Equivariant deep learning architectures exploit symmetries in learning
problems to improve the sample efficiency of neural-network-based models and
their ability to generalise. However, when modelling real-world data, learning
problems are often not exactly equivariant, but only approximately. For
example, when estimating the global temperature field from weather station
observations, local topographical features like mountains break translation
equivariance. In these scenarios, it is desirable to construct architectures
that can flexibly depart from exact equivariance in a data-driven way. In this
paper, we develop a general approach to achieving this using existing
equivariant architectures. Our approach is agnostic to both the choice of
symmetry group and model architecture, making it widely applicable. We consider
the use of approximately equivariant architectures in neural processes (NPs), a
popular family of meta-learning models. We demonstrate the effectiveness of our
approach on a number of synthetic and real-world regression experiments,
demonstrating that approximately equivariant NP models can outperform both
their non-equivariant and strictly equivariant counterparts.

通过使用已有的等变体系结构，我们在构建神经过程中使用近似等变性的方法，证明了该模型在合成和真实回归实验中的有效性。

近似等变神经过程

Approximately Equivariant Neural Processes

We present Roto-Translation Equivariant Spherical Deconvolution (RT-ESD), an
$E(3)\times SO(3)$ equivariant framework for sparse deconvolution of volumes
where each voxel contains a spherical signal. Such 6D data naturally arises in
diffusion MRI (dMRI), a medical imaging modality widely used to measure
microstructure and structural connectivity. As each dMRI voxel is typically a
mixture of various overlapping structures, there is a need for blind
deconvolution to recover crossing anatomical structures such as white matter
tracts. Existing dMRI work takes either an iterative or deep learning approach
to sparse spherical deconvolution, yet it typically does not account for
relationships between neighboring measurements. This work constructs
equivariant deep learning layers which respect to symmetries of spatial
rotations, reflections, and translations, alongside the symmetries of voxelwise
spherical rotations. As a result, RT-ESD improves on previous work across
several tasks including fiber recovery on the DiSCo dataset,
deconvolution-derived partial volume estimation on real-world \textit{in vivo}
human brain dMRI, and improved downstream reconstruction of fiber tractograms
on the Tractometer dataset. Our implementation is available at
this https URL

本文介绍一种 $E (3)	imes SO (3)$ 等变的分别对球形信号进行稀疏反卷积的框架，并使用等变深度学习层来改进交叉解剖结构如白质束的测量，从而提高 dMRI 的成像质量。

$E (3) \times SO (3)$- 等变网络在扩散磁共振球面去卷积中的应用

$E(3) \times SO(3)$-Equivariant Networks for Spherical Deconvolution in  Diffusion MRI

A simultaneously accurate and computationally efficient parametrization of
the energy and atomic forces of molecules and materials is a long-standing goal
in the natural sciences. In pursuit of this goal, neural message passing has
lead to a paradigm shift by describing many-body correlations of atoms through
iteratively passing messages along an atomistic graph. This propagation of
information, however, makes parallel computation difficult and limits the
length scales that can be studied. Strictly local descriptor-based methods, on
the other hand, can scale to large systems but do not currently match the high
accuracy observed with message passing approaches. This work introduces
Allegro, a strictly local equivariant deep learning interatomic potential that
simultaneously exhibits excellent accuracy and scalability of parallel
computation. Allegro learns many-body functions of atomic coordinates using a
series of tensor products of learned equivariant representations, but without
relying on message passing. Allegro obtains improvements over state-of-the-art
methods on the QM9 and revised MD-17 data sets. A single tensor product layer
is shown to outperform existing deep message passing neural networks and
transformers on the QM9 benchmark. Furthermore, Allegro displays remarkable
generalization to out-of-distribution data. Molecular dynamics simulations
based on Allegro recover structural and kinetic properties of an amorphous
phosphate electrolyte in excellent agreement with first principles
calculations. Finally, we demonstrate the parallel scaling of Allegro with a
dynamics simulation of 100 million atoms.

本文介绍了一种基于 equivariant deep learning 的新算法，Allegro，该算法使用了学习到的等变性表示的张量积序列，实现了对于分子和材料的能量和原子力的高精度预测，并能进行大规模的并行计算和分子动力学仿真。

学习局部等变表示在大规模微观动力学中的应用

Learning Local Equivariant Representations for Large-Scale Atomistic  Dynamics

Recent work in equivariant deep learning bears strong similarities to
physics. Fields over a base space are fundamental entities in both subjects, as
are equivariant maps between these fields. In deep learning, however, these
maps are usually defined by convolutions with a kernel, whereas they are
partial differential operators (PDOs) in physics. Developing the theory of
equivariant PDOs in the context of deep learning could bring these subjects
even closer together and lead to a stronger flow of ideas. In this work, we
derive a $G$-steerability constraint that completely characterizes when a PDO
between feature vector fields is equivariant, for arbitrary symmetry groups
$G$. We then fully solve this constraint for several important groups. We use
our solutions as equivariant drop-in replacements for convolutional layers and
benchmark them in that role. Finally, we develop a framework for equivariant
maps based on Schwartz distributions that unifies classical convolutions and
differential operators and gives insight about the relation between the two.

研究在相等变深度学习和物理领域中共享一致的数学概念，并通过极化微分算子和基于调和分析的方法提出了一个新的框架，使我们能够更好地将基于显式对称性应变的捕捉无论是在深度学习还是物理学中都可以更精确地进行表达。