[论文解读] Self-adaptive loss balanced Physics-informed neural networks for the incompressible Navier-Stokes equations

Zixue Xiang, Wei Peng+3 more|arXiv (Cornell University)|Apr 13, 2021

Model Reduction and Neural Networks参考文献 52被引用 26

一句话总结

论文提出 lbPINNs，一种自适应、损失平衡的物理信息神经网络框架，用于求解不可压Navier-Stokes方程，通过在训练中自适应损失项权重来提高精度。与基线PINNs相比，在二维Kovasznay流、二维圆柱尾迹和三维Beltrami流中显示出更优结果。

ABSTRACT

There have been several efforts to Physics-informed neural networks (PINNs) in the solution of the incompressible Navier-Stokes fluid. The loss function in PINNs is a weighted sum of multiple terms, including the mismatch in the observed velocity and pressure data, the boundary and initial constraints, as well as the residuals of the Navier-Stokes equations. In this paper, we observe that the weighted combination of competitive multiple loss functions plays a significant role in training PINNs effectively. We establish Gaussian probabilistic models to define the loss terms, where the noise collection describes the weight parameter for each loss term. We propose a self-adaptive loss function method, which automatically assigns the weights of losses by updating the noise parameters in each epoch based on the maximum likelihood estimation. Subsequently, we employ the self-adaptive loss balanced Physics-informed neural networks (lbPINNs) to solve the incompressible Navier-Stokes equations,\hspace{-1pt} including\hspace{-1pt} two-dimensional\hspace{-1pt} steady Kovasznay flow, two-dimensional unsteady cylinder wake, and three-dimensional unsteady Beltrami flow. Our results suggest that the accuracy of PINNs for effectively simulating complex incompressible flows is improved by adaptively appropriate weights in the loss terms. The outstanding adaptability of lbPINNs is not irrelevant to the initialization choice of noise parameters, which illustrates the robustness. The proposed method can also be employed in other problems where PINNs apply besides fluid problems.

研究动机与目标

• 动机：在不可压Navier-Stokes问题中需要改进多物理损失项的训练平衡。

提出的方法

• 在PINN框架中对NS方程进行表述，残差为 f1 和 f2，并包含多种损失项（PDE、边界、初始、数据）。
• 引入对损失项的高斯概率建模，配备每项观测噪声参数，通过最大似然实现对损失的平衡。

实验结果

研究问题

• RQ1 动态图的数据驱动权重的损失项能否提高不可压NS流动中PINN的精度？
• RQ2 自适应损失平衡对2D和3D流动问题的收敛性与鲁棒性有何影响？
• RQ3 初始噪声配置对 lbPINNs 性能和鲁棒性有何影响？
• RQ4 lbPINNs 是否能在有代表性的层流与非定常 NS 情况下取得低于基线PINNs的误差？

主要发现

• lbPINNs 在测试案例中的速度和压力相对误差显著低于基线PINNs（在某些设置下低至 1e-4 到 1e-5）。
• 自适应噪声参数在训练中演化以平衡 PDE 与边界/数据损失，加速收敛。
• lbPINNs 对不同初始噪声设置具有鲁棒性，并在测试的流场中维持比基线PINNs更高的准确性。
• 对于二维Kovasznay流，lbPINNs 在相似条件下达到接近 6.4e-4 的 L2 误差，而在类似条件下基线 PINNs 为 4.4e-3。
• lbPINNs 在二维圆柱尾迹和三维 Beltrami 流演示中体现出更高的速度和压力精度，PDE 与数据损失在自适应噪声的作用下同步收敛。
• 该方法提供了一种有原则的损失权重分配方法，扩展到流体动力学以外的其他 PINN 应用。