Fully invertible hyperbolic neural networks for segmenting large-scale surface and sub-surface data

TL;DR

The paper tackles memory bottlenecks in deep segmentation of large-scale 3D/4D geoscience data by advancing fully invertible hyperbolic neural networks based on a conservative leapfrog discretization of the non-linear Telegraph equation. It introduces a block-low-rank layer to curb exponential kernel growth and combines invertible pooling with orthogonal transforms to enable input–output dimensionality and resolution changes while preserving invertibility. The authors demonstrate the approach on time-lapse hyperspectral data, regional aquifer mapping, and 3D seismic interpretation, achieving substantial memory savings and enabling end-to-end processing of large volumes. This work expands the applicability of invertible networks to complex, large-scale geoscience tasks, offering practical memory efficiency and flexibility for processing full-volume data blocks.

Abstract

The large spatial/temporal/frequency scale of geoscience and remote-sensing datasets causes memory issues when using convolutional neural networks for (sub-) surface data segmentation. Recently developed fully reversible or fully invertible networks can mostly avoid memory limitations by recomputing the states during the backward pass through the network. This results in a low and fixed memory requirement for storing network states, as opposed to the typical linear memory growth with network depth. This work focuses on a fully invertible network based on the telegraph equation. While reversibility saves the major amount of memory used in deep networks by the data, the convolutional kernels can take up most memory if fully invertible networks contain multiple invertible pooling/coarsening layers. We address the explosion of the number of convolutional kernels by combining fully invertible networks with layers that contain the convolutional kernels in a compressed form directly. A second challenge is that invertible networks output a tensor the same size as its input. This property prevents the straightforward application of invertible networks to applications that map between different input-output dimensions, need to map to outputs with more channels than present in the input data, or desire outputs that decrease/increase the resolution compared to the input data. However, we show that by employing invertible networks in a non-standard fashion, we can still use them for these tasks. Examples in hyperspectral land-use classification, airborne geophysical surveying, and seismic imaging illustrate that we can input large data volumes in one chunk and do not need to work on small patches, use dimensionality reduction, or employ methods that classify a patch to a single central pixel.

Figures (13)

• Figure 1: Diagram of the flow of the multi-level hyperbolic network in \ref{['network']}. Shown for 3D input data with one channel and two levels. The network contains seven layers, with pooling after layer four and unpooling after layer six.
• Figure 2: Memory requirements (Gigabyte) for network states (activations) and $3 \times 3 \times 3$ convolutional kernels. Left: as a function of input size and a fixed $50$ layer network with two coarsening stages. Middle: as a function of an increasing number of layers but with fixed input size ($300^3$) and fixed number of two coarsenings. Right: as a function of an increasing number of coarsening steps but with a fixed number of layers ($50$) and input size ($300^3$). Our proposed Block-Low-Rank (BLR) layers avoid an exploding number of convolutional kernels with increased coarsening in invertible networks.
• Figure 3: Empirically observed properties of an untrained and randomly initialized invertible hyperbolic network, as a function of the 'time-step' $h$. The perturbation for the middle figure is also chosen randomly for every test point.
• Figure 4: Hyperspectral data collected at two different times.
• Figure 5: (a) Plan view of all true labels with point-annotation locations for training and validation overlaid. (b) prediction, and (c) error map. Most errors are boundary effects, and just a few farm fields are identified as changed/not-changed incorrectly (red arrows highlight two examples).

Theorems & Definitions (1)

• Theorem 2.1: Stability of the fully invertible hyperbolic network \ref{['telegraph2']} and \ref{['network']}.