GANs improve stochastic parameterization of the Lorenz '96 model.
problem Improving stochastic parameterizations for sub-grid processes.
method Developed a GAN-based stochastic parameterization for the Lorenz '96 model.
result GAN configurations outperform a bespoke parameterization in skillful forecasts and climate simulations.
Climate projections suffer from uncertain equilibrium climate sensitivity. The reason behind this uncertainty is the resolution of global climate models, which is too coarse to resolve key processes such as clouds and convection. These processes are approximated using heuristics in a process called parameterization. Th…
DD-SP uses ML to improve SP for Lorenz 96 systems, outperforming LR and DD-P.
problem Improving computational efficiency in weather/climate modeling.
method Data-driven super-parameterization using recurrent neural networks.
result DD-SP is more accurate and cheaper than SP, especially with scale separation.
Combining LETKF and RC improves chaotic system prediction from noisy, sparse data.
problem Improving chaotic system prediction from imperfect observations and models.
method Combining LETKF and RC to predict spatio-temporal chaotic systems from noisy and sparsely distributed observations.
result The proposed method using LETKF and RC outperforms LETKF in predicting chaotic systems from noisy and sparse observations.
EnEMF uses Epanechnikov kernel for high-dimensional filtering, improving accuracy and robustness.
problem Suboptimal Gaussian mixture kernel density estimates in high-dimensional settings.
method Ensemble Epanechnikov mixture filter (EnEMF) using optimal Epanechnikov kernel.
result EnEMF reduces error per particle on high-dimensional systems like Lorenz '96.
In this paper, the performance of three deep learning methods for predicting short-term evolution and for reproducing the long-term statistics of a multi-scale spatio-temporal Lorenz 96 system is examined. The methods are: echo state network (a type of reservoir computing, RC-ESN), deep feed-forward artificial neural n…
New method calibrates predictions in chaotic systems using variational inference.
problem Uncertainty in data assimilation for chaotic systems.
method Variational inference applied to multivariate Gaussian distribution.
result Nearly perfectly calibrated predictions in chaotic Lorenz-96 dynamics.
Analog methods improve forecast accuracy in complex models.
problem Improving forecast accuracy in complex models like Lorenz-96.
method Constructing analogs using variational autoencoders for ensemble data assimilation.
result Constructed analogs perform as well as a full ensemble square root filter.
CG-EnKF and NS-EnKF outperform deep learning-based SF in data assimilation.
problem Data assimilation with non-linear perturbations.
method Two non-linear extensions of EnKF: CG-EnKF and NS-EnKF.
result CG-EnKF and NS-EnKF outperform SF in high-dimensional multiscale data assimilation.
Deep learning improves chaotic dynamics filtering without ensemble.
problem Discovering efficient DA schemes for chaotic dynamics.
method Residual Convolutional Neural Network for the analysis step.
result Deep learning achieves ensemble filtering accuracy without an ensemble.
A new method predicts non-Markovian closure terms for complex systems.
problem Predicting the effect of unresolved variables on resolved dynamics in high-dimensional systems.
method Mamba-Assisted Closure (MAC) framework: sequence model trained to predict closure from resolved trajectory, coupled with reduced-order equations.
result Substantially outperforms existing methods in predictive accuracy and long-time stability.
E&E uses contrastive learning to speed up SBI for high-dimensional systems.
problem Challenges in training high-dimensional emulators for complex systems.
method Contrastive learning for low-dimensional latent embedding and fast emulator.
result Superior performance in non-identifiable parameter estimation tasks.
A hybrid method combines data assimilation and machine learning to predict chaotic dynamics from sparse noisy data.
problem Predicting chaotic dynamics from sparse and noisy observations.
method Iterative application of ensemble Kalman filter for data assimilation and neural network for model emulation.
result The hybrid method successfully predicts chaotic dynamics up to two Lyapunov times, retrieves positive Lyapunov exponents, and more energetic frequencies.
Develops theory for data-driven methods in dynamical systems.
problem Lack of analysis for data-driven methods in dynamical systems.
method Establishes existence of mapping and properties of operator learning architecture.
result Novel universal approximation theorems for smoothing and forecasting.
New parameterization tackles stochasticity in weather models.
problem Uncertainty in small-scale processes in weather models.
method Bayesian neural network with Hamiltonian Monte Carlo for uncertainty quantification and memory.
result Shows skillful forecasts and trustworthy uncertainty quantifications.
In this work, a novel sequential Monte Carlo filter is introduced which aims at efficient sampling of high-dimensional state spaces with a limited number of particles. Particles are pushed forward from the prior to the posterior density using a sequence of mappings that minimizes the Kullback-Leibler divergence between…
Proposes a new method for data assimilation using closed-form conditional diffusion models.
problem Data assimilation for systems with complex, non-Gaussian probability distributions.
method Uses kernel density estimation to model joint distributions and leverages the score function for efficient evaluation.
result Outperforms ensemble Kalman and particle filters in nonlinear data assimilation problems.
A new ML-based filter improves data assimilation for nonlinear systems.
problem Improving data assimilation for nonlinear systems using ensemble methods.
method Developed a machine learning-based conditional mean filter (ML-EnCMF) integrating ANN and linear functions.
result ML-EnCMF outperforms EnKF and likelihood-based EnCMF in nonlinear systems.
Enhanced ensemble filters use machine learning to improve accuracy in filtering models.
problem Accuracy limitations of traditional ensemble Kalman filters.
method Introduces a measure neural mapping (MNM) to map joint predicted state and observation to updated state estimates.
result Superior root-mean-square-error performance compared to leading methods in filtering models.
We consider filtering in high-dimensional non-Gaussian state-space models with intractable transition kernels, nonlinear and possibly chaotic dynamics, and sparse observations in space and time. We propose a novel filtering methodology that harnesses transportation of measures, convex optimization, and ideas from proba…
New satellite knots counter a conjecture about Lorenz knots.
problem A conjecture about satellite knots and Lorenz knots was disproven.
method Constructed infinitely many counterexamples of satellite knots that are not cables.
result The conjecture was amended and shown to hold for many Lorenz knots.
While most classical approaches to Granger causality detection repose upon linear time series assumptions, many interactions in neuroscience and economics applications are nonlinear. We develop an approach to nonlinear Granger causality detection using multilayer perceptrons where the input to the network is the past t…
EnSF improves accuracy in tracking high-dimensional nonlinear systems.
problem Low accuracy in high-dimensional, nonlinear filtering problems.
method Score-based diffusion model, mini-batch Monte Carlo estimator.
result EnSF outperforms state-of-the-art methods in tracking high-dimensional systems.
We describe the Williams zeta functions and the twist zeta functions of sub-Lorenz templates generated by renormalizable Lorenz maps, in terms of the corresponding zeta-functions of the sub-Lorenz templates generated by the renormalized map and by the map that determines the renormalization type.
We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots code…
Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…
This is a review article on Lorenz knots.
The unknotting number of a positive braid with n strands and k intersections is known to be equal to (k-n+1)/2. We consider Lorenz knots (which are positive braids) and, using a different method, find their unknotting numbers in terms of their positions on the Lorenz attractor.
This article is a survey on Lorenz knots. We describe the original construction, prove several classical properties, in particular the fact that the closure of a positive braid is a fibered knot, and describe Ghys'correspondance between modular knots and Lorenz knots. We also prove two new properties, namely that follo…
We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant (Kf−,Kf+)=(X,Y)∗(S,W), in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit form…
Study of Lorenz links and T-links, showing equivalence and unique presentations.
problem Understanding equivalence and uniqueness of T-link presentations for Lorenz links.
method Minimal-braid approach, V-link braids, T-link parameters, geometric type criteria.
result Explicit correspondence between V-links and T-links, criteria for equivalence and uniqueness.
We exhibit low-dilatation families of surface homeomorphisms among monodromies of Lorenz knots.
AD-EnKFs use machine learning to improve data assimilation in high-dimensional systems.
problem Data assimilation in high-dimensional, unknown dynamics systems.
method Auto-differentiable ensemble Kalman filters blending machine learning and ensemble Kalman filters.
result AD-EnKFs outperform existing methods in the Lorenz-96 model.
In paper "A new twist on Lorenz links" (Journal of Topology 2(2009), 227-248) Joan Birman and Ilya Kofman prove the coincidence of the class of Lorenz links and the class of twisted links. The proof in that work is algebraic. We will identify this class in terms of grid diagrams and provide a transparent geometric argu…
Computed linking number of modular knots and Lorenz links.
problem Computing the linking number of modular knots in a specific space.
method Using correspondence between modular links and Lorenz links, and intersection number involving Conway topographs.
result Computed linking number of modular knots and compared to a previous formula.
In this article the Lorenz dynamical system is revived and revisited and the current state of the art results for one step ahead forecasting for the Lorenz trajectories are published. Multitask learning is shown to help learning the hard to learn z trajectory. The article is a reflection upon the evolution of neural ne…
New findings on T-links derived from torus links.
problem Difficulty in determining geometric type of T-links.
method Examined T-links obtained by full twists along torus links.
result T-links obtained by full twists are not torus links.
Unified framework for blending ML and mechanistic models in dynamical systems.
problem Learning dynamical systems from noisy, partially observed data.
method A unifying framework that combines mechanistic and machine learning approaches.
result Proves that hybrid models can learn memory-dependent model error.
New satellite knots found that can't be represented by positive braids with full twists.
problem Satellite knots that cannot be represented by positive braids with full twists.
method Analyzing positive minimal braids and their closures to find satellite knots.
result Infinitely many satellite knots with Lorenz patterns and companions are not Lorenz knots.
This paper learns state, dynamics, and filtering algorithms together for data assimilation.
problem Costly parameter tuning and inaccurate dynamics models hinder data assimilation algorithms.
method Auto-differentiable data assimilation framework that learns state, dynamics, and parameters via gradient-based optimization.
result Several data assimilation methods can be learned or tuned within this framework.
During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic p…
Deep density methods improve filtering in high-dimensional systems.
problem Nonlinear filtering in high-dimensional systems.
method Two deep density methods based on Feynman-Kac formulas and neural networks.
result Logarithmic deep backward stochastic differential equation filter outperforms classical methods in high dimensions.
We show that the zeroes of the Alexander polynomial of a Lorenz knot all lie in some annulus whose width depends explicitly on the genus and the braid index of the considered knot.
Improved algorithm for modular links provides upper volume bounds.
problem Understanding the geometry of modular links and Lorenz links.
method Bunch algorithm to study modular links and provide upper volume bounds.
result First upper volume bound independent of word exponents and quadratic in braid index.
New method uses neural networks to efficiently approximate Bayesian inference for complex models.
problem Efficiently approximating Bayesian inference for complex models with varying temperatures.
method Fully amortized neural posterior estimator trained on a single forward pass.
result Achieves competitive posterior approximations across various temperatures and benchmarks.
GC-KAN uses KANs to detect Granger causality in time series data.
problem Detecting causal relationships in nonlinear time series data.
method Developed GC-KAN framework using Kolmogorov-Arnold networks for Granger causality detection.
result KANs outperform MLPs in identifying sparse Granger causal relationships.
A new framework reduces inconsistencies in chaotic surrogate modeling.
problem Consistency issues between probabilistic objectives and dynamical system dynamics.
method KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter.
result KAFFEE mitigates the dynamic-probabilistic consistency gap, improving reconstruction and predictive scores.
Long-lead forecasting for spatio-temporal systems can often entail complex nonlinear dynamics that are difficult to specify it a priori. Current statistical methodologies for modeling these processes are often highly parameterized and thus, challenging to implement from a computational perspective. One potential parsim…