Learning emergent PDEs in a learned emergent space

• URL: http://arxiv.org/abs/2012.12738v1
• Date: Wed, 23 Dec 2020 15:17:21 GMT
• Title: Learning emergent PDEs in a learned emergent space
• Authors: Felix P. Kemeth, Tom Bertalan, Thomas Thiem, Felix Dietrich, Sung Joon Moon, Carlo R. Laing and Ioannis G. Kevrekidis
• Abstract summary: We learn predictive models in the form of partial differential equations (PDEs) for the collective description of a coupled-agent system. We show that the collective dynamics on a slow manifold can be approximated through a learned model based on local "spatial" partial derivatives in the emergent coordinates. The proposed approach thus integrates the automatic, data-driven extraction of emergent space coordinates parametrizing the agent dynamics.
• Score: 0.6157382820537719
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We extract data-driven, intrinsic spatial coordinates from observations of the dynamics of large systems of coupled heterogeneous agents. These coordinates then serve as an emergent space in which to learn predictive models in the form of partial differential equations (PDEs) for the collective description of the coupled-agent system. They play the role of the independent spatial variables in this PDE (as opposed to the dependent, possibly also data-driven, state variables). This leads to an alternative description of the dynamics, local in these emergent coordinates, thus facilitating an alternative modeling path for complex coupled-agent systems. We illustrate this approach on a system where each agent is a limit cycle oscillator (a so-called Stuart-Landau oscillator); the agents are heterogeneous (they each have a different intrinsic frequency $\omega$) and are coupled through the ensemble average of their respective variables. After fast initial transients, we show that the collective dynamics on a slow manifold can be approximated through a learned model based on local "spatial" partial derivatives in the emergent coordinates. The model is then used for prediction in time, as well as to capture collective bifurcations when system parameters vary. The proposed approach thus integrates the automatic, data-driven extraction of emergent space coordinates parametrizing the agent dynamics, with machine-learning assisted identification of an "emergent PDE" description of the dynamics in this parametrization.

Related papers

• 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)
• On the Trajectory Regularity of ODE-based Diffusion Sampling [79.17334230868693]
  Diffusion-based generative models use differential equations to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models.
  arXiv  Detail & Related papers  (2024-05-18T15:59:41Z)
• Synthetic location trajectory generation using categorical diffusion models [50.809683239937584]
  Diffusion models (DPMs) have rapidly evolved to be one of the predominant generative models for the simulation of synthetic data. We propose using DPMs for the generation of synthetic individual location trajectories (ILTs) which are sequences of variables representing physical locations visited by individuals.
  arXiv  Detail & Related papers  (2024-02-19T15:57:39Z)
• Data-Driven Model Selections of Second-Order Particle Dynamics via Integrating Gaussian Processes with Low-Dimensional Interacting Structures [0.9821874476902972]
  We focus on the data-driven discovery of a general second-order particle-based model. We present applications to modeling two real-world fish motion datasets.
  arXiv  Detail & Related papers  (2023-11-01T23:45:15Z)
• Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems [0.0]
  Autoencoder framework combines implicit regularization with internal linear layers and $L$ regularization (weight decay) We show that this framework can be naturally extended for applications of state-space modeling and forecasting.
  arXiv  Detail & Related papers  (2023-05-01T21:14:47Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Random Feature Models for Learning Interacting Dynamical Systems [2.563639452716634]
  We consider the problem of constructing a data-based approximation of the interacting forces directly from noisy observations of the paths of the agents in time. The learned interaction kernels are then used to predict the agents behavior over a longer time interval. In addition, imposing sparsity reduces the kernel evaluation cost which significantly lowers the simulation cost for forecasting the multi-agent systems.
  arXiv  Detail & Related papers  (2022-12-11T20:09:36Z)
• Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach [0.0]
  We present an Equation/Variable free machine learning (EVFML) framework for the control of the collective dynamics of complex/multiscale systems. The proposed implementation consists of three steps: (A) from high-dimensional agent-based simulations, machine learning (in particular, non-linear manifold learning (DMs)) We exploit the Equation-free approach to perform numerical bifurcation analysis of the emergent dynamics. We design data-driven embedded wash-out controllers that drive the agent-based simulators to their intrinsic, imprecisely known, emergent open-loop unstable steady-states.
  arXiv  Detail & Related papers  (2022-07-12T18:16:22Z)
• Capturing Actionable Dynamics with Structured Latent Ordinary Differential Equations [68.62843292346813]
  We propose a structured latent ODE model that captures system input variations within its latent representation. Building on a static variable specification, our model learns factors of variation for each input to the system, thus separating the effects of the system inputs in the latent space.
  arXiv  Detail & Related papers  (2022-02-25T20:00:56Z)
• Multi-objective discovery of PDE systems using evolutionary approach [77.34726150561087]
  In the paper, a multi-objective co-evolution algorithm is described. The single equations within the system and the system itself are evolved simultaneously to obtain the system. In contrast to the single vector equation, a component-wise system is more suitable for expert interpretation and, therefore, for applications.
  arXiv  Detail & Related papers  (2021-03-11T15:37:52Z)
• Learning Interaction Kernels for Agent Systems on Riemannian Manifolds [9.588842746998486]
  We generalize the theory and algorithms in [1] introduced in the Euclidean setting. We show that our estimators converge at a rate that is independent of the dimension of the manifold. We demonstrate highly accurate performance of the learning algorithm on three classical first order interacting systems.
  arXiv  Detail & Related papers  (2021-01-30T22:15:50Z)

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.