Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks

By Alexandre M. Tartakovsky, Carlos Ortiz Marrero, Paris Perdikaris, Guzel Tartakovsky, David Alonso Barajas-Solano. 2018

110 citations recorded by the source corpus.

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. For the parameter estimation problem, we assume that partial measurements of the coefficient and states are available and demonstrate that under these conditions, the proposed method is more accurate than state-of-the-art methods. For the non-linear diffusion PDE model with a fully unknown constitutive relationship (i.e., no measurements of constitutive relationship are available), the physics informed DNN method can accurately estimate the non-linear constitutive relationship based on state measurements only. Finally, we demonstrate that the proposed method remains accurate in the presence of measurement noise.

Subfields: Artificial Intelligence, Computational Mechanics, Statistical and Nonlinear Physics

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Citations in the corpus

Understanding and mitigating gradient pathologies in physics-informed neural networks (2020, background): “This is the main reason why the majority of recent works focused on applying deep learning tools to modeling and simulating physical systems governed by PDEs, still relies on generic fully connected architectures [12, 24, 23, 43, 26].”
Adversarial Uncertainty Quantification in Physics-Informed Neural Networks (2018, methodology): “To this end, we revisit the Darcy ﬂow example put forth in [51] corresponding to a two-dimensional nonlinear diﬀusion equation with an unknown state-dependent diﬀusion coeﬃcient where q = 8 .”
Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data (2019, methodology): “A similar approach is also applied to learn the constitutive relationship in a Darcy ﬂow [40].”
DeepXDE: A Deep Learning Library for Solving Differential Equations (2019, background): “An attractive feature of PINNs is that it can be used to solve inverse problems with minimum change of the code for forward problems [47, 48, 51, 21, 13].”
A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems (2019, background): “…PINNs include (1) learning the velocity and pressure ﬁelds based on partial observations of spatial-temporal visualizations of a passive scalar, i.e., solute concentration [23], and (2) estimation of the unknown constitutive relationship in the nonlinear diﬀusion equation for unsaturated ﬂows [24].”
A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate (2021, methodology): “Based on the physics informed deep neural networks, Tartakovsky et al. studied the estimation of parameters and unknown physics in PDE models [43].”
When and why PINNs fail to train: A neural tangent kernel perspective (2020, background): “…[6, 7], meta-material design [8, 9, 10], free boundary problems [11], Bayesian networks and uncertainty quantiﬁcation [12, 13, 14, 15, 16], high-dimensional PDEs [17, 18], stochastic differential equations [19], fractional differential equations [20, 21], and beyond [22, 23, 24, 25].”
Integrating machine learning and multiscale modeling—perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences (2019, unknown): “[68] Tartakovsky, A.”
Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport (2019, methodology): “In [25], we demonstrated that the Darcy law can be used as a constraint for solving the minimization problem (4) and learning hydraulic conductivity from sparse measurements of conductivity and hydraulic head using the PINN approach [22].”
Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems (2018, methodology): “…similar methods have been advanced, e.g., Bayesian estimation [13] and variational Bayes inference [14], and have been proposed for a wide variety of objectives, from parameter estimation [15] to discovering partial diﬀerential equations [16, 17, 18, 19] to learning constitutive relationships [20].”
Multiscale modeling meets machine learning: What can we learn? (2019, methodology): “In areas where multiscale models are well-developed, simulation across many parameter has been used as a supplement to existing training data for nonlinear diffusion models to provide physics-informed machine learning [100, 121, 122].”
Machine learning in cardiovascular flows modeling: Predicting pulse wave propagation from non-invasive clinical measurements using physics-informed deep learning (2019, background): “Physics-informed neural networks Physics informed neural networks (PINNs) is a new paradigm that leverages recent advances in deep learning to infer solutions, parameters and/or constitutive laws invlolving partial differential equations (PDEs) [37, 38, 35, 53].”