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.

翻译：在偏微分方程研究中，长时间范围的物理信号预测是最具挑战性的任务之一。为突破传统求解器的局限，人们提出了多种深度学习方法。这些方法均基于自回归模型，但存在稳定性问题。受隐式数值格式稳定性特性的启发，我们提出了一种稳定的自回归隐式神经网络。基于数值格式的稳定性定义，我们建立了一套理论框架以确保该网络预测的稳定性。这要求我们对网络权重施加严格约束，并在潜在空间中传播动力学过程。实验结果表明，我们的方法验证了稳定性特性，在两类输运型偏微分方程的长期预测中展现了更优的性能。