Implicit-depth neural networks have grown as powerful alternatives to
traditional networks in various applications in recent years. However, these
models often lack guarantees of existence and uniqueness, raising stability,
performance, and reproducibility issues. In this paper, we present a new
analysis of the existence and uniqueness of fixed points for implicit-depth
neural networks based on the concept of subhomogeneous operators and the
nonlinear Perron-Frobenius theory. Compared to previous similar analyses, our
theory allows for weaker assumptions on the parameter matrices, thus yielding a
more flexible framework for well-defined implicit networks. We illustrate the
performance of the resulting subhomogeneous networks on feed-forward,
convolutional, and graph neural network examples.

本文基于亚向同调算子和非线性 Perron-Frobenius 理论，对隐式深度神经网络的不动点的存在性和唯一性进行了新的分析。相较于先前的类似分析，我们的理论对参数矩阵的假设更弱，从而为隐式网络提供了更灵活的框架。我们通过前馈、卷积和图神经网络示例展示了所得到的亚向同调网络的性能。

次齐深度均衡模型

Subhomogeneous Deep Equilibrium Models

A theory of existence and uniqueness is developed for general stochastic
differential mean field games with common noise. The concepts of strong and
weak solutions are introduced in analogy with the theory of stochastic
differential equations, and existence of weak solutions for mean field games is
shown to hold under very general assumptions. Examples and counter-examples are
provided to enlighten the underpinnings of the existence theory. Finally, an
analog of the famous result of Yamada and Watanabe is derived, and it is used
to prove existence and uniqueness of a strong solution under additional
assumptions.

本文研究具有普遍意义的随机微分均场博弈理论的存在性和唯一性，引入强解和弱解的概念并使用弱解证明了均场博弈存在性的广泛条件，提供了案例和反案例以阐明本文存在理论的基础，最后派生出 Yamada 和 Watanabe 著名结果的类比，并用于证明附加条件下强解的存在性和唯一性。

具有共同噪声的均场博弈

Mean field games with common noise

In this article, we generalize the Wasserstein distance to measures with
different masses. We study the properties of such distance. In particular, we
show that it metrizes weak convergence for tight sequences.
We use this generalized Wasserstein distance to study a transport equation
with source, in which both the vector field and the source depend on the
measure itself. We prove existence and uniqueness of the solution to the Cauchy
problem when the vector field and the source are Lipschitzian with respect to
the generalized Wasserstein distance.

用广义的 Wasserstein 距离研究了源项的输运方程，证明了解的唯一性和存在性。

广义瓦瑟斯坦距离及其在带源输运方程中的应用

Generalized Wasserstein distance and its application to transport  equations with source

This paper revisits the concept of rough paths of inhomogeneous degree of
smoothness (geometric \Pi-rough paths in our terminology) sketched by Lyons
("Differential equations driven by rough signals", Revista Mathematica Iber.
Vol 14, Nr. 2,215-310, 1998). Although geometric \Pi-rough paths can be treated
as p-rough paths for a sufficiently large p and the theory of integration of
Lip-\gamma one-forms (\gamma>p-1) along geometric p-rough paths applies, we
prove the existence of integrals of one-forms under weaker conditions.
Moreover, we consider differential equations driven by geometric \Pi-rough
paths and give sufficient conditions for existence and uniqueness of solution.

本文重新思考了由 Lyons 构思的不同光滑度的不均匀粗糙路径（在我们的术语中称为几何 π- 粗糙路径），并证明了在更弱的条件下可能存在一阶形式的积分，同时考虑由几何 π- 粗糙路径驱动的微分方程并提供了解的充分条件。