robust than the SPH for uneven particle distributions. [ABSTRACT FROM AUTHOR]