Stability of implicit neural networks for long-term forecasting in dynamical systems
This work addresses stability challenges in deep learning-based forecasting for PDEs, which is important for researchers in computational physics and machine learning, though it appears incremental as it builds on implicit numerical schemes.
The paper tackled the problem of long-term forecasting in dynamical systems, particularly for Partial Differential Equations (PDEs), by introducing a stable auto-regressive implicit neural network that addresses stability issues in existing deep learning methods, resulting in improved long-term forecasting performance for two transport PDEs.
Forecasting physical signals in long time range is among the most challenging tasks in Partial Differential Equations (PDEs) research. To circumvent limitations of traditional solvers, many different Deep Learning methods have been proposed. They are all based on auto-regressive methods and exhibit stability issues. Drawing inspiration from the stability property of implicit numerical schemes, we introduce a stable auto-regressive implicit neural network. We develop a theory based on the stability definition of schemes to ensure the stability in forecasting of this network. It leads us to introduce hard constraints on its weights and propagate the dynamics in the latent space. Our experimental results validate our stability property, and show improved results at long-term forecasting for two transports PDEs.