Wei Lin

Jifan Shi, Yang Li, Juan Zhao et al.

Detecting and quantifying causality is a focal topic in the fields of science, engineering, and interdisciplinary studies. However, causal studies on non-intervention systems attract much attention but remain extremely challenging. Delay-embedding technique provides a promising approach. In this study, we propose a framework named Interventional Dynamical Causality (IntDC) in contrast to the traditional Constructive Dynamical Causality (ConDC). ConDC, including Granger causality, transfer entropy and convergence of cross-mapping, measures the causality by constructing a dynamical model without considering interventions. A computational criterion, Interventional Embedding Entropy (IEE), is proposed to measure causal strengths in an interventional manner. IEE is an intervened causal information flow but in the delay-embedding space. Further, the IEE theoretically and numerically enables the deciphering of IntDC solely from observational (non-interventional) time-series data, without requiring any knowledge of dynamical models or real interventions in the considered system. In particular, IEE can be applied to rank causal effects according to their importance and construct causal networks from data. We conducted numerical experiments to demonstrate that IEE can find causal edges accurately, eliminate effects of confounding, and quantify causal strength robustly over traditional indices. We also applied IEE to real-world tasks. IEE performed as an accurate and robust tool for causal analyses solely from the observational data. The IntDC framework and IEE algorithm provide an efficient approach to the study of causality from time series in diverse non-intervention complex systems.

Jingdong Zhang, Qunxi Zhu, Wei Lin

Accurately finding and predicting dynamics based on the observational data with noise perturbations is of paramount significance but still a major challenge presently. Here, for the Hamiltonian mechanics, we propose the Hamiltonian Neural Koopman Operator (HNKO), integrating the knowledge of mathematical physics in learning the Koopman operator, and making it automatically sustain and even discover the conservation laws. We demonstrate the outperformance of the HNKO and its extension using a number of representative physical systems even with hundreds or thousands of freedoms. Our results suggest that feeding the prior knowledge of the underlying system and the mathematical theory appropriately to the learning framework can reinforce the capability of machine learning in solving physical problems.

Qunxi Zhu, Yao Guo, Wei Lin

Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with some representative datasets. Recently, an augmented framework has been successfully developed for conquering some limitations emergent in application of the original framework. Here we propose a new class of continuous-depth neural networks with delay, named as Neural Delay Differential Equations (NDDEs), and, for computing the corresponding gradients, we use the adjoint sensitivity method to obtain the delayed dynamics of the adjoint. Since the differential equations with delays are usually seen as dynamical systems of infinite dimension possessing more fruitful dynamics, the NDDEs, compared to the NODEs, own a stronger capacity of nonlinear representations. Indeed, we analytically validate that the NDDEs are of universal approximators, and further articulate an extension of the NDDEs, where the initial function of the NDDEs is supposed to satisfy ODEs. More importantly, we use several illustrative examples to demonstrate the outstanding capacities of the NDDEs and the NDDEs with ODEs' initial value. Specifically, (1) we successfully model the delayed dynamics where the trajectories in the lower-dimensional phase space could be mutually intersected, while the traditional NODEs without any argumentation are not directly applicable for such modeling, and (2) we achieve lower loss and higher accuracy not only for the data produced synthetically by complex models but also for the real-world image datasets, i.e., CIFAR10, MNIST, and SVHN. Our results on the NDDEs reveal that appropriately articulating the elements of dynamical systems into the network design is truly beneficial to promoting the network performance.