Learning Physics-Informed Neural Networks without Stacked Back-propagation
This addresses a computational bottleneck for researchers and practitioners using PINNs in scientific computing, representing an incremental improvement over existing methods.
The paper tackles the scalability issue of Physics-Informed Neural Networks (PINNs) in high-dimensional second-order PDE problems by developing a novel approach that accelerates training without stacked back-propagation, achieving competitive error and significantly faster performance.
Physics-Informed Neural Network (PINN) has become a commonly used machine learning approach to solve partial differential equations (PDE). But, facing high-dimensional secondorder PDE problems, PINN will suffer from severe scalability issues since its loss includes second-order derivatives, the computational cost of which will grow along with the dimension during stacked back-propagation. In this work, we develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. We further discuss the model capacity and provide variance reduction methods to address key limitations in the derivative estimation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is significantly faster. Our code is released at https://github.com/LithiumDA/PINN-without-Stacked-BP.