Adversarial Uncertainty Quantification in Physics-Informed Neural Networks

By Yibo Yang, Paris Perdikaris. 2018

424 citations recorded by the source corpus.

Abstract We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physics-informed neural networks. Specifically, we employ latent variable models to construct probabilistic representations for the system states, and put forth an adversarial inference procedure for training them on data, while constraining their predictions to satisfy given physical laws expressed by partial differential equations. Such physics-informed constraints provide a regularization mechanism for effectively training deep generative models as surrogates of physical systems in which the cost of data acquisition is high, and training data-sets are typically small. This provides a flexible framework for characterizing uncertainty in the outputs of physical systems due to randomness in their inputs or noise in their observations that entirely bypasses the need for repeatedly sampling expensive experiments or numerical simulators. We demonstrate the effectiveness of our approach through a series of examples involving uncertainty propagation in non-linear conservation laws, and the discovery of constitutive laws for flow through porous media directly from noisy data.

Subfields: Computer Vision and Pattern Recognition, Statistical and Nonlinear Physics, Statistics and Probability, Safety, Robustness & Interpretability, AI for Physical Sciences

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Citations in the corpus

Physics informed deep learning for computational elastodynamics without labeled data (2020, background): “, crack detection [62–64] and material damage model calibration [65]) due to the optimization nature of training PINN [9], and (3) the capability to conduct uncertainty quantification (UQ) on physical systems [43, 66].”
Understanding and mitigating gradient pathologies in physics-informed neural networks (2020, methodology): “A representative example of this approach is physics-informed neural networks [12, 13, 14, 15], in which the outputs of a neural network are constrained to approximately satisfy a system of partial differential equations (PDE) by using a regularization functionalR[fθ(x)] that typically corresponds t”
PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs (2019, methodology): “The CG PINN is constructed as a [1] + [4] × 2 + [1] DNN encoded with the sODE du/dt = a and IC u(0) = 0, while each fine PINN is constructed as a [1] + [16] × 3 + [1] DNN encoded with the exact ODE du/dt = a +ω cos(ωt) and IC u(0) = 0.”
On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks (2020, background): “PINNs in particular, have demonstrated remarkable power in applications including fluid dynamics [12, 13, 14], biomedical engineering [15, 16, 17], meta-material design [18, 19], free boundary problems [20], Bayesian networks and uncertainty quantification [21, 22], high dimensional PDEs [23, 24, 25”
Integrating machine learning and multiscale modeling—perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences (2019, neutral): “[84] Yang, Y.”
Weak Adversarial Networks for High-dimensional Partial Differential Equations (2019, methodology): “In [28], an adversarial inference procedure is used for quantifying and propagating uncertainty in systems governed by non-linear differential equations, where the discriminator distinguishes the real observation and the approximation provided by the generative network through the given physical law”
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems (2020, background): “9) [290] [303] [288] [304] [318] [102] [298] [184] [68] [300] [288] [274] [77]”
Informed Machine Learning – A Taxonomy and Survey of Integrating Prior Knowledge into Learning Systems (2019, background): “An extension to generative models is possible, too [82], [89].”
Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, background): “In these cases, physics-guided constraints provide effective regularization for training deep generative models to serve as surrogates of physical systems where the cost of acquiring data is high and the data sets are small [282].”
PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, methodology): “The total uncertainty quantification (UQ) in PINN was also investigated based on adversarial inference [39] and arbitrary polynomial chaos in conjunction with dropout [40].”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, background): “Another important extension is the uncertainty quantification for the PDE solutions inferred by neural networks [29,30,31,32,33].”
Physics-informed machine learning: case studies for weather and climate modelling (2021, methodology): “[88] use adversarial UQ in NNs to construct a probabilistic ML model for a system governed by PDEs and use the model to characterize uncertainty to due noisy inputs.”