1,682 research outputs found
Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
Surrogate modeling and uncertainty quantification tasks for PDE systems are
most often considered as supervised learning problems where input and output
data pairs are used for training. The construction of such emulators is by
definition a small data problem which poses challenges to deep learning
approaches that have been developed to operate in the big data regime. Even in
cases where such models have been shown to have good predictive capability in
high dimensions, they fail to address constraints in the data implied by the
PDE model. This paper provides a methodology that incorporates the governing
equations of the physical model in the loss/likelihood functions. The resulting
physics-constrained, deep learning models are trained without any labeled data
(e.g. employing only input data) and provide comparable predictive responses
with data-driven models while obeying the constraints of the problem at hand.
This work employs a convolutional encoder-decoder neural network approach as
well as a conditional flow-based generative model for the solution of PDEs,
surrogate model construction, and uncertainty quantification tasks. The
methodology is posed as a minimization problem of the reverse Kullback-Leibler
(KL) divergence between the model predictive density and the reference
conditional density, where the later is defined as the Boltzmann-Gibbs
distribution at a given inverse temperature with the underlying potential
relating to the PDE system of interest. The generalization capability of these
models to out-of-distribution input is considered. Quantification and
interpretation of the predictive uncertainty is provided for a number of
problems.Comment: 51 pages, 18 figures, submitted to Journal of Computational Physic
Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data
Numerical simulations on fluid dynamics problems primarily rely on spatially
or/and temporally discretization of the governing equation into the
finite-dimensional algebraic system solved by computers. Due to complicated
nature of the physics and geometry, such process can be computational
prohibitive for most real-time applications and many-query analyses. Therefore,
developing a cost-effective surrogate model is of great practical significance.
Deep learning (DL) has shown new promises for surrogate modeling due to its
capability of handling strong nonlinearity and high dimensionality. However,
the off-the-shelf DL architectures fail to operate when the data becomes
sparse. Unfortunately, data is often insufficient in most parametric fluid
dynamics problems since each data point in the parameter space requires an
expensive numerical simulation based on the first principle, e.g.,
Naiver--Stokes equations. In this paper, we provide a physics-constrained DL
approach for surrogate modeling of fluid flows without relying on any
simulation data. Specifically, a structured deep neural network (DNN)
architecture is devised to enforce the initial and boundary conditions, and the
governing partial differential equations are incorporated into the loss of the
DNN to drive the training. Numerical experiments are conducted on a number of
internal flows relevant to hemodynamics applications, and the forward
propagation of uncertainties in fluid properties and domain geometry is studied
as well. The results show excellent agreement on the flow field and
forward-propagated uncertainties between the DL surrogate approximations and
the first-principle numerical simulations.Comment: 43 pages, 12 figure
Theory-guided Auto-Encoder for Surrogate Construction and Inverse Modeling
A Theory-guided Auto-Encoder (TgAE) framework is proposed for surrogate
construction and is further used for uncertainty quantification and inverse
modeling tasks. The framework is built based on the Auto-Encoder (or
Encoder-Decoder) architecture of convolutional neural network (CNN) via a
theory-guided training process. In order to achieve the theory-guided training,
the governing equations of the studied problems can be discretized and the
finite difference scheme of the equations can be embedded into the training of
CNN. The residual of the discretized governing equations as well as the data
mismatch constitute the loss function of the TgAE. The trained TgAE can be used
to construct a surrogate that approximates the relationship between the model
parameters and responses with limited labeled data. In order to test the
performance of the TgAE, several subsurface flow cases are introduced. The
results show the satisfactory accuracy of the TgAE surrogate and efficiency of
uncertainty quantification tasks can be improved with the TgAE surrogate. The
TgAE also shows good extrapolation ability for cases with different correlation
lengths and variances. Furthermore, the parameter inversion task has been
implemented with the TgAE surrogate and satisfactory results can be obtained
PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain
Recently, the advent of deep learning has spurred interest in the development
of physics-informed neural networks (PINN) for efficiently solving partial
differential equations (PDEs), particularly in a parametric setting. Among all
different classes of deep neural networks, the convolutional neural network
(CNN) has attracted increasing attention in the scientific machine learning
community, since the parameter-sharing feature in CNN enables efficient
learning for problems with large-scale spatiotemporal fields. However, one of
the biggest challenges is that CNN only can handle regular geometries with
image-like format (i.e., rectangular domains with uniform grids). In this
paper, we propose a novel physics-constrained CNN learning architecture, aiming
to learn solutions of parametric PDEs on irregular domains without any labeled
data. In order to leverage powerful classic CNN backbones, elliptic coordinate
mapping is introduced to enable coordinate transforms between the irregular
physical domain and regular reference domain. The proposed method has been
assessed by solving a number of PDEs on irregular domains, including heat
equations and steady Navier-Stokes equations with parameterized boundary
conditions and varying geometries. Moreover, the proposed method has also been
compared against the state-of-the-art PINN with fully-connected neural network
(FC-NN) formulation. The numerical results demonstrate the effectiveness of the
proposed approach and exhibit notable superiority over the FC-NN based PINN in
terms of efficiency and accuracy.Comment: 57 pages, 26 figure
Transfer learning based multi-fidelity physics informed deep neural network
For many systems in science and engineering, the governing differential
equation is either not known or known in an approximate sense. Analyses and
design of such systems are governed by data collected from the field and/or
laboratory experiments. This challenging scenario is further worsened when
data-collection is expensive and time-consuming. To address this issue, this
paper presents a novel multi-fidelity physics informed deep neural network
(MF-PIDNN). The framework proposed is particularly suitable when the physics of
the problem is known in an approximate sense (low-fidelity physics) and only a
few high-fidelity data are available. MF-PIDNN blends physics informed and
data-driven deep learning techniques by using the concept of transfer learning.
The approximate governing equation is first used to train a low-fidelity
physics informed deep neural network. This is followed by transfer learning
where the low-fidelity model is updated by using the available high-fidelity
data. MF-PIDNN is able to encode useful information on the physics of the
problem from the {\it approximate} governing differential equation and hence,
provides accurate prediction even in zones with no data. Additionally, no
low-fidelity data is required for training this model. Applicability and
utility of MF-PIDNN are illustrated in solving four benchmark reliability
analysis problems. Case studies to illustrate interesting features of the
proposed approach are also presented
Simulator-free Solution of High-Dimensional Stochastic Elliptic Partial Differential Equations using Deep Neural Networks
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
\textit{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 the traditional squared residual (SR) based loss
function for training the ResNet, we introduce a novel 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.Comment: 63 pages, 32 figure
PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs
Partial differential equations (PDEs) play a fundamental role in modeling and
simulating problems across a wide range of disciplines. Recent advances in deep
learning have shown the great potential of physics-informed neural networks
(PINNs) to solve PDEs as a basis for data-driven modeling and inverse analysis.
However, the majority of existing PINN methods, based on fully-connected NNs,
pose intrinsic limitations to low-dimensional spatiotemporal parameterizations.
Moreover, since the initial/boundary conditions (I/BCs) are softly imposed via
penalty, the solution quality heavily relies on hyperparameter tuning. To this
end, we propose the novel physics-informed convolutional-recurrent learning
architectures (PhyCRNet and PhyCRNet-s) for solving PDEs without any labeled
data. Specifically, an encoder-decoder convolutional long short-term memory
network is proposed for low-dimensional spatial feature extraction and temporal
evolution learning. The loss function is defined as the aggregated discretized
PDE residuals, while the I/BCs are hard-encoded in the network to ensure
forcible satisfaction (e.g., periodic boundary padding). The networks are
further enhanced by autoregressive and residual connections that explicitly
simulate time marching. The performance of our proposed methods has been
assessed by solving three nonlinear PDEs (e.g., 2D Burgers' equations, the
- and FitzHugh Nagumo reaction-diffusion equations), and
compared against the start-of-the-art baseline algorithms. The numerical
results demonstrate the superiority of our proposed methodology in the context
of solution accuracy, extrapolability and generalizability.Comment: 22 page
A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables
The data-centric construction of inexpensive surrogates for fine-grained,
physical models has been at the forefront of computational physics due to its
significant utility in many-query tasks such as uncertainty quantification.
Recent efforts have taken advantage of the enabling technologies from the field
of machine learning (e.g. deep neural networks) in combination with simulation
data. While such strategies have shown promise even in higher-dimensional
problems, they generally require large amounts of training data even though the
construction of surrogates is by definition a Small Data problem. Rather than
employing data-based loss functions, it has been proposed to make use of the
governing equations (in the simplest case at collocation points) in order to
imbue domain knowledge in the training of the otherwise black-box-like
interpolators. The present paper provides a flexible, probabilistic framework
that accounts for physical structure and information both in the training
objectives as well as in the surrogate model itself. We advocate a
probabilistic (Bayesian) model in which equalities that are available from the
physics (e.g. residuals, conservation laws) can be introduced as virtual
observables and can provide additional information through the likelihood. We
further advocate a generative model i.e. one that attempts to learn the joint
density of inputs and outputs that is capable of making use of unlabeled data
(i.e. only inputs) in a semi-supervised fashion in order to promote the
discovery of lower-dimensional embeddings which are nevertheless predictive of
the fine-grained model's output
Physics-Constrained Bayesian Neural Network for Fluid Flow Reconstruction with Sparse and Noisy Data
In many applications, flow measurements are usually sparse and possibly
noisy. The reconstruction of a high-resolution flow field from limited and
imperfect flow information is significant yet challenging. In this work, we
propose an innovative physics-constrained Bayesian deep learning approach to
reconstruct flow fields from sparse, noisy velocity data, where equation-based
constraints are imposed through the likelihood function and uncertainty of the
reconstructed flow can be estimated. Specifically, a Bayesian deep neural
network is trained on sparse measurement data to capture the flow field. In the
meantime, the violation of physical laws will be penalized on a large number of
spatiotemporal points where measurements are not available. A non-parametric
variational inference approach is applied to enable efficient
physics-constrained Bayesian learning. Several test cases on idealized vascular
flows with synthetic measurement data are studied to demonstrate the merit of
the proposed method.Comment: 17 pages, 5 figure
Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks
In recent years, deep learning has proven to be a viable methodology for
surrogate modeling and uncertainty quantification for a vast number of physical
systems. However, in their traditional form, such models can require a large
amount of training data. This is of particular importance for various
engineering and scientific applications where data may be extremely expensive
to obtain. To overcome this shortcoming, physics-constrained deep learning
provides a promising methodology as it only utilizes the governing equations.
In this work, we propose a novel auto-regressive dense encoder-decoder
convolutional neural network to solve and model non-linear dynamical systems
without training data at a computational cost that is potentially magnitudes
lower than standard numerical solvers. This model includes a Bayesian framework
that allows for uncertainty quantification of the predicted quantities of
interest at each time-step. We rigorously test this model on several non-linear
transient partial differential equation systems including the turbulence of the
Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D
Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For
each system, the predictive results and uncertainty are presented and discussed
together with comparisons to the results obtained from traditional numerical
analysis methods.Comment: 48 pages, 30 figures, Accepted to Journal of Computational Physic
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