Inferring Underwater Topography with FINN

TL;DR

The paper tackles inferring underwater topography from wave dynamics governed by the shallow-water equations (SWE). It adapts the finite-volume neural network (FINN) to solve SWE and treats the bottom depth $H$ as a learnable field, comparing against DISTANA and PhyDNet. FINN delivers superior topography reconstruction with lower full and inner reconstruction errors and robust edge handling due to its embedded physical structure, outperforming the baselines. The results suggest that physics-structured neural solvers can recover latent bathymetry from spatiotemporal SWE data and hold promise for real-world coastal inference and broader PDE-based applications. The work also outlines avenues for extending FINN to more complex flow models and larger-scale scenarios.

Abstract

Spatiotemporal partial differential equations (PDEs) find extensive application across various scientific and engineering fields. While numerous models have emerged from both physics and machine learning (ML) communities, there is a growing trend towards integrating these approaches to develop hybrid architectures known as physics-aware machine learning models. Among these, the finite volume neural network (FINN) has emerged as a recent addition. FINN has proven to be particularly efficient in uncovering latent structures in data. In this study, we explore the capabilities of FINN in tackling the shallow-water equations, which simulates wave dynamics in coastal regions. Specifically, we investigate FINN's efficacy to reconstruct underwater topography based on these particular wave equations. Our findings reveal that FINN exhibits a remarkable capacity to infer topography solely from wave dynamics, distinguishing itself from both conventional ML and physics-aware ML models. Our results underscore the potential of FINN in advancing our understanding of spatiotemporal phenomena and enhancing parametrization capabilities in related domains.

Figures (10)

• Figure 1: FINN architecture. The composition of the modules to solve an advection-diffusion PDE for one control volume in FVM mesh with the help of adjacent volumes. Black lines show the forward pass, whereas red lines indicate the gradient flow. Figure from karlbauer2021composing.
• Figure 2: Illustration of SWE.$x$, $y$ and $z$ correspond to width, length and depth, respectively. In the case of flat topography, $H$ is equal in every location. However, a non-flat topography is used in this study and $H$ has different values in different locations. $\eta$ is the deviance of the free surface (thick blue line) from the average depth (thick black line) and it is the unknown state. In topography reconstruction, $H$ is reproduced from $\eta$.
• Figure 3: Modification of FINN to solve SWE. Dark black arrows show the forward pass. Loss is computed between $\eta^{(t+1)}$ and $\hat{\eta}^{(t+1)}$ as the mean squared error. The prediction is fed back into the model in a closed-loop fashion (dashed arrow).
• Figure 4: Topography of the training and inference set. $H$ is located over a $1000km^2$ area. The training topography (left) is smooth, has a large depth range, and has different rotations and scales for each sequence in order to create a large variety in data. The inference topography (right) is more non-linear, bumpy, and is not rotated. It is the only topography for all sequences as it needs to be inferred by the models. The depth scale for the particular topography was randomly chosen as $\beta = 0.68$. It is the same in the left plot for a meaningful comparison.
• Figure 5: Different topographies over a $1000km^2$ area with varying depths. $-H$ is used for plotting to make the visualizations more realistic because physically higher depth means a bigger distance from the free surface. FINN's $H$ inference progress is given in \ref{['appendix:swe_top_inf']}