88 research outputs found
Locally optimal unstructured finite element meshes in 3 dimensions
This paper investigates the adaptive finite element solution of a general class of variational problems in three dimensions using a combination of node movement, edge swapping, face swapping and node insertion. The adaptive strategy proposed is a generalization of previous work in two dimensions and is based upon the construction of a hierarchy of locally optimal meshes.
Results presented, both for a single equation and a system of coupled equations, suggest that this approach is able to produce better meshes of tetrahedra than those obtained by more conventional adaptive strategies and in a relatively efficient manner
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
Understanding the Efficacy of U-Net & Vision Transformer for Groundwater Numerical Modelling
This paper presents a comprehensive comparison of various machine learning
models, namely U-Net, U-Net integrated with Vision Transformers (ViT), and
Fourier Neural Operator (FNO), for time-dependent forward modelling in
groundwater systems. Through testing on synthetic datasets, it is demonstrated
that U-Net and U-Net + ViT models outperform FNO in accuracy and efficiency,
especially in sparse data scenarios. These findings underscore the potential of
U-Net-based models for groundwater modelling in real-world applications where
data scarcity is prevalent
On spatial adaptivity and interpolation when using the method of lines
The solution of time-dependent partial differential equations with discrete time static remeshing is considered within a method of lines framework. Numerical examples in one and two space dimensions are used to show that spatial interpolation error may have an important impact on the efficiency of integration. Analysis of a simple problem and of the time integration method is used to confirm the experimental results and a computational test for monitoring the impact of this error is derived and tested
Attention U-Net as a surrogate model for groundwater prediction
Numerical simulations of groundwater flow are used to analyze and predict the
response of an aquifer system to its change in state by approximating the
solution of the fundamental groundwater physical equations. The most used and
classical methodologies, such as Finite Difference (FD) and Finite Element (FE)
Methods, use iterative solvers which are associated with high computational
cost. This study proposes a physics-based convolutional encoder-decoder neural
network as a surrogate model to quickly calculate the response of the
groundwater system. Holding strong promise in cross-domain mappings,
encoder-decoder networks are applicable for learning complex input-output
mappings of physical systems. This manuscript presents an Attention U-Net model
that attempts to capture the fundamental input-output relations of the
groundwater system and generates solutions of hydraulic head in the whole
domain given a set of physical parameters and boundary conditions. The model
accurately predicts the steady state response of a highly heterogeneous
groundwater system given the locations and piezometric head of up to 3 wells as
input. The network learns to pay attention only in the relevant parts of the
domain and the generated hydraulic head field corresponds to the target samples
in great detail. Even relative to coarse finite difference approximations the
proposed model is shown to be significantly faster than a comparative
state-of-the-art numerical solver, thus providing a base for further
development of the presented networks as surrogate models for groundwater
prediction
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