Learning Specialized Activation Functions for Physics-Informed Neural Networks (Communications in Computational Physics, 2023)
Physics-informed neural networks (PINNs) are known to suffer from optimization difficulty. In this work, we reveal the connection between the optimizationdifficulty of PINNs and activation functions. Specifically, we show that PINNs exhibithigh sensitivity to activation functions when solving PDEs with distinct properties.Existing works usually choose activation functions by inefficient trial-and-error. Toavoid the inefficient manual selection and to alleviate the optimization difficulty ofPINNs, we introduce adaptive activation functions to search for the optimal functionwhen solving different problems. We compare different adaptive activation functionsand discuss their limitations in the context of PINNs. Furthermore, we propose to tailor the idea of learning combinations of candidate activation functions to the PINNsoptimization, which has a higher requirement for the smoothness and diversity onlearned functions. This is achieved by removing activation functions which cannotprovide higher-order derivatives from the candidate set and incorporating elementaryfunctions with different properties according to our prior knowledge about the PDE athand. We further enhance the search space with adaptive slopes. The proposed adaptive activation function can be used to solve different PDE systems in an interpretableway. Its effectiveness is demonstrated on a series of benchmarks. Code is available athttps://github.com/LeapLabTHU/AdaAFforPINNs.