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

TL;DR

This paper introduces a stable auto-regressive implicit neural network inspired by implicit numerical schemes, ensuring long-term stability and improved forecasting accuracy for PDEs by applying stability constraints and latent space dynamics.

Contribution

The paper proposes a novel stable neural network architecture for long-term PDE forecasting, incorporating stability constraints inspired by numerical schemes and latent space dynamics.

Findings

Validated stability property through experiments

Achieved improved long-term forecasting results

Applicable to transport PDEs

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.