21 research outputs found
LNO: Laplace Neural Operator for Solving Differential Equations
We introduce the Laplace neural operator (LNO), which leverages the Laplace
transform to decompose the input space. Unlike the Fourier Neural Operator
(FNO), LNO can handle non-periodic signals, account for transient responses,
and exhibit exponential convergence. LNO incorporates the pole-residue
relationship between the input and the output space, enabling greater
interpretability and improved generalization ability. Herein, we demonstrate
the superior approximation accuracy of a single Laplace layer in LNO over four
Fourier modules in FNO in approximating the solutions of three ODEs (Duffing
oscillator, driven gravity pendulum, and Lorenz system) and three PDEs
(Euler-Bernoulli beam, diffusion equation, and reaction-diffusion system).
Notably, LNO outperforms FNO in capturing transient responses in undamped
scenarios. For the linear Euler-Bernoulli beam and diffusion equation, LNO's
exact representation of the pole-residue formulation yields significantly
better results than FNO. For the nonlinear reaction-diffusion system, LNO's
errors are smaller than those of FNO, demonstrating the effectiveness of using
system poles and residues as network parameters for operator learning. Overall,
our results suggest that LNO represents a promising new approach for learning
neural operators that map functions between infinite-dimensional spaces.Comment: 18 pages, 8 figures, 2 table
A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation
The phase-field fracture free-energy functional is non-convex with respect to
the displacement and the phase field. This results in a poor performance of the
conventional monolithic solvers like the Newton-Raphson method. In order to
circumvent this issue, researchers opt for the alternate minimization
(staggered) solvers. Staggered solvers are robust for the phase-field based
fracture simulations as the displacement and the phase-field sub-problems are
convex in nature. Nevertheless, the staggered solver requires very large number
of iterations (of the order of thousands) to converge. In this work, a robust
monolithic solver is presented for the phase-field fracture problem. The solver
adopts a fracture energy-based arc-length method and an adaptive
under-relaxation scheme. The arc-length method enables the simulation to
overcome critical points (snap-back, snap-through instabilities) during the
loading of a specimen. The use of an under-relaxation scheme stabilizes the
solver by preventing the divergence due to an ill-behaving stiffness matrix.
The efficiency of the proposed solver is further amplified with an adaptive
mesh refinement scheme based on PHT-splines within the framework of
isogeometric analysis. The numerical examples presented in the manuscript
demonstrates the efficacy of the solver. All the codes and data-sets
accompanying this work will be made available on GitHub
(https://github.com/rbharali/IGAFrac)
Physics-Informed Deep Neural Operator Networks
Standard neural networks can approximate general nonlinear operators,
represented either explicitly by a combination of mathematical operators, e.g.,
in an advection-diffusion-reaction partial differential equation, or simply as
a black box, e.g., a system-of-systems. The first neural operator was the Deep
Operator Network (DeepONet), proposed in 2019 based on rigorous approximation
theory. Since then, a few other less general operators have been published,
e.g., based on graph neural networks or Fourier transforms. For black box
systems, training of neural operators is data-driven only but if the governing
equations are known they can be incorporated into the loss function during
training to develop physics-informed neural operators. Neural operators can be
used as surrogates in design problems, uncertainty quantification, autonomous
systems, and almost in any application requiring real-time inference. Moreover,
independently pre-trained DeepONets can be used as components of a complex
multi-physics system by coupling them together with relatively light training.
Here, we present a review of DeepONet, the Fourier neural operator, and the
graph neural operator, as well as appropriate extensions with feature
expansions, and highlight their usefulness in diverse applications in
computational mechanics, including porous media, fluid mechanics, and solid
mechanics.Comment: 33 pages, 14 figures. arXiv admin note: text overlap with
arXiv:2204.00997 by other author
Deep neural operators can predict the real-time response of floating offshore structures under irregular waves
The use of neural operators in a digital twin model of an offshore floating
structure can provide a paradigm shift in structural response prediction and
health monitoring, providing valuable information for real-time control. In
this work, the performance of three neural operators is evaluated, namely, deep
operator network (DeepONet), Fourier neural operator (FNO), and Wavelet neural
operator (WNO). We investigate the effectiveness of the operators to accurately
capture the responses of a floating structure under six different sea state
codes based on the wave characteristics described by the World
Meteorological Organization (WMO). The results demonstrate that these
high-precision neural operators can deliver structural responses more
efficiently, up to two orders of magnitude faster than a dynamic analysis using
conventional numerical solvers. Additionally, compared to gated recurrent units
(GRUs), a commonly used recurrent neural network for time-series estimation,
neural operators are both more accurate and efficient, especially in situations
with limited data availability. To further enhance the accuracy, novel
extensions, such as wavelet-DeepONet and self-adaptive WNO, are proposed. Taken
together, our study shows that FNO outperforms all other operators for
approximating the mapping of one input functional space to the output space as
well as for responses that have small bandwidth of the frequency spectrum,
whereas for learning the mapping of multiple functions in the input space to
the output space as well as for capturing responses within a large frequency
spectrum, DeepONet with historical states provides the highest accuracy.Comment: 27 pages, 24 figures, 11 table
Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks
Deep neural networks are an attractive alternative for simulating complex
dynamical systems, as in comparison to traditional scientific computing
methods, they offer reduced computational costs during inference and can be
trained directly from observational data. Existing methods, however, cannot
extrapolate accurately and are prone to error accumulation in long-time
integration. Herein, we address this issue by combining neural operators with
recurrent neural networks to construct a novel and effective architecture,
resulting in superior accuracy compared to the state-of-the-art. The new hybrid
model is based on operator learning while offering a recurrent structure to
capture temporal dependencies. The integrated framework is shown to stabilize
the solution and reduce error accumulation for both interpolation and
extrapolation of the Korteweg-de Vries equation.Comment: 12 pages, 5 figure
Deep transfer learning for partial differential equations under conditional shift with DeepONet
Traditional machine learning algorithms are designed to learn in isolation,
i.e. address single tasks. The core idea of transfer learning (TL) is that
knowledge gained in learning to perform one task (source) can be leveraged to
improve learning performance in a related, but different, task (target). TL
leverages and transfers previously acquired knowledge to address the expense of
data acquisition and labeling, potential computational power limitations, and
the dataset distribution mismatches. Although significant progress has been
made in the fields of image processing, speech recognition, and natural
language processing (for classification and regression) for TL, little work has
been done in the field of scientific machine learning for functional regression
and uncertainty quantification in partial differential equations. In this work,
we propose a novel TL framework for task-specific learning under conditional
shift with a deep operator network (DeepONet). Inspired by the conditional
embedding operator theory, we measure the statistical distance between the
source domain and the target feature domain by embedding conditional
distributions onto a reproducing kernel Hilbert space. Task-specific operator
learning is accomplished by fine-tuning task-specific layers of the target
DeepONet using a hybrid loss function that allows for the matching of
individual target samples while also preserving the global properties of the
conditional distribution of target data. We demonstrate the advantages of our
approach for various TL scenarios involving nonlinear PDEs under conditional
shift. Our results include geometry domain adaptation and show that the
proposed TL framework enables fast and efficient multi-task operator learning,
despite significant differences between the source and target domains.Comment: 19 pages, 3 figure
Developing a cost-effective emulator for groundwater flow modeling using deep neural operators
Current groundwater models face a significant challenge in their
implementation due to heavy computational burdens. To overcome this, our work
proposes a cost-effective emulator that efficiently and accurately forecasts
the impact of abstraction in an aquifer. Our approach uses a deep neural
operator (DeepONet) to learn operators that map between infinite-dimensional
function spaces via deep neural networks. The goal is to infer the distribution
of hydraulic head in a confined aquifer in the presence of a pumping well. We
successfully tested the DeepONet on four problems, including two forward
problems, an inverse analysis, and a nonlinear system. Additionally, we propose
a novel extension of the DeepONet-based architecture to generate accurate
predictions for varied hydraulic conductivity fields and pumping well locations
that are unseen during training. Our emulator's predictions match the target
data with excellent performance, demonstrating that the proposed model can act
as an efficient and fast tool to support a range of tasks that require
repetitive forward numerical simulations or inverse simulations of groundwater
flow problems. Overall, our work provides a promising avenue for developing
cost-effective and accurate groundwater models
Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysms
[EN] Thoracic aortic aneurysm (TAA) is a localized dilatation of the aorta that can lead to life-threatening dissection or rupture. In vivo assessments of TAA progression are largely limited to measurements of aneurysm size and growth rate. There is promise, however, that computational modelling of the evolving biomechanics of the aorta could predict future geometry and properties from initiating mechanobiological insults. We present an integrated framework to train a deep operator network (DeepONet)-based surrogate model to identify TAA contributing factors using synthetic finite-element-based datasets. For training, we employ a constrained mixture model of aortic growth and remodelling to generate maps of local aortic dilatation and distensibility for multiple TAA risk factors. We evaluate the performance of the surrogate model for insult distributions varying from fusiform (analytically defined) to complex (randomly generated). We propose two frameworks, one trained on sparse information and one on full-field greyscale images, to gain insight into a preferred neural operator-based approach. We show that this continuous learning approach can predict the patient-specific insult profile associated with any given dilatation and distensibility map with high accuracy, particularly when based on full-field images. Our findings demonstrate the feasibility of applying DeepONet to support transfer learning of patient-specific inputs to predict TAA progression.This work was supported by the National Institutes of Health (grant nos. P01 HL134605 and U01 HL142518)Goswami, S.; Li, DS.; Rego, BV.; Latorre, M.; Humphrey, JD.; Karniadakis, GE. (2022). Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysms. Journal of The Royal Society Interface. 19(193):1-16. https://doi.org/10.1098/rsif.2022.04101161919