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，长期物理信号预测是最具挑战性的任务之一。为规避传统求解器的局限性，研究者提出了多种深度学习方法。这些方法均基于自回归模式，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。基于格式稳定性定义建立理论框架，确保该网络在预测过程中的稳定性。由此引入对网络权重的硬约束，并在隐空间中传播动力学特征。实验验证了稳定性属性，并表明该方法在两个输运偏微分方程的长期预测中取得了更优结果。