Enhanced Physics-Informed Neural Networks with Optimized Sensor Placement via Multi-Criteria Adaptive Sampling

Chenhong Zhou, Jie Chen, Zaifeng Yang, Alexander Matyasko, Ching Eng Png

Abstract: Physics-informed neural networks (PINNs) have emerged as promising and powerful tools for solving partial differential equations (PDEs). To enforce PINNs that yield accurate solutions to PDEs, a set of scattered spatiotemporal points, known as collocation points, are typically sampled within the computational domain. The choices of collocation points significantly impact the performance of PINNs. However, existing sampling methods primarily rely on the PDE residual, which is insufficient for solutions with steep gradients. To enhance the accuracy of PINNs, we propose a novel multi-criteria adaptive sampling (MCAS) approach to optimally select appropriate collocation points. The MCAS approach integrates three sampling criteria: PDE’s residual, the gradient of residual, and the gradient of solutions, enabling us to capture the PDE violations and the sharpness of solutions. Experimental results demonstrate that the proposed MCAS approach is not only applicable for collocation points but also for optimizing sensor placement, consistently outperforming the residual-based sampling methods.