Temel Kavramlar
Adaptive resampling of collocation points based on the mixed second derivative of the residual can significantly improve the accuracy of Physics-Informed Neural Network solutions compared to fixed collocation point distributions.
Özet
This paper investigates strategies for selecting the collocation points used in Physics-Informed Neural Networks (PINNs) to solve partial differential equations. The quality of PINN solutions depends heavily on the number and distribution of these collocation points.
The authors consider several adaptive resampling methods that redistribute the collocation points based on different information sources, including the local PDE residual and the mixed spatial and temporal derivatives of the residual and the solution estimate. These adaptive methods are compared against fixed uniform and pseudo-random (Hammersley) collocation point distributions.
The results show that the adaptive methods, especially those using the mixed second derivative of the residual as the guiding metric, can significantly outperform the fixed distributions, particularly when the number of collocation points is relatively small. This suggests that the adaptive approaches can achieve a given level of accuracy with fewer collocation points, potentially reducing the overall computational cost.
The performance of the different methods is evaluated on two benchmark problems - the 1D Burgers' equation and the Allen-Cahn equation. The authors explore the impact of varying the problem parameters, such as the initial conditions and the diffusion coefficient, on the relative effectiveness of the sampling strategies.
Overall, the paper demonstrates that the choice of collocation point distribution can have a substantial impact on the accuracy of PINN solutions, and that adaptive resampling methods, particularly those leveraging information about the mixed derivatives of the residual and solution, can be an effective approach for improving PINN performance.
İstatistikler
uux + ut = ν uxx
u(−1, t) = u(1, t) = 0
u(x, 0) = −sin(πx)
∂u
∂t = D∂2u
∂x2 + 5(u −u3)
u(−1, t) = u(1, t) = −1
u(x, 0) = x2 cos(πx)