Bayesian filtering of partially and noisily observed dynamical systems seeks to infer the evolving conditional distribution of the state of a dynamical system, given observations, in an online fashion. This Bayesian filtering distribution is the natural object for uncertainty quantification, but it is rarely available as a supervised learning target. However, one can often use the forecast model to generate synthetic system trajectories, along with synthetic observations. We introduce the proper scoring ensemble filter (PSEF), an ensemble data assimilation method based on training an analysis map to approximate the filtering distribution using only synthetic state--observation trajectories. The analysis step is represented as a permutation-invariant, transformer-based map that takes as input a forecast ensemble and observations, producing an analysis ensemble. Training is based on strictly proper scoring rules -- with the energy score used in our implementation -- so that probabilistic accuracy is rewarded over the whole probability distribution. We prove that, under a realizability assumption, the population objective is minimized by the true Bayesian filtering distribution. We also derive the finite-ensemble empirical objective used in training and relate its single state--observation trajectory form to the population objective, using a mean-field consistency argument. Numerical experiments show that the learned filter accurately approximates challenging filtering distributions, including nonlinear, non-Gaussian, and multi-modal posteriors, and achieves stronger performance in data assimilation tasks than classical methods or learning-based methods with mean-squared-error objectives. For close-to-Gaussian problems, learning a correction to the EnKF is the best approach, while for highly non-Gaussian problems an end-to-end approach that discards this inductive bias is superior.

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.

Reduced-order modeling of high-dimensional dynamical systems is often hindered by the non-Markovian closure term that represents the effect of unresolved variables on the resolved dynamics. Inspired by the Mori--Zwanzig formalism, in which the closure takes the form of a memory functional of the resolved trajectory, we recast closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework: a Mamba-based sequence model, trained to predict the closure from the resolved trajectory, is coupled with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. A key feature of the framework is its exploitation of the dual representation of state-space models -- the model is trained in a sequence-to-sequence fashion via the convolutional form, and deployed for step-by-step autoregressive rollout via the recurrent form, yielding both efficient long-trajectory training and constant per-step inference cost. On the viscous Burgers' equation and the chaotic two-scale Lorenz '96 system, the MAC model substantially outperforms the Markovian reduced-order model, the GRU-based sequence model, and the Wilks method in predictive accuracy and long-time rollout stability.

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

Machine learning has opened new frontiers in purely data-driven algorithms for data assimilation in, and for forecasting of, dynamical systems; the resulting methods are showing some promise. However, in contrast to model-driven algorithms, analysis of these data-driven methods is poorly developed. In this paper we address this issue, developing a theory to underpin data-driven methods to solve smoothing problems arising in data assimilation and forecasting problems. The theoretical framework relies on two key components: (i) establishing the existence of the mapping to be learned; (ii) the properties of the operator learning architecture used to approximate this mapping. By studying these two components in conjunction, we establish novel universal approximation theorems for purely data driven algorithms for both smoothing and forecasting of dynamical systems. We work in the continuous time setting, hence deploying neural operator architectures. The theoretical results are illustrated with experiments studying the Lorenz `63, Lorenz `96 and Kuramoto-Sivashinsky dynamical systems.

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.