Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs
Published in SIAM Journal of Computational Physics (under review), 2023
Recommended citation: Z. Hu, Z. Yang, Y. Wang, G.E. Karniadakis, K. Kawaguchi, Bias-variance trade-off in physics-informed neural networks with randomized smoothing for high-dimensional PDEs, 2023, arXiv preprint arXiv:2311.15283.
In this paper, we present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We also propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version.