Many functions characterising physical systems are additively separable. This
is the case, for instance, of mechanical Hamiltonian functions in physics,
population growth equations in biology, and consumer preference and utility
functions in economics. We consider the scenario in which a surrogate of a
function is to be tested for additive separability. The detection that the
surrogate is additively separable can be leveraged to improve further learning.
Hence, it is beneficial to have the ability to test for such separability in
surrogates. The mathematical approach is to test if the mixed partial
derivative of the surrogate is zero; or empirically, lower than a threshold. We
present and comparatively and empirically evaluate the eight methods to compute
the mixed partial derivative of a surrogate function.

测试虚拟函数的可加分离性对于进一步的学习是有益的，可以通过数学方法或实证方法来计算虚拟函数的混合偏导数。

基于物理知识的机器学习的加性可分性测试的比较评估

A Comparative Evaluation of Additive Separability Tests for  Physics-Informed Machine Learning

The modelling of dynamical systems from discrete observations is a challenge
faced by modern scientific and engineering data systems. Hamiltonian systems
are one such fundamental and ubiquitous class of dynamical systems. Hamiltonian
neural networks are state-of-the-art models that unsupervised-ly regress the
Hamiltonian of a dynamical system from discrete observations of its vector
field under the learning bias of Hamilton's equations. Yet Hamiltonian dynamics
are often complicated, especially in higher dimensions where the state space of
the Hamiltonian system is large relative to the number of samples. A recently
discovered remedy to alleviate the complexity between state variables in the
state space is to leverage the additive separability of the Hamiltonian system
and embed that additive separability into the Hamiltonian neural network.
Following the nomenclature of physics-informed machine learning, we propose
three separable Hamiltonian neural networks. These models embed additive
separability within Hamiltonian neural networks. The first model uses additive
separability to quadratically scale the amount of data for training Hamiltonian
neural networks. The second model embeds additive separability within the loss
function of the Hamiltonian neural network. The third model embeds additive
separability through the architecture of the Hamiltonian neural network using
conjoined multilayer perceptions. We empirically compare the three models
against state-of-the-art Hamiltonian neural networks, and demonstrate that the
separable Hamiltonian neural networks, which alleviate complexity between the
state variables, are more effective at regressing the Hamiltonian and its
vector field.

建议了三种具备可加性可分性的 Hamilton 神经网络模型，通过减少状态变量之间的复杂性，更有效地回归 Hamilton 的向量场。

分离型哈密尔顿神经网络

Separable Hamiltonian Neural Networks

Reallocating resources to get mutually beneficial outcomes is a fundamental
problem in various multi-agent settings. While finding an arbitrary Pareto
optimal allocation is generally easy, checking whether a particular allocation
is Pareto optimal can be much more difficult. This problem is equivalent to
checking that the allocated objects cannot be reallocated in such a way that at
least one agent prefers her new share to his old one, and no agent prefers her
old share to her new one. We consider the problem for two related types of
preference relations over sets of objects. In the first part of the paper we
focus on the setting in which agents express additive cardinal utilities over
objects. We present computational hardness results as well as polynomial-time
algorithms for testing Pareto optimality under different restrictions such as
two utility values or lexicographic utilities. In the second part of the paper
we assume that agents express only their (ordinal) preferences over single
objects, and that their preferences are additively separable. In this setting,
we present characterizations and polynomial-time algorithms for possible and
necessary Pareto optimality.

在多智能体问题中，通过新的资源配置方式实现互惠互利是一个根本性问题。本文探讨了两种不同偏好关系的情况下，如何以不同的限制条件测试 Pareto 最优性。第一部分着重于代理人对物体的加法计数效用的表达方法，提出了计算复杂性结果和多项式时间算法；第二部分假设代理人只表达其对单个对象的偏好，并将其偏好降解为相加的形式，提出了可能和必要 Pareto 最优性的特征和多项式时间算法。