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Found 14 papers, 6 papers with code
Multiplicative Dynamic Mode Decomposition
Koopman operators are infinite-dimensional operators that linearize nonlinear dynamical systems, facilitating the study of their spectral properties and enabling the prediction of the time evolution of observable quantities.
LLMs learn governing principles of dynamical systems, revealing an in-context neural scaling law
Pretrained large language models (LLMs) are surprisingly effective at performing zero-shot tasks, including time-series forecasting.
Operator learning without the adjoint
There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint?
On the Convergence of Hermitian Dynamic Mode Decomposition
We show that, under suitable conditions, the eigenvalues and eigenfunctions of HDMD converge to the spectral properties of the underlying Koopman operator.
A Mathematical Guide to Operator Learning
We explain the types of problems and PDEs amenable to operator learning, discuss various neural network architectures, and explain how to employ numerical PDE solvers effectively.
Elliptic PDE learning is provably data-efficient
PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data.
Data-driven discovery of Green's functions
Finally, theoretical results on Green's functions and rational NNs are combined to design a human-understandable deep learning method for discovering Green's functions from data.
Learning Green's functions associated with time-dependent partial differential equations
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data.
A generalization of the randomized singular value decomposition
The randomized singular value decomposition (SVD) is a popular and effective algorithm for computing a near-best rank $k$ approximation of a matrix $A$ using matrix-vector products with standard Gaussian vectors.
Data-driven discovery of Green's functions with human-understandable deep learning
There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner.
Accurate numerical simulation of electrodiffusion and water movement in brain tissue
Mathematical modelling of ionic electrodiffusion and water movement is emerging as a powerful avenue of investigation to provide new physiological insight into brain homeostasis.
Numerical Analysis Computational Engineering, Finance, and Science Numerical Analysis
Learning elliptic partial differential equations with randomized linear algebra
Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$.
Classification of chaotic time series with deep learning
We use standard deep neural networks to classify univariate time series generated by discrete and continuous dynamical systems based on their chaotic or non-chaotic behaviour.
