Data-driven discovery of coarse-grained equations

Bakarji, Joseph; Tartakovsky, Daniel M.

doi:10.1016/j.jcp.2021.110219

Cited by 31 publications

(20 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When the potential terms of the differential equations we seek are not known a priori, or if they include integral operators (e.g. non-local closures [82]), a naive implementation of SINDy gives sub-optimal results. A particularly versatile solution is to use autoencoders to transform the coordinates of the input variable before discovering a differential equation, as in Champion et al [36].…”

Section: Introductionmentioning

confidence: 99%

Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders

Bakarji¹,

Champion²,

Kutz³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

A central challenge in data-driven model discovery is the presence of hidden, or latent, variables that are not directly measured but are dynamically important. Takens' theorem provides conditions for when it is possible to augment these partial measurements with time delayed information, resulting in an attractor that is diffeomorphic to that of the original full-state system. However, the coordinate transformation back to the original attractor is typically unknown, and learning the dynamics in the embedding space has remained an open challenge for decades. Here, we design a custom deep autoencoder network to learn a coordinate transformation from the delay embedded space into a new space where it is possible to represent the dynamics in a sparse, closed form. We demonstrate this approach on the Lorenz, R össler, and Lotka-Volterra systems, learning dynamics from a single measurement variable. As a challenging example, we learn a Lorenz analogue from a single scalar variable extracted from a video of a chaotic waterwheel experiment. The resulting modeling framework combines deep learning to uncover effective coordinates and the sparse identification of nonlinear dynamics (SINDy) for interpretable modeling. Thus, we show that it is possible to simultaneously learn a closedform model and the associated coordinate system for partially observed dynamics.

show abstract

Section: Introductionmentioning

confidence: 99%

Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders

Bakarji¹,

Champion²,

Kutz³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The number of different approaches pursued here is vast. It includes the use of simple neural networks [12], recurrent neural networks [13], deep neural networks with dimension reduction via diffusion maps [14], recurrent neural network architectures with dimension reduction accomplished via an autoencoder scheme [15], multilayer perceptrons for generating reduced-order metamodels [2], and direct / constrained equation learning methods [16]. Because these models are designed to learn the appropriate macroscale behavior from an appropriate suite of ML approaches, the end result is frequently high-fidelity predictive ability from the generated network, but usually without the generation of an explicit macroscale equation for the process.…”

Section: Introductionmentioning

confidence: 99%

Explicit physics-informed neural networks for non-linear upscaling closure: the case of transport in tissues

Taghizadeh,

Byrne,

Wood

2021

Preprint

View full text Add to dashboard Cite

In this work, we use a combination of formal upscaling and data-driven machine learning for explicitly closing a nonlinear transport and reaction process in a multiscale tissue. The classical effectiveness factor model is used to formulate the macroscale reaction kinetics. We train a multilayer perceptron network using training data generated by direct numerical simulations over microscale examples. Once trained, the network is used for numerically solving the upscaled (coarse-grained) differential equation describing mass transport and reaction in two example tissues. The network is described as being explicit in the sense that the network is trained using macroscale concentrations and gradients of concentration as components of the feature space.Network training and solutions to the macroscale transport equations were computed for two different tissues. The two tissue types (brain and liver) exhibit markedly different geometrical complexity and spatial scale (cell size and sample size). The upscaled solutions for the average concentration are compared with numerical solutions derived from the microscale concentration fields by a posteriori averaging. There are two outcomes of this work of particular note: 1) we find that the trained network exhibits good generalizability, and it is able to predict the effectiveness factor with high fidelity for realistically-structured tissues despite the significantly different scale and geometry of the two example tissue types; and 2) the approach results in an upscaled PDE with an effectiveness factor that is predicted (implicitly) via the trained neural network. This latter result emphasizes our purposeful connection between conventional averaging methods with the use of machine learning for closure; this contrasts with some machine learning methods for upscaling

show abstract

“…A notable development in this pursuit is the sparse identification of nonlinear dynamics (SINDy) algorithm ( [3]), a general framework for discovering dynamical systems using sparse regression. Since its inception in the context of autonomous ordinary differential equations (ODEs), SINDy has been extended to autonomous partial differential equations (PDEs) ( [28,30]), stochastic differential equations (SDEs) ( [2]), non-autonomous systems ( [27]), and coarse-grained equations ( [1]), to name a few. A significant challenge in using SINDy to solve real-world problems is the computation of derivatives from noisy data.…”

mentioning

confidence: 99%

“…Within the last few years, the consensus has emerged that weak-form SINDy (WSINDy, see [23,24,22]), where integration against test functions replaces numerical differentiation, is a powerful method that is significantly more robust to noisy data, particularly in the context of PDEs. Furthermore, WSINDy's efficient convolutional formulation makes it a viable method for identifying PDEs under the constraints of limited memory capacity and computing power that exist in the online setting 1 .…”

mentioning

confidence: 99%

Online Weak-form Sparse Identification of Partial Differential Equations

Messenger¹,

Dall’Anese²,

Bortz³

2022

Preprint

View full text Add to dashboard Cite

This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weakform discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem.In particular, we do not regularize the 0 -pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.

show abstract

Data-driven discovery of coarse-grained equations

Cited by 31 publications

References 45 publications

Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders

Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders

Explicit physics-informed neural networks for non-linear upscaling closure: the case of transport in tissues

Online Weak-form Sparse Identification of Partial Differential Equations

Contact Info

Product

Resources

About