State-space models improve dynamical system predictions efficiently and accurately.
problem Challenges in predicting dynamical systems, including long-time integration and long-range dependencies.
method State-space models implemented in Mamba, addressing limitations of existing architectures.
result Mamba outperforms other models in interpolation and challenging extrapolation tasks.
Cohesion uses deep Koopman operators to generate long-range forecasts of chaotic dynamics.
problem Challenges in data-driven emulation of chaotic dynamics, especially long-range skill decay.
method Generative modeling with coherent priors estimated using reduced-order models.
result Superior long-range forecasting skill on chaotic systems, including climate dynamics.
New methods avoid spectral pollution in transfer operators for accurate analysis.
problem Spectral pollution in finite-dimensional approximations of transfer operators.
method Algorithms for computing spectral properties of transfer operators without spectral pollution.
result Accurate spectral estimation across various applications, including protein folding models.
TK-GCN forecasts spatiotemporal dynamics using Koopman-enhanced graph convolutional networks.
problem Forecasting complex spatiotemporal dynamics over irregular domains.
method Two-stage framework: Koopman-enhanced Graph Convolutional Network (K-GCN) for spatial encoding and Transformer for temporal modeling.
result TK-GCN outperforms state-of-the-art methods in spatiotemporal cardiac dynamics forecasting.
Kernel Dynamic Mode Decomposition reconstructs dynamical systems using Laplacian kernel.
problem Reconstructing spatial-temporal dynamics of complex systems.
method Kernel Dynamic Mode Decomposition with Laplacian kernel.
result Laplacian kernel allows for the closability of Koopman operators in RKHS, enabling reconstruction.
QBSD optimizes KPI forecasting for RAN networks with fast runtime and accuracy.
problem Efficiently forecasting KPIs for RAN networks with dynamic operating ranges.
method Quartile-Based Seasonality Decomposition (QBSD) for live single-step forecasting.
result QBSD outperforms other methods in runtime efficiency and forecast accuracy.
Fourier Neural Operators accurately predict dynamics of high-dimensional ionic models.
problem Approximating stiff, multiscale ionic models using neural networks.
method Fourier Neural Operators for learning dynamics of high-dimensional ionic models.
result Fourier Neural Operators can accurately predict dynamics of high-dimensional ionic models.
The wide adoption of DNNs has given birth to unrelenting computing requirements, forcing datacenter operators to adopt domain-specific accelerators to train them. These accelerators typically employ densely packed full precision floating-point arithmetic to maximize performance per area. Ongoing research efforts seek t…
Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.
problem Determining when the Calabi operator can identify the range of the Killing operator in Lorentzian locally symmetric spaces.
method Developed criteria for a connection to be in the range of a connection, applied to the Killing connection.
result For indecomposable spaces, the Calabi operator suffices to identify the range of the Killing operator; for products, it fails.
This paper extends transfer operator theory to McKean-Vlasov equations.
problem Analyzing the behavior of complex dynamical systems using transfer operators.
method Extended dynamic mode decomposition and Galerkin projection.
result Finite-dimensional approximations of transfer operators computed.
Machine learning recently has been used to identify the governing equations for dynamics in physical systems. The promising results from applications on systems such as fluid dynamics and chemical kinetics inspire further investigation of these methods on complex engineered systems. Dynamics of these systems play a cru…
We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity…
We explore nature of price formation in financial markets and develop a theory of bid and ask price dynamics in which the two prices form due to quantum-chaotic interaction between buy and sell orders. In this model bid and ask prices are represented by eigenvalues of a 2x2 price operator corresponding to 'bid' and 'as…
TGNs learn from dynamic graphs efficiently and outperform previous methods.
problem Learning from graphs that evolve over time.
method Temporal Graph Networks (TGNs) combining memory and graph operators.
result Significantly outperforms previous approaches on dynamic graphs.
Infrastructure monitoring is critical for safe operations and sustainability. Water distribution networks (WDNs) are large-scale networked critical systems with complex cascade dynamics which are difficult to predict. Ubiquitous monitoring is expensive and a key challenge is to infer the contaminant dynamics from parti…
Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear is a central challenge in modern dynamical systems. These transformations have the potential to enable prediction, estimation, and control of nonlinear systems using standard linear theory. The Koopman operator has emerged…
This paper describes results characterizing the range of the time-t heat operator on various manifolds, including Euclidean spaces, spheres, and hyperbolic spaces. The guiding principle behind these results is this: The functions in the range of the heat operator should be, roughly, those functions having an analytic c…
New method clusters directed graphs using Koopman operators.
problem Challenges in clustering directed graphs, especially complex eigenvalues and lack of cluster definition.
method Relate graph Laplacians to transfer operators and metastable sets in stochastic systems, derive clustering algorithms for directed and time-evolving graphs.
result Clusters can be interpreted as coherent sets, useful for analyzing transport and mixing processes.
Neural signals are characterized by rich temporal and spatiotemporal dynamics that reflect the organization of cortical networks. Theoretical research has shown how neural networks can operate at different dynamic ranges that correspond to specific types of information processing. Here we present a data analysis framew…
New conditions for weighted composition operators in group homomorphisms.
problem Conditions for weighted composition operators in group homomorphisms.
method Range decreasing group homomorphisms.
result New insights into weighted composition operators and their algebraic structure.
CViT learns complex physical systems using vision transformer techniques.
problem Learning maps between infinite-dimensional function spaces in scientific machine learning.
method Combines vision transformer encoder, grid-based coordinate embedding, and cross-attention mechanism.
result Achieves state-of-the-art performance on multiple benchmarks, often surpassing larger models.
The paper tackles data-driven optimal control of unknown nonlinear systems using RKHS.
problem Unknown nonlinear dynamics and stage cost functions.
method Embed state densities into RKHS, learn Markov operators, solve Hamilton-Jacobi-Bellman recursions.
result Solves a wide range of nonlinear control problems, including depth regulation.
Conventional hardware-friendly quantization methods, such as fixed-point or integer, tend to perform poorly at very low word sizes as their shrinking dynamic ranges cannot adequately capture the wide data distributions commonly seen in sequence transduction models. We present AdaptivFloat, a floating-point inspired num…
New insights into X-ray transform on hyperbolic disk, with functional relations and range characterizations.
problem Understanding the X-ray transform on hyperbolic geometry.
method Derived new singular value decompositions, range characterizations, and intertwining relations with wedge-type differential operators.
result Sharp understanding of boundary behavior and invertibility settings for the X-ray transform.
Two heuristics solve dynamic multiple travelling salesmen problems.
problem Dynamic routing with unknown customers.
method Balanced dynamic closest vehicle heuristic and balanced dynamic assignment vehicle heuristic.
result Continuous approximation models for strategic dynamic routing.
New algorithms extract Koopman invariant subspaces from large-scale data.
problem Difficulty in discerning the Koopman invariant subspace from many Koopman eigenmodes.
method Multi-task feature learning and pruning procedure to remove spurious modes.
result Effective in approximating Koopman operator for complex flows.
PBC improves AI and dynamical subseasonal forecasts by reducing biases.
problem Subseasonal forecast accuracy drops due to model biases and compounding errors.
method Probabilistic bias correction (PBC) using machine learning to correct historical forecasts.
result PBC doubles AI Forecasting System's subseasonal skill and improves dynamical model skill.
We study degenerate complex Monge-Ampère equations on a compact Kähler manifold (X,ω). We show that the complex Monge-Ampère operator (ω+ddc⋅)n is well-defined on the class E(X,ω) of ω-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class E(X,ω) is the la…
DOODL learns shared spectral dynamics across related dynamical systems.
problem Learning independent dynamical operators for each system limits discovery of shared structure.
method DOODL learns a dictionary of characteristic spectral dynamics on a manifold of related systems.
result DOODL achieves errors one to two orders of magnitude lower than independent operator estimation methods.
DeepSVM learns SVMs without PDE solving, achieving high pricing accuracy.
problem Computational bottleneck in real-time calibration of stochastic volatility models.
method Physics-informed Deep Operator Network (PI-DeepONet) that enforces terminal payoffs and no-arbitrage conditions.
result DeepSVM achieves high pricing accuracy across various market dynamics.
Understanding temporal dynamics has proved to be highly valuable for accurate recommendation. Sequential recommenders have been successful in modeling the dynamics of users and items over time. However, while different model architectures excel at capturing various temporal ranges or dynamics, distinct application cont…
Geometrically studies Moore-Penrose inverse and polar decomposition continuity.
problem Continuity of Moore-Penrose inverse for perturbations by operator ideals.
method Geometric construction using essential codimension and Banach-Lie group action.
result Moore-Penrose inverse is a real analytic map between manifolds.
Neural Power Unit (NPU) learns arbitrary power functions on real numbers.
problem Neural Networks struggle with generalizing beyond seen data and arithmetic operations.
method Introduces Neural Power Unit (NPU) that operates on real numbers and learns arbitrary power functions.
result NPU outperforms competitors in accuracy and sparsity on arithmetic datasets and discovers governing equations from data.
Approximate dynamic programming (ADP) has proven itself in a wide range of applications spanning large-scale transportation problems, health care, revenue management, and energy systems. The design of effective ADP algorithms has many dimensions, but one crucial factor is the stepsize rule used to update a value functi…
GraphKKE learns fixed-length feature vectors from time-evolving graphs of human microbiome data.
problem Understanding dynamic changes in human microbiome graphs over time.
method Spectral analysis of transfer operators and graph kernels.
result GraphKKE captures temporal changes in human microbiome graphs.
Revisits orbital minimization for neural operator decomposition.
problem Training neural networks to approximate eigenfunctions of operators.
method Adapts orbital minimization method (OMM) for neural networks.
result Justifies broader applicability of OMM in modern learning pipelines.
ABC method improves subseasonal weather forecasting by 60-90%.
problem Improving subseasonal temperature and precipitation forecasting accuracy.
method Combines dynamical forecasts with machine learning-based bias correction.
result Significant improvement in temperature and precipitation forecasting skills.
Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
We consider non-self-adjoint Schrödinger operators Δ+V where Δ is the Laplace-Beltrami operator on a Zoll manifold X and V∈C∞(X,C). We obtain asymptotic results on the pseudo-spectrum and numerical range of such operators.
MRIF models dynamic user interests at multiple temporal-ranges.
problem Capturing dynamic and multi-resolution user interests in recommendation.
method Multi-resolution Interest Fusion (MRIF) model that considers both temporal-ranges and drifts in user interests.
result MRIF outperforms state-of-the-art recommendation methods consistently.
Enhances neural operators with physics knowledge for more accurate simulations.
problem Improving accuracy and generalization of neural operators for physical systems.
method Jointly learns from original PDEs and simplified forms, incorporating fundamental physics.
result Significant improvement in nRMSE across various PDE problems.
Deep reinforcement learning algorithms have been successfully applied to a range of challenging control tasks. However, these methods typically struggle with achieving effective exploration and are extremely sensitive to the choice of hyperparameters. One reason is that most approaches use a noisy version of their oper…
The paper addresses uncertainty in demand prediction for dynamic pricing.
problem Uncertainty quantification in the demand function for dynamic pricing.
method Developed a debiased approach to construct accurate confidence intervals for the demand function.
result Asymptotic normality guarantee of the debiased estimator for the demand function.
dynoNet learns dynamical systems using linear operators.
problem Learning complex dynamical systems.
method dynoNet uses linear dynamical operators for sequence modeling and system identification.
result dynoNet effectively identifies systems on benchmarks.
The paper constructs a complex for the Dirac operator in 4 dimensions.
problem Constructing a complex for the Dirac operator in 4 dimensions.
method Using the Penrose transform, the paper constructs a relative BGG complex and its direct image.
result An explicit construction of a complex starting with the Dirac operator in any number of variables.
A new metric compares dynamical systems using operator eigenvalues.
problem Comparing and interpolating nonlinear dynamical systems from trajectory data.
method Representing systems as distributions of operator eigenvalues and projectors, defining a spectral-Grassmann Wasserstein metric.
result The proposed metric outperforms standard operator-based distances in machine learning applications.
Kernel method approximates dynamical operators from data.
problem Estimating eigenfunctions of dynamical operators from data.
method Kernel-based approach in reproducing kernel Hilbert spaces.
result Eigenfunctions estimated via matrix eigenvalue problems.
New algorithm learns Koopman operator online, with complexity control and convergence guarantees.
problem Online learning of Koopman operator for general nonlinear systems.
method Sparse online learning via stochastic approximation, RKHS action, CME operator.
result Provably convergent algorithm with finite-time guarantees in mis-specified setting.