A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks
A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks
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DOI：
10.1016/j.jcp.2023.112263
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发表时间：
2022-10
期刊：
影响因子：
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通讯作者：
S. Dong;Yiran Wang
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文献类型：
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作者：
S. Dong;Yiran Wang
提出了一种基于随机神经网络的参数偏微分方程（PDE）逆解的求解方法。这将最初为正向偏微分方程开发的局部极端学习机技术扩展到逆问题。我们开发了三种算法来训练神经网络来解决偏微分方程逆问题。第一种算法（称为NLLSQ）确定的逆参数和可训练的网络参数的非线性最小二乘法与扰动（NLLSQ-perturbb）一起。第二种算法（称为VarPro-F1）通过变量投影从整个问题中消除逆参数，以获得仅关于可训练网络参数的简化问题。该算法首先用NLLSQ-扰动算法求解可训练网络参数，然后用线性最小二乘法求逆参数。第三种算法（称为VarPro-F2）通过变量投影从整个问题中消除可训练的网络参数，以获得仅关于逆参数的简化问题。该算法首先求解逆参数的简化问题，然后计算可训练的网络参数。VarPro-F1和VarPro-F2在某种意义上是互逆的。所提出的方法产生精确的结果为逆偏微分方程问题，如本文的数值例子所示。对于无噪声数据，逆参数和解场的误差随着配置点数量或可训练网络参数数量的增加而呈指数下降，并可以达到接近机器精度的水平。对于噪声数据，与无噪声数据的情况相比，精度降低，但该方法仍然相当准确。所提出的方法进行了比较物理通知的神经网络方法。
We present a method for computing the inverse parameters and the solution field to inverse parametric partial differential equations (PDE) based on randomized neural networks. This extends the local extreme learning machine technique originally developed for forward PDEs to inverse problems. We develop three algorithms for training the neural network to solve the inverse PDE problem. The first algorithm (termed NLLSQ) determines the inverse parameters and the trainable network parameters all together by the nonlinear least squares method with perturbations (NLLSQ-perturb). The second algorithm (termed VarPro-F1) eliminates the inverse parameters from the overall problem by variable projection to attain a reduced problem about the trainable network parameters only. It solves the reduced problem first by the NLLSQ-perturb algorithm for the trainable network parameters, and then computes the inverse parameters by the linear least squares method. The third algorithm (termed VarPro-F2) eliminates the trainable network parameters from the overall problem by variable projection to attain a reduced problem about the inverse parameters only. It solves the reduced problem for the inverse parameters first, and then computes the trainable network parameters afterwards. VarPro-F1 and VarPro-F2 are reciprocal to each other in some sense. The presented method produces accurate results for inverse PDE problems, as shown by the numerical examples herein. For noise-free data, the errors of the inverse parameters and the solution field decrease exponentially as the number of collocation points or the number of trainable network parameters increases, and can reach a level close to the machine accuracy. For noisy data, the accuracy degrades compared with the case of noise-free data, but the method remains quite accurate. The presented method has been compared with the physics-informed neural network method.