Advection Augmented Convolutional Neural Networks

TL;DR

This work proposes to augment CNNs with advection by designing a novel semi-Lagrangian push operator and complements it with Reaction and Diffusion neural components to form a network that mimics the Reaction-Advection-Diffusion equation, in high dimensions.

Abstract

Many problems in physical sciences are characterized by the prediction of space-time sequences. Such problems range from weather prediction to the analysis of disease propagation and video prediction. Modern techniques for the solution of these problems typically combine Convolution Neural Networks (CNN) architecture with a time prediction mechanism. However, oftentimes, such approaches underperform in the long-range propagation of information and lack explainability. In this work, we introduce a physically inspired architecture for the solution of such problems. Namely, we propose to augment CNNs with advection by designing a novel semi-Lagrangian push operator. We show that the proposed operator allows for the non-local transformation of information compared with standard convolutional kernels. We then complement it with Reaction and Diffusion neural components to form a network that mimics the Reaction-Advection-Diffusion equation, in high dimensions. We demonstrate the effectiveness of our network on a number of spatio-temporal datasets that show their merit.

Figures (6)

• Figure 1: A simple task of moving information from one side of the image to the other. The source image in A is moved to the target image in B. The convergence of a simple ResNet and an ADRnet proposed in this work is in (C).
• Figure 2: An illustration of the advection-diffusion reaction process. In the first step (advection), a pixel on the lower left of the image is transported into the middle of the mesh. In the second step (diffusion), the information is diffused to its neighbours and finally, in the last step (reaction) each pixel interact locally to change its value.
• Figure 3: Discretization of the push operator. (a) Left: Semi-Lagrangian mass preserving transport, discretizing the continuity. (b) Right: Semi-Lagrangian color preserving transport.
• Figure 4: Prediction and error for the SWE problem using our ADRNet.
• Figure 5: Example of the forecast for ADRNet relative to ground truth for four time steps.