Learning viscoelasticity models from indirect data using deep neural networks

Abstract We propose a novel approach to model viscoelasticity materials, where rate-dependent and non-linear constitutive relationships are approximated with deep neural networks . We assume that inputs and outputs of the neural networks are not directly observable, and therefore common training techniques with input–output pairs for the neural networks are inapplicable. To that end, we develop a novel computational approach to both calibrate parametric and learn neural-network-based constitutive relations of viscoelasticity materials from indirect displacement data in the context of multiple-physics systems. We show that limited displacement data holds sufficient information to quantify the viscoelasticity behavior. We formulate the inverse computation – modeling viscoelasticity properties from observed displacement data – as a PDE-constrained optimization problem and minimize the error functional using a gradient-based optimization method. The gradients are computed by a combination of automatic differentiation and implicit function differentiation rules. The effectiveness of our method is demonstrated through numerous benchmark problems in geomechanics and porous media transport.

[1]  Martín Abadi,et al.  TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems , 2016, ArXiv.

[2]  Jorge Nocedal,et al.  A Limited Memory Algorithm for Bound Constrained Optimization , 1995, SIAM J. Sci. Comput..

[3]  Eric Darve,et al.  Learning Constitutive Relations using Symmetric Positive Definite Neural Networks , 2020, J. Comput. Phys..

[4]  E Weinan,et al.  Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning , 2017, ArXiv.

[5]  E Weinan,et al.  Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations , 2017, Communications in Mathematics and Statistics.

[6]  Harvey Thomas Banks,et al.  A Brief Review of Elasticity and Viscoelasticity for Solids , 2011 .

[7]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[8]  Sepp Hochreiter,et al.  Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs) , 2015, ICLR.

[9]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[10]  Qinwu Xu,et al.  An inverse model and mathematical solution for inferring viscoelastic properties and dynamic deformations of heterogeneous structures , 2016 .

[11]  Charbel Farhat,et al.  Learning constitutive relations from indirect observations using deep neural networks , 2020, J. Comput. Phys..

[12]  Paris Perdikaris,et al.  Physics‐Informed Deep Neural Networks for Learning Parameters and Constitutive Relationships in Subsurface Flow Problems , 2020, Water Resources Research.

[13]  V. Solomatov,et al.  Scaling of temperature‐ and stress‐dependent viscosity convection , 1995 .

[14]  Roscoe A. Bartlett,et al.  Sacado: Automatic Differentiation Tools for C++ Codes. , 2009 .

[15]  A. Tartakovsky,et al.  Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport , 2019, Advances in Water Resources.

[16]  Denis Demidov,et al.  AMGCL: An Efficient, Flexible, and Extensible Algebraic Multigrid Implementation , 2018, Lobachevskii Journal of Mathematics.

[17]  M Fink,et al.  Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: An inverse problem approach , 2004 .

[18]  Luca Antiga,et al.  Automatic differentiation in PyTorch , 2017 .

[19]  Geoffrey E. Hinton,et al.  Deep Learning , 2015, Nature.

[20]  Eduards Skuķis,et al.  Characterisation of Viscoelastic Layers in Sandwich Panels via an Inverse Technique , 2009 .

[21]  Marcela Perrone-Bertolotti,et al.  Identifying task-relevant spectral signatures of perceptual categorization in the human cortex , 2020, Scientific Reports.

[22]  Jung Kim,et al.  Characterization of Viscoelastic Soft Tissue Properties from In Vivo Animal Experiments and Inverse FE Parameter Estimation , 2005, MICCAI.

[23]  Harold R. Parks,et al.  The Implicit Function Theorem , 2002 .

[24]  James F. Greenleaf,et al.  Inverse estimation of viscoelastic material properties for solids immersed in fluids using vibroacoustic techniques , 2007 .

[25]  Barak A. Pearlmutter,et al.  Automatic differentiation in machine learning: a survey , 2015, J. Mach. Learn. Res..

[26]  Jinkook Kim,et al.  Experimental and Numerical Sensitivity Assessment of Viscoelasticity for Polymer Composite Materials , 2020, Scientific Reports.

[27]  Eric Darve,et al.  The Neural Network Approach to Inverse Problems in Differential Equations , 2019, 1901.07758.

[28]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[29]  Andrew J. Wathen,et al.  Optimal Solvers for PDE-Constrained Optimization , 2010, SIAM J. Sci. Comput..

[30]  William B. Zimmerman,et al.  Multiphysics Modeling with Finite Element Methods , 2006 .

[31]  Paris Perdikaris,et al.  Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , 2019, J. Comput. Phys..

[32]  Zachary Chase Lipton.  A Critical Review of Recurrent Neural Networks for Sequence Learning , 2015, ArXiv.

[33]  Joonas Sorvari,et al.  Modelling Methods for Viscoelastic Constitutive Modelling of Paper , 2009 .

[34]  Sepp Hochreiter,et al.  Self-Normalizing Neural Networks , 2017, NIPS.

[35]  Eric Darve,et al.  Physics Constrained Learning for Data-driven Inverse Modeling from Sparse Observations , 2020, J. Comput. Phys..

[36]  Gabriel Peyr'e,et al.  Super-efficiency of automatic differentiation for functions defined as a minimum , 2020, ICML.

[37]  M. Arnold,et al.  Convergence of the generalized-α scheme for constrained mechanical systems , 2007 .

[38]  Miles Lubin,et al.  Forward-Mode Automatic Differentiation in Julia , 2016, ArXiv.

[39]  Nihat Özkaya,et al.  Mechanical Properties of Biological Tissues , 2012 .

[40]  J. Janno,et al.  Inverse problems for identification of memory kernels in viscoelasticity , 1997 .

[41]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[42]  E. Pagnacco,et al.  Inverse strategies for the identification of elastic and viscoelastic material parameters using full-field measurements , 2007 .

[43]  Petr N. Vabishchevich,et al.  Splitting scheme for poroelasticity and thermoelasticity problems , 2013, FDM.

[44]  Arnulf Jentzen,et al.  Solving high-dimensional partial differential equations using deep learning , 2017, Proceedings of the National Academy of Sciences.

[45]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

[46]  Paris Perdikaris,et al.  A comparative study of physics-informed neural network models for learning unknown dynamics and constitutive relations , 2019, ArXiv.

[47]  R. Christensen.  Theory of viscoelasticity : an introduction , 1971 .

[48]  Yoshua Bengio,et al.  Understanding the difficulty of training deep feedforward neural networks , 2010, AISTATS.

[49]  Jian Sun,et al.  Identity Mappings in Deep Residual Networks , 2016, ECCV.

[50]  Joel Nothman,et al.  SciPy 1.0-Fundamental Algorithms for Scientific Computing in Python , 2019, ArXiv.

[51]  David J. Thuente,et al.  Line search algorithms with guaranteed sufficient decrease , 1994, TOMS.

[52]  M. Heinkenschloss,et al.  Large-Scale PDE-Constrained Optimization: An Introduction , 2003 .

[53]  Andi Merxhani.  An introduction to linear poroelasticity , 2016, 1607.04274.

[54]  Paris Perdikaris,et al.  Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks , 2018, 1808.03398.