The authors introduce a novel adaptive sampling technique called Annealed Adaptive Importance Sampling (AAIS) to improve the performance of Physics-Informed Neural Networks (PINNs) in solving high-dimensional partial differential equations (PDEs).
The key highlights are:
The AAIS method is inspired by the Expectation Maximization (EM) algorithm and aims to approximate complex, multi-modal target distributions derived from PDE residuals. It employs finite mixtures, including both Gaussian and Student's t-distributions, to mimic the target density.
The authors propose a straightforward yet robust resampling framework for PINNs that maintains a controlled training dataset size while strategically incorporating adaptive points to mitigate the risk of local minima.
Numerical experiments on various high-dimensional Poisson problems demonstrate the superior performance of the AAIS-PINN approach compared to conventional PINNs and other adaptive sampling methods, especially in scenarios with singular or multi-modal solutions.
The AAIS algorithm shows promising capabilities in solving high-dimensional PDEs, where it outperforms the Residual-based Adaptive Distribution (RAD) method, particularly when the number of search points is limited.
The authors observe a consistent "frequency-increasing" phenomenon in the residuals across different PDE problems when using the proposed adaptive sampling methods, indicating their efficiency in capturing multi-scale solutions.
Overall, the AAIS-PINN framework represents a significant advancement in addressing the challenges of solving high-dimensional PDEs, with potential applications in a wide range of scientific and engineering domains.
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