Stability of implicit neural networks for long-term forecasting in dynamical systems

Léon Migus, Julien Salomon, patrick gallinari

Keywords: Deep Learning, Stability, Partial Differential Equations, Forecasting dynamical systems, Numerical analysis

TL;DR: We propose a new deep learning approach that guarantees stability when forecasting physical signals over a long time range.

Abstract: 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.