Time is not a dimension as the others. In Physics-Informed Neural Networks (PINN) several proposals attempted to adapt the time sampling or time weighting to take into account the specifics of this special dimension. But these proposals are not principled and need guidance to be used. We explain here theoretically why the Lyapunov exponents give actionable insights and propose a weighting scheme to automatically adapt to chaotic, periodic or stable dynamics. We characterize theoretically the best weighting scheme under computational constraints as a cumulative exponential integral of the local Lyapunov exponent estimators and show that it performs well in practice under the regimes mentioned above.

翻译：时间并非普通维度。在物理信息神经网络(PINN)中，已有若干研究尝试通过调整时间采样或时间加权策略来应对这一特殊维度的特性。然而这些方法缺乏理论依据，实际应用时需要人工指导。本文从理论上阐释了李雅普诺夫指数为何能提供可操作的洞见，并提出一种能自动适应混沌、周期或稳定动力学的加权方案。我们在计算约束条件下从理论上证明了最优加权方案可表示为局部李雅普诺夫指数估计量的累积指数积分，并通过实验验证了该方案在上述动力学机制下均具有优异性能。