Efficient Continuous-Depth Modeling with GRU Equivalents

Continuous-Depth Neural Networks (CDNNs), including Neural Ordinary Differential Equations (ODEs) and Liquid-Time-Constant Neural Networks (LTC-NN), suffer from high computational costs due to solving numerous nonlinear ODEs during training and inference. We introduce Continuous Depth Acceleration (CoDA), a framework that leverages Mori–Zwanzig/Koopman operator theory to replace continuous-depth layers requiring multiple nonlinear ODEs with a compact GRU module, a single low-dimensional linear ODE, and a dense layer. We prove PAC learnability of CoDA, establishing that this transformation preserves accuracy and can be applied repeatedly across multiple layers with unified backpropagation. Experiments on the Liquid Foundation Model (LFM-1.2B) demonstrate $6.7\times$ training speedup and $1.8\times$ inference speedup without loss of accuracy. Across six real-world LTC-NN applications, CoDA consistently outperforms state-of-the-art acceleration techniques—including neural flows, model order reduction, and variational formulations—in both training and inference time while maintaining competitive or superior accuracy. The implementation and datasets are publicly available at https://github.com/ImpactLabASU/CoDA-ICML2026.