The dynamics of flexible filaments entrained in flow, important for
understanding many biological and industrial processes, are computationally
expensive to model with full-physics simulations. This work describes a
data-driven technique to create high-fidelity low-dimensional models of
flexible fiber dynamics using machine learning; the technique is applied to
sedimentation in a quiescent, viscous Newtonian fluid, using results from
detailed simulations as the data set. The approach combines an autoencoder
neural network architecture to learn a low-dimensional latent representation of
the filament shape, with a neural ODE that learns the evolution of the particle
in the latent state. The model was designed to model filaments of varying
flexibility, characterized by an elasto-gravitational number $\mathcal{B}$, and
was trained on a data set containing the evolution of fibers beginning at set
angles of inclination. For the range of $\mathcal{B}$ considered here
(100-10000), the filament shape dynamics can be represented with high accuracy
with only four degrees of freedom, in contrast to the 93 present in the
original bead-spring model used to generate the dynamic trajectories. We
predict the evolution of fibers set at arbitrary angles and demonstrate that
our data-driven model can accurately forecast the evolution of a fiber at both
trained and untrained elasto-gravitational numbers.

通过使用机器学习，本研究描述了一种数据驱动技术来创建高保真度的低维模型，用于灵活纤维动力学的沉积过程预测。

基于数据的沉积弯曲纤维的低维模型

Data-driven low-dimensional model of a sedimenting flexible fiber

Self-training techniques have shown remarkable value across many deep
learning models and tasks. However, such techniques remain largely unexplored
when considered in the context of learning fast solvers for systems of partial
differential equations (Eg: Neural Operators). In this work, we explore the use
of self-training for Fourier Neural Operators (FNO). Neural Operators emerged
as a data driven technique, however, data from experiments or traditional
solvers is not always readily available. Physics Informed Neural Operators
(PINO) overcome this constraint by utilizing a physics loss for the training,
however the accuracy of PINO trained without data does not match the
performance obtained by training with data. In this work we show that
self-training can be used to close this gap in performance. We examine
canonical examples, namely the 1D-Burgers and 2D-Darcy PDEs, to showcase the
efficacy of self-training. Specifically, FNOs, when trained exclusively with
physics loss through self-training, approach 1.07x for Burgers and 1.02x for
Darcy, compared to FNOs trained with both data and physics loss. Furthermore,
we discover that pseudo-labels can be used for self-training without
necessarily training to convergence in each iteration. A consequence of this is
that we are able to discover self-training schedules that improve upon the
baseline performance of PINO in terms of accuracy as well as time.

本研究探索了在求解偏微分方程组（如神经算子）中应用自训练技术的可行性，通过使用自训练技术，四尔叶神经算子（FNO）仅通过物理损失的训练就取得了比同时使用数据和物理损失训练的情况下在 Burgers 和 Darcy 方程中更好的结果，此外，我们发现可以利用伪标签进行自训练，同时优化了 PINO 的性能。

物理知情的神经运算符是否能够自我改善？

Can Physics Informed Neural Operators Self Improve?

Equilibrium modeling is common in a variety of fields such as game theory and
transportation science. The inputs for these models, however, are often
difficult to estimate, while their outputs, i.e., the equilibria they are meant
to describe, are often directly observable. By combining ideas from inverse
optimization with the theory of variational inequalities, we develop an
efficient, data-driven technique for estimating the parameters of these models
from observed equilibria. We use this technique to estimate the utility
functions of players in a game from their observed actions and to estimate the
congestion function on a road network from traffic count data. A distinguishing
feature of our approach is that it supports both parametric and
\emph{nonparametric} estimation by leveraging ideas from statistical learning
(kernel methods and regularization operators). In computational experiments
involving Nash and Wardrop equilibria in a nonparametric setting, we find that
a) we effectively estimate the unknown demand or congestion function,
respectively, and b) our proposed regularization technique substantially
improves the out-of-sample performance of our estimators.

本文介绍了一种基于逆优化及变分不等式理论的数据驱动方法，用于估计在平衡模型中难以估计的参数，包括游戏中玩家的收益函数和道路网络中的拥挤函数，并使用统计学习方法进行参数化和非参数化估计。经过计算实验，结果表明该方法能够有效地估算出未知的需求或拥挤函数，并且我们提出的正则化技术大大提高了我们估算器的外样性能。