Paper Notes: Unsupervised Learning for Full Waveform Inversion: Making CNNs and Partial Differential Equations a Loop

Abstract : To share the understanding of the paper, the original text can be found in Peng Jin, Xitong Zhang, Yinpeng Chen, Sharon Xiaolei Huang, Zicheng Liu, Youzuo Lin, Unsupervised learning of full-waveform inversion: connecting CNN and partial differential equation in a loop. The paper was published
in The top meeting in computer science is ICLR.

1. Paper Contribution

• An unsupervised FWI network is proposed. In fact, it is a bit far-fetched to say "unsupervised", because its supervisory information (velocity model) can calculate the loss between the data obtained through forward modeling and the original data.
• Made a data set OpenFWI, which is specially introduced in another paper. It is very important for researchers in this direction.

2. Thesis work

Figure 1. Method framework

• CNN for inversion
• Forward modeling with PDE

2.1 Forward modeling

$${\nabla }^{2}p(\mathbf{r},t)-\frac{1}{v(\mathbf{r}{)}^2}\frac{{\partial }^{2}p(r,t)}{\partial t^2}=s(\mathbf{r},t)\text{\hspace{1em}( 1 )}$$
where$p(\mathbf{r},t)$ is at$Time\; t$ , positionr \mathbf{r}The pressure wavefield of$r$$v\left(r\right)$ is the velocity graph,$s(r,t)$ is the source item.
The forward modeling process is
$$\tilde{\mathbf{p}}=f(\hat{\mathbf{v}})\text{\hspace{1em}(2)}$$
标准的有限差分法
$$\frac{{\partial }^{2}p(\mathbf{r},t)}{\partial t^2}\approx \frac{1}{(\Delta t{)}^2}\left(p_{\mathbf{r}}^{t+1}-2p_r^t+p_r^{t-1}\right)+O\left(\right(t)\_{}^2)\text{\hspace{1em}( 5 )}$$
where$p_{\mathbf{r}}^t$means ttThe wave field at time$t$$p_r^{t+1}$means $t+\Delta ttime$ .$O$ indicates the same order, and the corresponding data is discarded.

According to the chain rule, the loss $\mathcal{L}$ 对应对速度的梯度
$$\frac{\partial \mathcal{L}}{\partial v(\mathbf{r})}=\sum _{t=0}^T\left[\frac{\partial L}{\partial p(\mathbf{r},t)}\right]\frac{\partial p(r,t)}{\partial v(\mathbf{r})}\text{\hspace{1em}(7)}$$

2.2 Inversion loss function

$$\mathcal{L}(\mathbf{p},\tilde{\mathbf{p}})={\mathcal{L}}_{\text{pixel}}(\mathbf{p},\tilde{p})+L_{\text{perceptual}}(p,\tilde{p})\text{\hspace{1em}( 8 )}$$
where$p$和$\tilde{\mathbf{p}}$represent the input and reconstructed seismic wave data respectively.
You can use $l_1$与$l_2$Norm weighted sum to define the pixel loss
$${\mathcal{L}}_{\text{pixel}}(\mathbf{p},\tilde{p})=l_1{l}_1(p,\tilde{p})+l_2{l}_2(p,\tilde{p})\text{\hspace{1em}( 9 )}$$
$${\mathcal{L}}_{\text{perceptual}}(\mathbf{p},\tilde{\mathbf{p}})=l_3{l}_1\left(\varphi \right(p),\varphi (\tilde{p}))+l_4{l}_2\left(\varphi \right(p),\varphi (\tilde{p}\left)\right)\text{\hspace{1em}( 9 )}$$
where$\varphi (\cdot )$ represents the VGG-16 feature extraction network trained with ImageNet.

3. Related work

• physical drive
• data driven

4. Summary

Forward and reverse, walk on two legs.