Course Correcting Koopman Representations

• URL: http://arxiv.org/abs/2310.15386v2
• Date: Thu, 23 Nov 2023 06:32:10 GMT
• Title: Course Correcting Koopman Representations
• Authors: Mahan Fathi and Clement Gehring and Jonathan Pilault and David Kanaa and Pierre-Luc Bacon and Ross Goroshin
• Abstract summary: We study autoencoder formulations of this problem, and different ways they can be used to model dynamics. We propose an inference-time mechanism, which we refer to as Periodic Reencoding, for faithfully capturing long term dynamics.
• Score: 12.517740162118855
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Koopman representations aim to learn features of nonlinear dynamical systems (NLDS) which lead to linear dynamics in the latent space. Theoretically, such features can be used to simplify many problems in modeling and control of NLDS. In this work we study autoencoder formulations of this problem, and different ways they can be used to model dynamics, specifically for future state prediction over long horizons. We discover several limitations of predicting future states in the latent space and propose an inference-time mechanism, which we refer to as Periodic Reencoding, for faithfully capturing long term dynamics. We justify this method both analytically and empirically via experiments in low and high dimensional NLDS.

Related papers

• PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems [31.006807854698376]
  We propose a new graph learning approach, namely, Physics-encoded Message Passing Graph Network (PhyMPGN) We incorporate a GNN into a numerical integrator to approximate the temporal marching of partialtemporal dynamics for a given PDE system. PhyMPGN is capable of accurately predicting various types of operatortemporal dynamics on coarse unstructured meshes.
  arXiv  Detail & Related papers  (2024-10-02T08:54:18Z)
• Latent Space Energy-based Neural ODEs [73.01344439786524]
  This paper introduces a novel family of deep dynamical models designed to represent continuous-time sequence data. We train the model using maximum likelihood estimation with Markov chain Monte Carlo. Experiments on oscillating systems, videos and real-world state sequences (MuJoCo) illustrate that ODEs with the learnable energy-based prior outperform existing counterparts.
  arXiv  Detail & Related papers  (2024-09-05T18:14:22Z)
• Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint [12.638698799995815]
  We introduce the Dynamic System Prediction from Sparse Observations using Voronoi Tessellation (DSOVT) framework. By integrating Voronoi tessellations with deep learning models, DSOVT is adept at predicting dynamical systems with sparse, unstructured observations. Compared to purely data-driven models, our physics-based approach enables the model to learn physical laws within explicitly formulated dynamics.
  arXiv  Detail & Related papers  (2024-08-31T13:43:52Z)
• Koopman Invertible Autoencoder: Leveraging Forward and Backward Dynamics for Temporal Modeling [13.38194491846739]
  We propose a novel machine learning model based on Koopman operator theory, which we call Koopman Invertible Autoencoders (KIA) KIA captures the inherent characteristic of the system by modeling both forward and backward dynamics in the infinite-dimensional Hilbert space. This enables us to efficiently learn low-dimensional representations, resulting in more accurate predictions of long-term system behavior.
  arXiv  Detail & Related papers  (2023-09-19T03:42:55Z)
• Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics [97.38308257547186]
  Many NN approaches learn an end-to-end model that implicitly models both the governing PDE and material models. We argue that the governing PDEs are often well-known and should be explicitly enforced rather than learned. We introduce a new framework termed "Neural Constitutive Laws" (NCLaw) which utilizes a network architecture that strictly guarantees standard priors.
  arXiv  Detail & Related papers  (2023-04-27T17:42:24Z)
• Anamnesic Neural Differential Equations with Orthogonal Polynomial Projections [6.345523830122166]
  We propose PolyODE, a formulation that enforces long-range memory and preserves a global representation of the underlying dynamical system. Our construction is backed by favourable theoretical guarantees and we demonstrate that it outperforms previous works in the reconstruction of past and future data.
  arXiv  Detail & Related papers  (2023-03-03T10:49:09Z)
• Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series [18.885471782270375]
  NCDSSM employs auxiliary variables to disentangle recognition from dynamics, thus requiring amortized inference only for the auxiliary variables. We propose three flexible parameterizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference. Empirical results on multiple benchmark datasets show improved imputation and forecasting performance of NCDSSM over existing models.
  arXiv  Detail & Related papers  (2023-01-26T18:45:04Z)
• SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Vision [73.26414295633846]
  A recently proposed class of models attempts to learn latent dynamics from high-dimensional observations. Existing methods rely on image reconstruction quality, which does not always reflect the quality of the learnt latent dynamics. We develop a set of new measures, including a binary indicator of whether the underlying Hamiltonian dynamics have been faithfully captured.
  arXiv  Detail & Related papers  (2021-11-10T23:26:58Z)
• Likelihood-Free Inference in State-Space Models with Unknown Dynamics [71.94716503075645]
  We introduce a method for inferring and predicting latent states in state-space models where observations can only be simulated, and transition dynamics are unknown. We propose a way of doing likelihood-free inference (LFI) of states and state prediction with a limited number of simulations.
  arXiv  Detail & Related papers  (2021-11-02T12:33:42Z)
• Liquid Time-constant Networks [117.57116214802504]
  We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations.
  arXiv  Detail & Related papers  (2020-06-08T09:53:35Z)
• Forecasting Sequential Data using Consistent Koopman Autoencoders [52.209416711500005]
  A new class of physics-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamical systems. We propose a novel Consistent Koopman Autoencoder model which, unlike the majority of existing work, leverages the forward and backward dynamics. Key to our approach is a new analysis which explores the interplay between consistent dynamics and their associated Koopman operators.
  arXiv  Detail & Related papers  (2020-03-04T18:24:30Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.