Deep learning-based surrogate models have demonstrated remarkable advantages
over classical solvers in terms of speed, often achieving speedups of 10 to
1000 times over traditional partial differential equation (PDE) solvers.
However, a significant challenge hindering their widespread adoption in both
scientific and industrial domains is the lack of understanding about their
prediction uncertainties, particularly in scenarios that involve critical
decision making. To address this limitation, we propose a method that
integrates efficient and precise uncertainty quantification into a deep
learning-based surrogate model. Our method, termed Latent Evolution of PDEs
with Uncertainty Quantification (LE-PDE-UQ), endows deep learning-based
surrogate models with robust and efficient uncertainty quantification
capabilities for both forward and inverse problems. LE-PDE-UQ leverages latent
vectors within a latent space to evolve both the system's state and its
corresponding uncertainty estimation. The latent vectors are decoded to provide
predictions for the system's state as well as estimates of its uncertainty. In
extensive experiments, we demonstrate the accurate uncertainty quantification
performance of our approach, surpassing that of strong baselines including deep
ensembles, Bayesian neural network layers, and dropout. Our method excels at
propagating uncertainty over extended auto-regressive rollouts, making it
suitable for scenarios involving long-term predictions. Our code is available
at: this https URL

提出了一种将高效精确的不确定性量化整合到基于深度学习的代理模型中的方法，称为 LE-PDE-UQ，其具备了前向和反向问题中的鲁棒高效的不确定性量化能力。

通过潜在全局演化对 PDE 的正向和反向问题进行不确定性量化

Uncertainty Quantification for Forward and Inverse Problems of PDEs via  Latent Global Evolution

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.

我们介绍了一种基于概率扩散模型的新型生成框架，用于多样化生成时空湍流。我们的方法在贝叶斯框架内统一了无条件和条件抽样策略，并能适应各种条件情景，包括指定条件与生成的非稳定流动结果之间具有直接可微分链接的情况以及缺乏明确相关性的情况。我们的方法的一个显著特点是基于自回归梯度条件抽样的长时跨度流动序列生成方法，在没有繁琐的重新训练过程的情况下完成。通过一系列数值实验展示了我们框架的多样化湍流生成能力，包括：1）从 URANS 输入合成 LES 模拟的瞬态流动序列；2）从给定的初始条件、规定的湍流统计数据或完全从头开始生成非均匀的、各向异性的壁面湍流；3）从不同输入分辨率的低分辨率数据中实现高速湍流边界层流动的超分辨生成。综合而言，我们的数值实验突显了所提方法的优点和变革潜力，在湍流生成领域取得了重大进展。

贝叶斯条件扩散模型用于多功能时空湍流生成

Bayesian Conditional Diffusion Models for Versatile Spatiotemporal  Turbulence Generation

Deep learning-based surrogate models have been widely applied in geological
carbon storage (GCS) problems to accelerate the prediction of reservoir
pressure and CO2 plume migration. Large amounts of data from physics-based
numerical simulators are required to train a model to accurately predict the
complex physical behaviors associated with this process. In practice, the
available training data are always limited in large-scale 3D problems due to
the high computational cost. Therefore, we propose to use a multi-fidelity
Fourier Neural Operator to solve large-scale GCS problems with more affordable
multi-fidelity training datasets. The Fourier Neural Operator has a desirable
grid-invariant property, which simplifies the transfer learning procedure
between datasets with different discretization. We first test the model
efficacy on a GCS reservoir model being discretized into 110k grid cells. The
multi-fidelity model can predict with accuracy comparable to a high-fidelity
model trained with the same amount of high-fidelity data with 81% less data
generation costs. We further test the generalizability of the multi-fidelity
model on a same reservoir model with a finer discretization of 1 million grid
cells. This case was made more challenging by employing high-fidelity and
low-fidelity datasets generated by different geostatistical models and
reservoir simulators. We observe that the multi-fidelity FNO model can predict
pressure fields with reasonable accuracy even when the high-fidelity data are
extremely limited.

采用多保真度傅里叶神经操作器解决大规模地质碳储存问题，并减少高真实度数据生成成本。