The pseudo-code of plugging our method into the vanilla BO is summarised in Algorithm 1. Therefore, our method is applicable to any other variants of BO in a plug-in manner. In this section, we present the proofs associated with the theoretical assertions from Section 2. To Lemma 1. Assume the GP employs a stationary kernel Lemma 2. Given Lemma 1, determining Proposition 2. Leveraging Lemma 2, suppose Lemma 3. As per Srinivas et al., the optimization process in BO can be conceptualized as a sampling Pr null |f ( x) µ(x) | ωσ ( x) null > δ, (24) where δ > 0 signifies the confidence level adhered to by the UCB. This lemma is directly from Srinivas et al. . The proof can be found therein. Theorem 1. Leveraging Corollary 1, when employing the termination method proposed in this paper, As discussed in Remark 2 of Section 2.2 in the main manuscript, we suggest initializing L-BFGS Different subplots are (a) our proposed method, (b) Naïve method, (c) Nguyen's method, (d) Lorenz's Different subplots are (a) our proposed method, (b) Naïve method, (c) Nguyen's method, (d) Lorenz's Different subplots are (a) our proposed method, (b) Naïve method, (c) Nguyen's method, (d) Lorenz's Different subplots are (a) our proposed method, (b) Naïve method, (c) Nguyen's method, (d) Lorenz's

Uncovering cause-effect relationships from observational time series is fundamental to understanding complex systems. While many methods infer static causal graphs, real-world systems often exhibit dynamic causality-where relationships evolve over time. Accurately capturing these temporal dynamics requires time-resolved causal graphs. We propose UnCLe, a novel deep learning method for scalable dynamic causal discovery. UnCLe employs a pair of Uncoupler and Recoupler networks to disentangle input time series into semantic representations and learns inter-variable dependencies via auto-regressive Dependency Matrices. It estimates dynamic causal influences by analyzing datapoint-wise prediction errors induced by temporal perturbations. Extensive experiments demonstrate that UnCLe not only outperforms state-of-the-art baselines on static causal discovery benchmarks but, more importantly, exhibits a unique capability to accurately capture and represent evolving temporal causality in both synthetic and real-world dynamic systems (e.g., human motion). UnCLe offers a promising approach for revealing the underlying, time-varying mechanisms of complex phenomena.

Matrix sizes are listed for convenience in Table 2. Table 2: Matrix dimensions for incremental SVD. On subsequent iterations, the procedure is as follows: 1. Observe a new n b data matrix X What is most important to note in this is that the solution to (14) is not unique. Our implementation of Bubblewrap neither normalizes nor assumes a scale for incoming data. When predicting more than one time step ahead, we use sequential sampling for both models. SVD) to the experimental datasets used in the main text.

The filtering distribution in hidden Markov models evolves according to the law of a mean-field model in state--observation space. The ensemble Kalman filter (EnKF) approximates this mean-field model with an ensemble of interacting particles, employing a Gaussian ansatz for the joint distribution of the state and observation at each observation time. These methods are robust, but the Gaussian ansatz limits accuracy. This shortcoming is addressed by approximating the mean-field evolution using a novel form of neural operator taking probability distributions as input: a Measure Neural Mapping (MNM). A MNM is used to design a novel approach to filtering, the MNM-enhanced ensemble filter (MNMEF), which is defined in both the mean-fieldlimit and for interacting ensemble particle approximations. The ensemble approach uses empirical measures as input to the MNM and is implemented using the set transformer, which is invariant to ensemble permutation and allows for different ensemble sizes. The derivation of methods from a mean-field formulation allows a single parameterization of the algorithm to be deployed at different ensemble sizes. In practice fine-tuning of a small number of parameters, for specific ensemble sizes, further enhances the accuracy of the scheme. The promise of the approach is demonstrated by its superior root-mean-square-error performance relative to leading methods in filtering the Lorenz 96 and Kuramoto-Sivashinsky models.

Conventional notions of generalization often fail to describe the ability of learned models to capture meaningful information from dynamical data. A neural network that learns complex dynamics with a small test error may still fail to reproduce its \emph{physical} behavior, including associated statistical moments and Lyapunov exponents. To address this gap, we propose an ergodic theoretic approach to generalization of complex dynamical models learned from time series data. Our main contribution is to define and analyze generalization of a broad suite of neural representations of classes of ergodic systems, including chaotic systems, in a way that captures emulating underlying invariant, physical measures. Our results provide theoretical justification for why regression methods for generators of dynamical systems (Neural ODEs) fail to generalize, and why their statistical accuracy improves upon adding Jacobian information during training. We verify our results on a number of ergodic chaotic systems and neural network parameterizations, including MLPs, ResNets, Fourier Neural layers, and RNNs.

The Canadian fertilizer company Nutrien, for example, operates two dozen manufacturing and processing facilities spread across the globe and nearly 2,000 retail stores in the Americas and Australia. To collect underutilized data from its industrial operations, and gain greater visibility into its supply chain, the company relies on a combination of cloud technology and artificial intelligence/machine learning (AI/ML) capabilities. "A digital supply chain connects us from grower to manufacturer, providing visibility throughout the value chain," says Adam Lorenz, senior director for strategic fleet and indirect procurement at Nutrien. This visibility is critical when it comes to navigating the company's supply chain challenges, which include seasonal demands, weather dependencies, manufacturing capabilities, and product availability. The company requires real-time visibility into its fleets, for example, to identify the location of assets, see where products are moving, and determine inventory requirements.

In meteorology, finding similar weather patterns or analogs in historical datasets can be useful for data assimilation, forecasting, and postprocessing. In climate science, analogs in historical and climate projection data are used for attribution and impact studies. However, most of the time, those large weather and climate datasets are nearline. They must be downloaded, which takes a lot of bandwidth and disk space, before the computationally expensive search can be executed. We propose a dimension reduction technique based on autoencoder (AE) neural networks to compress those datasets and perform the search in an interpretable, compressed latent space. A distance-regularized Siamese twin autoencoder (DIRESA) architecture is designed to preserve distance in latent space while capturing the nonlinearities in the datasets. Using conceptual climate models of different complexities, we show that the latent components thus obtained provide physical insight into the dominant modes of variability in the system. Compressing datasets with DIRESA reduces the online storage and keeps the latent components uncorrelated, while the distance (ordering) preservation and reconstruction fidelity robustly outperform Principal Component Analysis (PCA) and other dimension reduction techniques such as UMAP or variational autoencoders.