Foundational Inference Models for Dynamical Systems

• URL: http://arxiv.org/abs/2402.07594v2
• Date: Fri, 04 Oct 2024 10:41:18 GMT
• Title: Foundational Inference Models for Dynamical Systems
• Authors: Patrick Seifner, Kostadin Cvejoski, Antonia Körner, Ramsés J. Sánchez,
• Abstract summary: We offer a fresh perspective on the classical problem of imputing missing time series data, whose underlying dynamics are assumed to be determined by ODEs. We propose a novel supervised learning framework for zero-shot time series imputation, through parametric functions satisfying some (hidden) ODEs. We empirically demonstrate that one and the same (pretrained) recognition model can perform zero-shot imputation across 63 distinct time series with missing values.
• Score: 5.549794481031468
• License:
• Abstract: Dynamical systems governed by ordinary differential equations (ODEs) serve as models for a vast number of natural and social phenomena. In this work, we offer a fresh perspective on the classical problem of imputing missing time series data, whose underlying dynamics are assumed to be determined by ODEs. Specifically, we revisit ideas from amortized inference and neural operators, and propose a novel supervised learning framework for zero-shot time series imputation, through parametric functions satisfying some (hidden) ODEs. Our proposal consists of two components. First, a broad probability distribution over the space of ODE solutions, observation times and noise mechanisms, with which we generate a large, synthetic dataset of (hidden) ODE solutions, along with their noisy and sparse observations. Second, a neural recognition model that is trained offline, to map the generated time series onto the spaces of initial conditions and time derivatives of the (hidden) ODE solutions, which we then integrate to impute the missing data. We empirically demonstrate that one and the same (pretrained) recognition model can perform zero-shot imputation across 63 distinct time series with missing values, each sampled from widely different dynamical systems. Likewise, we demonstrate that it can perform zero-shot imputation of missing high-dimensional data in 10 vastly different settings, spanning human motion, air quality, traffic and electricity studies, as well as Navier-Stokes simulations -- without requiring any fine-tuning. What is more, our proposal often outperforms state-of-the-art methods, which are trained on the target datasets. Our pretrained model will be available online soon.

Related papers

• FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model [5.748690310135373]
  We propose a novel multi-modal foundation model, named textbfFMint, to bridge the gap between human-designed and data-driven models. Built on a decoder-only transformer architecture with in-context learning, FMint utilizes both numerical and textual data to learn a universal error correction scheme. Our results demonstrate the effectiveness of the proposed model in terms of both accuracy and efficiency compared to classical numerical solvers.
  arXiv  Detail & Related papers  (2024-04-23T02:36:47Z)
• Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs [50.25683648762602]
  We introduce Koopman VAE, a new generative framework that is based on a novel design for the model prior. Inspired by Koopman theory, we represent the latent conditional prior dynamics using a linear map. KoVAE outperforms state-of-the-art GAN and VAE methods across several challenging synthetic and real-world time series generation benchmarks.
  arXiv  Detail & Related papers  (2023-10-04T07:14:43Z)
• 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)
• On the balance between the training time and interpretability of neural ODE for time series modelling [77.34726150561087]
  The paper shows that modern neural ODE cannot be reduced to simpler models for time-series modelling applications. The complexity of neural ODE is compared to or exceeds the conventional time-series modelling tools. We propose a new view on time-series modelling using combined neural networks and an ODE system approach.
  arXiv  Detail & Related papers  (2022-06-07T13:49:40Z)
• Neural ODEs with Irregular and Noisy Data [8.349349605334316]
  We discuss a methodology to learn differential equation(s) using noisy and irregular sampled measurements. In our methodology, the main innovation can be seen in the integration of deep neural networks with the neural ordinary differential equations (ODEs) approach. The proposed framework to learn a model describing the vector field is highly effective under noisy measurements.
  arXiv  Detail & Related papers  (2022-05-19T11:24:41Z)
• Mixed Effects Neural ODE: A Variational Approximation for Analyzing the Dynamics of Panel Data [50.23363975709122]
  We propose a probabilistic model called ME-NODE to incorporate (fixed + random) mixed effects for analyzing panel data. We show that our model can be derived using smooth approximations of SDEs provided by the Wong-Zakai theorem. We then derive Evidence Based Lower Bounds for ME-NODE, and develop (efficient) training algorithms.
  arXiv  Detail & Related papers  (2022-02-18T22:41:51Z)
• Using scientific machine learning for experimental bifurcation analysis of dynamic systems [2.204918347869259]
  This study focuses on training universal differential equation (UDE) models for physical nonlinear dynamical systems with limit cycles. We consider examples where training data is generated by numerical simulations, whereas we also employ the proposed modelling concept to physical experiments. We use both neural networks and Gaussian processes as universal approximators alongside the mechanistic models to give a critical assessment of the accuracy and robustness of the UDE modelling approach.
  arXiv  Detail & Related papers  (2021-10-22T15:43:03Z)
• Neural ODE Processes [64.10282200111983]
  We introduce Neural ODE Processes (NDPs), a new class of processes determined by a distribution over Neural ODEs. We show that our model can successfully capture the dynamics of low-dimensional systems from just a few data-points.
  arXiv  Detail & Related papers  (2021-03-23T09:32:06Z)
• Artificial neural network as a universal model of nonlinear dynamical systems [0.0]
  The map is built as an artificial neural network whose weights encode a modeled system. We consider the Lorenz system, the Roessler system and also Hindmarch-Rose neuron. High similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunovs exponents.
  arXiv  Detail & Related papers  (2021-03-06T16:02:41Z)
• Anomaly Detection of Time Series with Smoothness-Inducing Sequential Variational Auto-Encoder [59.69303945834122]
  We present a Smoothness-Inducing Sequential Variational Auto-Encoder (SISVAE) model for robust estimation and anomaly detection of time series. Our model parameterizes mean and variance for each time-stamp with flexible neural networks. We show the effectiveness of our model on both synthetic datasets and public real-world benchmarks.
  arXiv  Detail & Related papers  (2021-02-02T06:15:15Z)
• STEER: Simple Temporal Regularization For Neural ODEs [80.80350769936383]
  We propose a new regularization technique: randomly sampling the end time of the ODE during training. The proposed regularization is simple to implement, has negligible overhead and is effective across a wide variety of tasks. We show through experiments on normalizing flows, time series models and image recognition that the proposed regularization can significantly decrease training time and even improve performance over baseline models.
  arXiv  Detail & Related papers  (2020-06-18T17:44:50Z)

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.