Modeling an unknown dynamical system is crucial in order to predict the future behavior of the system. A standard approach is training recurrent models on measurement data. While these models typically provide exact short-term predictions, accumulating errors yield deteriorated long-term behavior. In contrast, models with reliable long-term predictions can often be obtained, either by training a robust but less detailed model, or by leveraging physics-based simulations. In both cases, inaccuracies in the models yield a lack of short-time details. Thus, different models with contrastive properties on different time horizons are available. This observation immediately raises the question: Can we obtain predictions that combine the best of both worlds? Inspired by sensor fusion tasks, we interpret the problem in the frequency domain and leverage classical methods from signal processing, in particular complementary filters. This filtering technique combines two signals by applying a high-pass filter to one signal, and low-pass filtering the other. Essentially, the high-pass filter extracts high-frequencies, whereas the low-pass filter extracts low frequencies. Applying this concept to dynamics model learning enables the construction of models that yield accurate long- and short-term predictions. Here, we propose two methods, one being purely learning-based and the other one being a hybrid model that requires an additional physics-based simulator.

翻译：建模未知动力系统对于预测系统未来行为至关重要。标准方法是在测量数据上训练循环模型。虽然此类模型通常能提供精确的短期预测，但累积误差会导致长期行为退化。相反，通过训练鲁棒但细节较少的模型，或利用基于物理的仿真，往往能够获得可靠长期预测的模型。然而，这两种方式都会因模型不精确而缺失短期细节。因此，我们获得了在不同时间尺度上具有对比特性的不同模型。这一现象自然引出一个问题：能否获得兼具两者优势的预测？受传感器融合任务的启发，我们在频域中解释该问题，并利用信号处理中的经典方法（特别是互补滤波器）。该滤波技术通过对一个信号施加高通滤波、对另一个信号施加低通滤波来融合两个信号。本质上，高通滤波器提取高频成分，而低通滤波器提取低频成分。将这一概念应用于动力学模型学习，能够构建出同时具有精准长期与短期预测能力的模型。本文提出两种方法：一种为纯学习方法，另一种为需要额外物理仿真器的混合模型。