MgNO: Efficient Parameterization of Linear Operators via Multigrid

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the $i$-th neuron in a nonlinear operator layer is defined by $\mathcal O_i(u) = \sigma\left( \sum_j \mathcal W_{ij} u + \mathcal B_{ij}\right)$. Here, $\mathcal W_{ij}$ denotes the bounded linear operator connecting $j$-th input neuron to $i$-th output neuron, and the bias $\mathcal B_{ij}$ takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs)

Low strain pile testing based on synchrosqueezing wavelet transformation analysis

Low strain detection, an indirect and nondestructive testing method, is one of the main pile integrity testing methods. We propose low strain testing analysis based on a synchrosqueezing wavelet transformation (SST). Through a typical model pile test, the SST is applied to identify pile bottom signal reflection time and to separate signal from noise. It is also compared with the conventional wavelet de-noising and the empirical mode decomposition (EMD) de-noising method. Results show that the SST technique can be used to identify the reflected signal of the pile bottom, achieve signal and noise separation, and improve signal-to-noise ratio. The method has significant advantage in low strain detection signal processing compared to other methods

Simulation analysis of low strain dynamic testing of pile with inhomogeneous elastic modulus

Low strain dynamic testing is an important nondestructive testing method in the engineering. However, the pile foundation material is usually assumed as having a uniform elastic modulus in low strain simulations. In this paper, we consider the elastic modulus of concrete as having an inhomogeneous elastic modulus that is described by the Weibull distribution model. An explicit algorithm was adopted in order to solve the model. The finite element method (FEM) was used to simulate the low strain dynamic test of a 3D pile. The response velocity characteristics of different shape parameters were obtained using this method, and the Daubechies wavelet transform was used to analyze the characteristics of the wavelet modulus. The result shows that simulation response velocity has a correlation with the different homogeneity of the elastic modulus