82 research outputs found
A Neural PDE Solver with Temporal Stencil Modeling
Numerical simulation of non-linear partial differential equations plays a
crucial role in modeling physical science and engineering phenomena, such as
weather, climate, and aerodynamics. Recent Machine Learning (ML) models trained
on low-resolution spatio-temporal signals have shown new promises in capturing
important dynamics in high-resolution signals, under the condition that the
models can effectively recover the missing details. However, this study shows
that significant information is often lost in the low-resolution down-sampled
features. To address such issues, we propose a new approach, namely Temporal
Stencil Modeling (TSM), which combines the strengths of advanced time-series
sequence modeling (with the HiPPO features) and state-of-the-art neural PDE
solvers (with learnable stencil modeling). TSM aims to recover the lost
information from the PDE trajectories and can be regarded as a temporal
generalization of classic finite volume methods such as WENO. Our experimental
results show that TSM achieves the new state-of-the-art simulation accuracy for
2-D incompressible Navier-Stokes turbulent flows: it significantly outperforms
the previously reported best results by 19.9% in terms of the highly-correlated
duration time and reduces the inference latency into 80%. We also show a strong
generalization ability of the proposed method to various out-of-distribution
turbulent flow settings. Our code is available at
"https://github.com/Edward-Sun/TSM-PDE"
Data-driven discovery of coordinates and governing equations
The discovery of governing equations from scientific data has the potential
to transform data-rich fields that lack well-characterized quantitative
descriptions. Advances in sparse regression are currently enabling the
tractable identification of both the structure and parameters of a nonlinear
dynamical system from data. The resulting models have the fewest terms
necessary to describe the dynamics, balancing model complexity with descriptive
ability, and thus promoting interpretability and generalizability. This
provides an algorithmic approach to Occam's razor for model discovery. However,
this approach fundamentally relies on an effective coordinate system in which
the dynamics have a simple representation. In this work, we design a custom
autoencoder to discover a coordinate transformation into a reduced space where
the dynamics may be sparsely represented. Thus, we simultaneously learn the
governing equations and the associated coordinate system. We demonstrate this
approach on several example high-dimensional dynamical systems with
low-dimensional behavior. The resulting modeling framework combines the
strengths of deep neural networks for flexible representation and sparse
identification of nonlinear dynamics (SINDy) for parsimonious models. It is the
first method of its kind to place the discovery of coordinates and models on an
equal footing.Comment: 25 pages, 6 figures; added acknowledgment
Machine learning algorithms for fluid mechanics
Neural networks have become increasingly popular in the field of fluid dynamics due to their ability to model complex, high-dimensional flow phenomena. Their flexibility in approximating continuous functions without any preconceived notion of functional form makes them a suitable tool for studying fluid dynamics. The main uses of neural networks in fluid dynamics include turbulence modelling, flow control, prediction of flow fields, and accelerating high-fidelity simulations. This thesis focuses on the latter two applications of neural networks.
First, the application of a convolutional neural network (CNN) to accelerate the solution of the Poisson equation step in the pressure projection method for incompressible fluid flows is investigated. The CNN learns to approximate the Poisson equation solution at a lower computational cost than traditional iterative solvers, enabling faster simulations of fluid flows. Results show that the CNN approach is accurate and efficient, achieving significant speedup in the Taylor-Green Vortex problem.
Next, predicting flow fields past arbitrarily-shaped bluff bodies from point sensor and plane velocity measurements using neural networks is focused on. A novel conformal-mapping-aided method is devised to embed geometry invariance for the outputs of the neural networks, which is shown to be critical for achieving good performance for flow datasets incorporating a diverse range of geometries. Results show that the proposed methods can accurately predict the flow field, demonstrating excellent agreement with simulation data. Moreover, the flow field predictions can be used to accurately predict lift and drag coefficients, making these methods useful for optimizing the shape of bluff bodies for specific applications.Open Acces
Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation
Turbulent flows have historically presented formidable challenges to
predictive computational modeling. Traditional numerical simulations often
require vast computational resources, making them infeasible for numerous
engineering applications. As an alternative, deep learning-based surrogate
models have emerged, offering data-drive solutions. However, these are
typically constructed within deterministic settings, leading to shortfall in
capturing the innate chaotic and stochastic behaviors of turbulent dynamics. We
introduce a novel generative framework grounded in probabilistic diffusion
models for versatile generation of spatiotemporal turbulence. Our method
unifies both unconditional and conditional sampling strategies within a
Bayesian framework, which can accommodate diverse conditioning scenarios,
including those with a direct differentiable link between specified conditions
and generated unsteady flow outcomes, and scenarios lacking such explicit
correlations. A notable feature of our approach is the method proposed for
long-span flow sequence generation, which is based on autoregressive
gradient-based conditional sampling, eliminating the need for cumbersome
retraining processes. We showcase the versatile turbulence generation
capability of our framework through a suite of numerical experiments,
including: 1) the synthesis of LES simulated instantaneous flow sequences from
URANS inputs; 2) holistic generation of inhomogeneous, anisotropic wall-bounded
turbulence, whether from given initial conditions, prescribed turbulence
statistics, or entirely from scratch; 3) super-resolved generation of
high-speed turbulent boundary layer flows from low-resolution data across a
range of input resolutions. Collectively, our numerical experiments highlight
the merit and transformative potential of the proposed methods, making a
significant advance in the field of turbulence generation.Comment: 37 pages, 31 figure
- …