2018

Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks

Tartakovsky, Alexandre M., Marrero, Carlos Ortiz, Perdikaris, Paris et al.

Understand

We present a physics informed deep neural network (DNN) method for estimating parameters and unknown physics (constitutive relationships) in partial differential equation (PDE) models.

We use PDEs in addition to measurements to train DNNs to approximate unknown parameters and constitutive relationships as well as states.
The proposed approach increases the accuracy of DNN approximations of partially known functions when a limited number of measurements is available and allows for training DNNs when no direct measurements of the functions of interest are available.
We employ physics informed DNNs to estimate the unknown space-dependent diffusion coefficient in a linear diffusion equation and an unknown constitutive relationship in a non-linear diffusion equation.

Built on

Comparison of three methods for selecting values of input variables in the analysis of output from a computer code
Michael D McKay, Richard J Beckman, and William J Conover · 1979
Earlier work this paper cites.
A closed-form equation for predicting the hydraulic conductivity of unsaturated soils 1
M Th Van Genuchten · 1980
Earlier work this paper cites.
Representations of non-linear systems: the narmax model
Sheng Chen and Steve A Billings · 1989
Earlier work this paper cites.
A limited memory algorithm for bound constrained optimization
Richard H Byrd, Peihuang Lu, Jorge Nocedal, and Ciyou Zhu · 1995
Earlier work this paper cites.
Modeling fluid flow and transport in variably saturated porous media with the stomp simulator. 1. nonvolatile three-phase model description
MD White, M Oostrom, and RJ Lenhard · 1995
Earlier work this paper cites.
Identification of distributed parameter systems: A neural net based approach
R Gonzalez-Garcia, R Rico-Martinez, and IG Kevrekidis · 1998
Earlier work this paper cites.
Equation-free, coarse-grained multiscale computation: Enabling mocroscopic simulators to perform system-level analysis
Ioannis G Kevrekidis, C William Gear, James M Hyman, Panagiotis G Kevrekidid, Olof Runborg, Constantinos Theodoropoulos, et al · 2003
Earlier work this paper cites.
Inverse problem in hydrogeology
Jesús Carrera, Andrés Alcolea, Agustín Medina, Juan Hidalgo, and Luit J. Slooten · 2005
Earlier work this paper cites.
Understanding the difficulty of training deep feedforward neural networks
Xavier Glorot and Yoshua Bengio · 2010
Earlier work this paper cites.
Parameter and state model reduction for large-scale statistical inverse problems
Chad Lieberman, Karen Willcox, and Omar Ghattas · 2010
Earlier work this paper cites.

Similar

Inverse problems: a bayesian perspective
Andrew M Stuart · 2010
Cited alongside, same era.
Nonlinear laplacian spectral analysis for time series with intermittency and low-frequency variability
Dimitrios Giannakis and Andrew J Majda · 2012
Cited alongside, same era.
Dynamics of fluids in porous media
Jacob Bear · 2013
Cited alongside, same era.
The method of volume averaging
Stephen Whitaker · 2013
Cited alongside, same era.
Asynchronous stochastic gradient descent for dnn training
Shanshan Zhang, Ce Zhang, Zhao You, Rong Zheng, and Bo Xu · 2013
Cited alongside, same era.
Linear functional minimization for inverse modeling
D. A. Barajas-Solano, B. E. Wohlberg, V. V. Vesselinov, and D. M. Tartakovsky · 2014
Cited alongside, same era.
A data–driven approximation of the koopman operator: Extending dynamic mode decomposition
Matthew O Williams, Ioannis G Kevrekidis, and Clarence W Rowley · 2015
Cited alongside, same era.
Chaos as an intermittently forced linear system
Steven L Brunton, Bingni W Brunton, Joshua L Proctor, Eurika Kaiser, and J Nathan Kutz · 2017
Cited alongside, same era.
Data-driven spectral decomposition and forecasting of ergodic dynamical systems
Dimitrios Giannakis · 2017
Cited alongside, same era.

Then

Deep learning for universal linear embeddings of nonlinear dynamics
Original
Bethany Lusch, J Nathan Kutz, and Steven L Brunton · 2017
Later among the works it cites.
Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations
Original
Maziar Raissi, Paris Perdikaris, and George Em Karniadakis · 2017
Later among the works it cites.
Physics informed deep learning (part ii): Data-driven discovery of nonlinear partial differential equations
Original
Maziar Raissi, Paris Perdikaris, and George Em Karniadakis · 2017
Later among the works it cites.
Learning deep neural network representations for koopman operators of nonlinear dynamical systems
Original
Enoch Yeung, Soumya Kundu, and Nathan Hodas · 2017
Later among the works it cites.
Variable projection methods for an optimized dynamic mode decomposition
Travis Askham and J Nathan Kutz · 2018
Closest in time.
Vampnets for deep learning of molecular kinetics
Andreas Mardt, Luca Pasquali, Hao Wu, and Frank Noé · 2018
Closest in time.
Deep hidden physics models: Deep learning of nonlinear partial differential equations
Original
Maziar Raissi · 2018
Closest in time.
Data-driven identification of parametric partial differential equations
Original
Samuel Rudy, Alessandro Alla, Steven L Brunton, and J Nathan Kutz · 2018
Closest in time.
Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
Christoph Wehmeyer and Frank Noé · 2018
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…