Neural Networks for Threshold Dynamics Reconstruction

Abstract

We introduce two convolutional neural network (CNN) architectures, inspired by the Merriman-Bence-Osher (MBO) algorithm and by cellular automatons, to model and learn threshold dynamics for front evolution from video data. The first model, termed the (single-dynamics) MBO network, learns a specific kernel and threshold for each input video without adapting to new dynamics, while the second, a meta-learning MBO network, generalizes across diverse threshold dynamics by adapting its parameters per input. Both models are evaluated on synthetic and real-world videos (ice melting and fire front propagation), with performance metrics indicating effective reconstruction and extrapolation of evolving boundaries, even under noisy conditions. Empirical results highlight the robustness of both networks across varied synthetic and real-world dynamics.

Figures (20)

• Figure 1: MBO network architecture
• Figure 2: Meta-learning MBO network architecture
• Figure 3: Example of kernels used for dataset generation. From left to right: standard Gaussian kernel, skewed Gaussian kernel, double Gaussian kernel, MNIST digit, and the indicator function of a disk.
• Figure 4: Example of dynamics obtained using the MBO scheme with standard Gaussian kernel and threshold 0.2 (top) and 0.5 (bottom).
• Figure 5: Example noise conditions on threshold dynamics generated with standard Gaussian kernel and threshold 0.5. Top: No noise. Middle: Gaussian blur. Bottom: Salt-and-pepper noise.