BO-SA-PINNs: Self-adaptive physics-informed neural networks based on Bayesian optimization for automatically designing PDE solvers
This work addresses the efficiency bottleneck for researchers and practitioners using PINNs to solve PDEs, though it is incremental as it builds on existing PINN methods with automated tuning.
The paper tackles the need for manual hyperparameter tuning in physics-informed neural networks (PINNs) for solving partial differential equations (PDEs) by proposing BO-SA-PINNs, a multi-stage framework that automatically designs hyperparameters and loss weights, achieving higher accuracy and fewer iterations in test cases like Helmholtz and Burgers equations.
Physics-informed neural networks (PINNs) is becoming a popular alternative method for solving partial differential equations (PDEs). However, they require dedicated manual modifications to the hyperparameters of the network, the sampling methods and loss function weights for different PDEs, which reduces the efficiency of the solvers. In this paper, we pro- pose a general multi-stage framework, i.e. BO-SA-PINNs to alleviate this issue. In the first stage, Bayesian optimization (BO) is used to select hyperparameters for the training process, and based on the results of the pre-training, the network architecture, learning rate, sampling points distribution and loss function weights suitable for the PDEs are automatically determined. The proposed hyperparameters search space based on experimental results can enhance the efficiency of BO in identifying optimal hyperparameters. After selecting the appropriate hyperparameters, we incorporate a global self-adaptive (SA) mechanism the second stage. Using the pre-trained model and loss information in the second-stage training, the exponential moving average (EMA) method is employed to optimize the loss function weights, and residual-based adaptive refinement with distribution (RAR-D) is used to optimize the sampling points distribution. In the third stage, L-BFGS is used for stable training. In addition, we introduce a new activation function that enables BO-SA-PINNs to achieve higher accuracy. In numerical experiments, we conduct comparative and ablation experiments to verify the performance of the model on Helmholtz, Maxwell, Burgers and high-dimensional Poisson equations. The comparative experiment results show that our model can achieve higher accuracy and fewer iterations in test cases, and the ablation experiments demonstrate the positive impact of every improvement.