Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks

By Sharmila Karumuri, Rohit Tripathy, Ilias Bilionis, Jitesh H. Panchal. 2019

154 citations recorded by the source corpus.

Abstract Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions, etc. Because of these functional uncertainties, the stochastic parameter space is often high-dimensional, requiring hundreds, or even thousands, of parameters to describe it. This poses an insurmountable challenge to response surface modeling since the number of forward model evaluations needed to construct an accurate surrogate grows exponentially with the dimension of the uncertain parameter space; a phenomenon referred to as the curse of dimensionality. State-of-the-art methods for high-dimensional uncertainty propagation seek to alleviate the curse of dimensionality by performing dimensionality reduction in the uncertain parameter space. However, one still needs to perform forward model evaluations that potentially carry a very high computational burden. We propose a novel methodology for high-dimensional uncertainty propagation of elliptic SPDEs which lifts the requirement for a deterministic forward solver. Our approach is as follows. We parameterize the solution of the elliptic SPDE using a deep residual network (ResNet). In a departure from traditional squared residual (SR) based loss function for training the ResNet, we introduce a physics-informed loss function derived from variational principles. Specifically, our loss function is the expectation of the energy functional of the PDE over the stochastic variables. We demonstrate our solver-free approach through various examples where the elliptic SPDE is subjected to different types of high-dimensional input uncertainties. Also, we solve high-dimensional uncertainty propagation and inverse problems.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Physics informed deep learning for computational elastodynamics without labeled data (2020, background): “The reason why DNN is well qualiﬁed as the solution approximator includes: (1) its extraordinary approximation capability proven by the universal approximation theorem [49], (2) its inﬁnite continuity property [14] with proper activation function, (3) the derivatives that appears in the PDEs can be ”
Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data (2019, methodology): “In the context of surrogate modeling, Nabian and Meidani [56] and Karumuri et al. [57] applied the PDE-constrained fully-connected neural network (FC-NN) for uncertainty propagation in steady heat equations.”
PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, methodology): “[55] applied the physics-informed FC-NN for efficient uncertainty propagation in steady heat equations.”
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On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks (2020, background): “We believe that this may be one of the fundamental reasons that cause failure of PINNs in learning high-frequency or multi-scale solutions of PDEs.”
Learning the solution operator of parametric partial differential equations with physics-informed DeepONets (2021, unknown): “More recently, ML tools are actively developed to infer solutions of PDEs [12, 13, 14, 15, 16, 17, 18], however most existing tools can only accommodate a fixed given set of input parameters or IBCs.”