Unsupervised learning with physics informed graph networks for partial differential equations

Natural physical phenomena are commonly expressed using partial differential equations (PDEs), in domains such as fluid dynamics, electromagnetism, and atmospheric science. These equations typically require numerical solutions under given boundary conditions. There is a burgeoning interest in the exploration of neural network methodologies for solving PDEs, mainly based on automatic differentiation methods to learn the PDE-solving process, which means that the model needs to be retrained when the boundary conditions of PDE are changed. However, automatic differentiation requires substantial memory resources to facilitate the training regimen. Moreover, a learning objective that is tailored to the solution process often lacks the flexibility to extend to boundary conditions; thereby limiting the solution’s overall precision. The method proposed in this paper introduces a graph neural network approach, embedded with physical information, mainly for solving Poisson’s equation. An approach is introduced that reduces memory usage and enhances training efficiency through an unsupervised learning methodology based on numerical differentiation. Concurrently, by integrating boundary conditions directly into the neural network as supplementary physical information, this approach ensures that a singular model is capable of solving PDEs across a variety of boundary conditions. To address the challenges posed by more complex network inputs, the introduction of graph residual connections serves as a strategic measure to prevent network overfitting and to elevate the accuracy of the solutions provided. Experimental findings reveal that, despite having 30 times more training parameters than the Physics-Informed Neural Networks (PINN) model, the proposed model consumes 2.2% less memory than PINN. Additionally, generalization in boundary conditions has been achieved to a certain extent. This enables the model to solve partial differential equations with different boundary conditions, a capability that PINN currently lacks. To validate the solving capability of the proposed method, it has been applied to the model equation, the Sod shock tube problem, and the two-dimensional inviscid airfoil problem. In terms of the solution accuracy of the model equations, the proposed method outperforms PINN by 30% to four orders of magnitude. Compared to the traditional numerical method, the Finite Element Method (FEM), the proposed method also shows an order of magnitude improvement. Additionally, when compared to the improved version of PINN, TSONN, our method demonstrates certain advantages. The forward problem of the Sod shock tube, which PINN is currently unable to solve, is successfully handled by the proposed method. For the airfoil problem, the results are comparable to those of PINN.