Physics-informed ConvNet: Learning physical field from a shallow neural network

We present a new approach for solving nonlinear partial differential equation (PDE) on regular or irregular domains based on physics-informed ConvNet, which we call the PICN. The network structure consists of three parts: 1) a convolutional neural network for physical field generation, 2) a pre-trained convolutional layer corresponding to the finite-difference filters to estimate differential fields of the generated physical field, and 3) an interpolation network for loss analysis in irregular geometry domains. From a CNN perspective, the physical field is generated by a deconvolution layer and a convolution layer. Unlike the standard Physics-informed Neural Network (PINN) approach, the convolutions corresponding to the finite-difference filters estimate the spatial gradients forming the physical operator and then construct the PDE residual in a PINN-like loss function. The total loss function involving boundary conditions and the physical constraints in irregular geometry domains can be calculated from an efficient linear interpolation network. The theoretical analysis of PICN convergence is performed on a simplified case for solving a one-dimensional physical field, and several examples of nonlinear PDE of solutions with multifrequency characteristics are executed. The theory and examples confirm the effective learning capability of PICN for the physical field solution with high-frequency components, compared to the standard PINN. A series of numerical cases are performed to validate the current PICN, including the solving (and estimation) of nonlinear physical operator equations and recovering physical information from noisy observations. First, the ability of PICN to solve nonlinear PDE has been verified by executing three nonlinear problems including ODE with sine nonlinearity, PDE involving nonlinear sine-square operators, and Schrödinger equation. The proposed PICN has been assessed by solving some nonlinear PDE on irregular domains such as star-shaped domain, bird-like domain, and starfish domain. Moreover, PICN is applied to identify the thermal diffusivity parameters in an anisotropic heat transfer problem from noisy data, and a denoising display of the temperature field from strong noisy data with standard deviations ranging from 0.1-0.4. The numerical results demonstrate the high accuracy approximation and fast convergence performance of PICN. The potential advantage in approximating complex physical fields with multi-frequency components indicates that PICN may become an alternative efficient neural network solver in physics-informed machine learning. This paper is adapted from the work originally posted on arXiv.com by the same authors (arXiv:2201.10967, Jan 26, 2022). The data and code accompanying this paper are publicly available at https://github.com/zengzhi2015/PICN.