Real world systems evolve in continuous-time according to their underlying causal relationships, yet their dynamics are often unknown. Existing approaches to learning such dynamics typically either discretize time ---leading to poor performance on irregularly sampled data--- or ignore the underlying causality. We propose CADYT, a novel method for causal discovery on dynamical systems addressing both these challenges. In contrast to state-of-the-art causal discovery methods that model the problem using discrete-time Dynamic Bayesian networks, our formulation is grounded in Difference-based causal models, which allow milder assumptions for modeling the continuous nature of the system. CADYT leverages exact Gaussian Process inference for modeling the continuous-time dynamics which is more aligned with the underlying dynamical process. We propose a practical instantiation that identifies the causal structure via a greedy search guided by the Algorithmic Markov Condition and Minimum Description Length principle. Our experiments show that CADYT outperforms state-of-the-art methods on both regularly and irregularly-sampled data, discovering causal networks closer to the true underlying dynamics.

Many physical systems can be described as a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. While this scheme is mathematically tempting, it can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. We tackle this task by leveraging higher-order numerical integrators. These integrators provide the necessary tools to discretize dynamical systems with arbitrary accuracy. However, most higher-order integrators require dynamics evaluations at intermediate time steps, making exact GP inference intractable. In previous work, this problem is often addressed by approximate inference techniques. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to enable direct inference, we propose to leverage multistep and Taylor integrators. We demonstrate how exact inference schemes can be derived for these types of integrators. Further, we derive tailored sampling schemes that allow one to draw consistent dynamics functions from the posterior. The learned model can thus be integrated with arbitrary integrators, just like a standard dynamical system. We show empirically and theoretically that our approach yields an accurate representation of the continuous-time system.

Dynamics model learning deals with the task of inferring unknown dynamics from measurement data and predicting the future behavior of the system. A typical approach to address this problem is to train recurrent models. However, predictions with these models are often not physically meaningful. Further, they suffer from deteriorated behavior over time due to accumulating errors. Often, simulators building on first principles are available being physically meaningful by design. However, modeling simplifications typically cause inaccuracies in these models. Consequently, hybrid modeling is an emerging trend that aims to combine the best of both worlds. In this paper, we propose a new approach to hybrid modeling, where we inform the latent states of a learned model via a black-box simulator. This allows to control the predictions via the simulator preventing them from accumulating errors. This is especially challenging since, in contrast to previous approaches, access to the simulator's latent states is not available. We tackle the task by leveraging observers, a well-known concept from control theory, inferring unknown latent states from observations and dynamics over time. In our learning-based setting, we jointly learn the dynamics and an observer that infers the latent states via the simulator. Thus, the simulator constantly corrects the latent states, compensating for modeling mismatch caused by learning. To maintain flexibility, we train an RNN-based residuum for the latent states that cannot be informed by the simulator.

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.