Physics-informed neural networks (PINNs) aim to constrain the outputs and gradients of deep learning models to satisfy specified governing physics equations, which have demonstrated significant potential for solving partial differential equations (PDEs). Although existing PINN methods have achieved pleasing performance, they always treat both easy and hard sample points indiscriminately, especially ones in the physical boundaries. This easily causes the PINN model to fall into undesirable local minima and unstable learning, thereby resulting in an Unbalanced Prediction Problem (UPP). To deal with this daunting problem, we propose a novel framework named Cognitive Physical Informed Neural Network (CoPINN) that imitates the human cognitive learning manner from easy to hard. Specifically, we first employ separable subnetworks to encode independent one-dimensional coordinates and apply an aggregation scheme to generate multi-dimensional predicted physical variables. Then, during the training phase, we dynamically evaluate the difficulty of each sample according to the gradient of the PDE residuals. Finally, we propose a cognitive training scheduler to progressively optimize the entire sampling regions from easy to hard, thereby embracing robustness and generalization against predicting physical boundary regions. Extensive experiments demonstrate that our CoPINN achieves state-of-the-art performance, particularly significantly reducing prediction errors in stubborn regions.

Partial differential equations (PDEs) can describe the laws of change of natural phenomena in disciplines such as physics, chemistry, and economics. Solving PDEs has a wide range of applications in many fields of science and engineering. Physics-Informed Neural Networks (PINN) is an effective method for solving PDEs and has attracted much attention. However, existing PINN methods always treat both easy and hard sample points indiscriminately, which causes the PINN model to fall into undesirable local minima and unstable learning, thereby resulting in an Unbalanced Prediction Problem (UPP). To deal with this, we propose Cognitive Physics-Informed Neural Networks CoPINN that imitate the human cognitive learning manner from easy to hard, thereby effectively mitigating UPP and promoting the application of the PINN method in real scenarios.