Metriplectic systems are learned from data in a way that scales quadratically in both the size of the state and the rank of the metriplectic operators. In addition to being provably energy-conserving and entropy-stable, the proposed neural metriplectic systems (NMS) approach includes approximation results that demonstrate its ability to accurately learn metriplectic dynamics from data, along with an error estimate that indicates its potential for generalization to unseen timescales when the approximation error is low. Examples are provided to illustrate performance both with full state information available and when entropic variables are unknown, confirming that the NMS approach exhibits superior accuracy and scalability without compromising on model expressivity.

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Raynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P$^2$INNs). P$^2$INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P$^2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known “failure modes”.

Long-term time series forecasting (LTSF) is a challenging task that has been investigated in various domains such as finance investment, health care, traffic, and weather forecasting. In recent years, Linear-based LTSF models showed better performance, pointing out the problem of Transformer-based approaches causing temporal information loss. However, Linear-based approach has also limitations that the model is too simple to comprehensively exploit the characteristics of the dataset. To solve these limitations, we propose LTSF-DNODE, which applies a model based on linear ordinary differential equations (ODEs) and a time series decomposition method according to data statistical characteristics. We show that LTSF-DNODE outperforms the baselines on various real-world datasets. In addition, for each dataset, we explore the impacts of regularization in the neural ordinary differential equation (NODE) framework.

The time-series anomaly detection is one of the most fundamental tasks for time-series. Unlike the time-series forecasting and classification, the time-series anomaly detection typically requires unsupervised (or self-supervised) training since collecting and labeling anomalous observations are difficult. In addition, most existing methods resort to limited forms of anomaly measurements and therefore, it is not clear whether they are optimal in all circumstances. To this end, we present a multivariate time-series anomaly detector based on score-based generative models, called MadSGM, which considers the broadest ever set of anomaly measurement factors: i) reconstruction-based, ii) density-based, and iii) gradient-based anomaly measurements. We also design a conditional score network and its denoising score matching loss for the time-series anomaly detection. Experiments on five real-world benchmark datasets illustrate that MadSGM achieves the most robust and accurate predictions.