Automating the Discovery of Partial Differential Equations in Dynamical Systems

• URL: http://arxiv.org/abs/2404.16444v2
• Date: Thu, 2 May 2024 09:31:10 GMT
• Title: Automating the Discovery of Partial Differential Equations in Dynamical Systems
• Authors: Weizhen Li, Rui Carvalho,
• Abstract summary: We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs automatically. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Identifying partial differential equations (PDEs) from data is crucial for understanding the governing mechanisms of natural phenomena, yet it remains a challenging task. We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs from limited prior knowledge automatically. Our method automates calculating partial derivatives, constructing a candidate library, and estimating a sparse model. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes, demonstrating its robustness in handling noisy and non-uniformly distributed data. We also test the algorithm's performance on datasets consisting solely of random noise to simulate scenarios with severely compromised data quality. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases. We highlight the potential of combining statistical methods, machine learning, and dynamical systems theory to automatically discover governing equations from collected data, streamlining the scientific modeling process.

Related papers

• Estimation of System Parameters Including Repeated Cross-Sectional Data through Emulator-Informed Deep Generative Model [5.3060535072023844]
  In politics, economics, and biology, available data are often independently collected at distinct time points from different subjects. Conventional optimization techniques struggle to accurately estimate DE parameters when RCS data exhibit various heterogeneities. We propose a new estimation method called the emulator-informed deep-generative model (EIDGM) EIDGM integrates a physics-informed neural network-based emulator that immediately generates DE solutions and a Wasserstein generative adversarial network-based parameter generator.
  arXiv  Detail & Related papers  (2024-12-27T08:19:23Z)
• A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systems [8.206685537936078]
  This study introduces a stepwise data-driven framework for discovering fractional differential equations (FDEs) directly from data. Our framework applies deep neural networks as surrogate models for denoising and reconstructing sparse and noisy observations. We validate the framework across various datasets, including synthetic anomalous diffusion data and experimental data on the creep behavior of frozen soils.
  arXiv  Detail & Related papers  (2024-12-05T08:38:30Z)
• 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)
• Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes [7.103713918313219]
  We develop conditional independence (CI) constraints on coordinate processes over selected intervals. We provide a sound and complete causal discovery algorithm, capable of handling both fully and partially observed data. We also propose a flexible, consistent signature kernel-based CI test to infer these constraints from data.
  arXiv  Detail & Related papers  (2024-02-28T16:58:31Z)
• Score-based Diffusion Models in Function Space [137.70916238028306]
  Diffusion models have recently emerged as a powerful framework for generative modeling. This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• 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)
• A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study [68.8204255655161]
  We investigate and compare the performance of several local and global smoothing techniques to a priori denoise the state measurements. We show that, in general, global methods, which use the entire measurement data set, outperform local methods, which employ a neighboring data subset around a local point.
  arXiv  Detail & Related papers  (2022-01-29T23:31:25Z)
• Score-Based Generative Modeling through Stochastic Differential Equations [114.39209003111723]
  We present a differential equation that transforms a complex data distribution to a known prior distribution by injecting noise. A corresponding reverse-time SDE transforms the prior distribution back into the data distribution by slowly removing the noise. By leveraging advances in score-based generative modeling, we can accurately estimate these scores with neural networks. We demonstrate high fidelity generation of 1024 x 1024 images for the first time from a score-based generative model.
  arXiv  Detail & Related papers  (2020-11-26T19:39:10Z)
• Identification of Probability weighted ARX models with arbitrary domains [75.91002178647165]
  PieceWise Affine models guarantees universal approximation, local linearity and equivalence to other classes of hybrid system. In this work, we focus on the identification of PieceWise Auto Regressive with eXogenous input models with arbitrary regions (NPWARX) The architecture is conceived following the Mixture of Expert concept, developed within the machine learning field.
  arXiv  Detail & Related papers  (2020-09-29T12:50:33Z)
• Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data [4.996878640124385]
  SINDy is a framework for the discovery of parsimonious dynamic models and equations from time-series data. We develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained by Rudy et al. We show the method can identify a diversity of probability distributions including Gaussian, uniform, Gamma, and Rayleigh.
  arXiv  Detail & Related papers  (2020-09-12T23:52:25Z)
• Weak SINDy For Partial Differential Equations [0.0]
  We extend our Weak SINDy (WSINDy) framework to the setting of partial differential equations (PDEs) The elimination of pointwise derivative approximations via the weak form enables effective machine-precision recovery of model coefficients from noise-free data. We demonstrate WSINDy's robustness, speed and accuracy on several challenging PDEs.
  arXiv  Detail & Related papers  (2020-07-06T16:03:51Z)

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.