Recent work has described the presence of the embedding gap in neural network
verification. On one side of the gap is a high-level specification about the
network's behaviour, written by a domain expert in terms of the interpretable
problem space. On the other side are a logically-equivalent set of
satisfiability queries, expressed in the uninterpretable embedding space in a
form suitable for neural network solvers. In this paper we describe an
algorithm for compiling the former to the latter. We explore and overcome
complications that arise from targeting neural network solvers as opposed to
standard SMT solvers.

最近的研究描述了神经网络验证中的嵌入间隙存在。在间隙的一侧是高级规范，由领域专家根据可解释的问题空间编写。在另一侧是逻辑上等价的一组可满足性查询，以适合神经网络求解器的不可解释的嵌入空间形式表达。本文描述了一种将前者编译为后者的算法。我们探索并克服了针对神经网络求解器而不是标准 SMT 求解器引发的复杂问题。

高效编译表达性问题空间规范为神经网络求解器

Efficient compilation of expressive problem space specifications to  neural network solvers

Neural network solvers represent an innovative and promising approach for
tackling time-fractional partial differential equations by utilizing deep
learning techniques. L1 interpolation approximation serves as the standard
method for addressing time-fractional derivatives within neural network
solvers. However, we have discovered that neural network solvers based on L1
interpolation approximation are unable to fully exploit the benefits of neural
networks, and the accuracy of these models is constrained to interpolation
errors. In this paper, we present the high-precision Hermite Neural Solver
(HNS) for solving time-fractional partial differential equations. Specifically,
we first construct a high-order explicit approximation scheme for fractional
derivatives using Hermite interpolation techniques, and rigorously analyze its
approximation accuracy. Afterward, taking into account the infinitely
differentiable properties of deep neural networks, we integrate the high-order
Hermite interpolation explicit approximation scheme with deep neural networks
to propose the HNS. The experimental results show that HNS achieves higher
accuracy than methods based on the L1 scheme for both forward and inverse
problems, as well as in high-dimensional scenarios. This indicates that HNS has
significantly improved accuracy and flexibility compared to existing L1-based
methods, and has overcome the limitations of explicit finite difference
approximation methods that are often constrained to function value
interpolation. As a result, the HNS is not a simple combination of numerical
computing methods and neural networks, but rather achieves a complementary and
mutually reinforcing advantages of both approaches. The data and code can be
found at https://github.com/hsbhc/HNS.

基于高阶 Hermite 插值和深度神经网络的高精度 Hermite Neural Solver（HNS）用于解决时间分数阶偏微分方程，取得较 L1 插值近似方法更高的准确性和灵活性。

HNS: 解决时间分数阶偏微分方程的高效 Hermite 神经求解器

HNS: An Efficient Hermite Neural Solver for Solving Time-Fractional  Partial Differential Equations

In this work we explore the combination of metaheuristics and learned neural
network solvers for combinatorial optimization. We do this in the context of
the transit network design problem, a uniquely challenging combinatorial
optimization problem with real-world importance. We train a neural network
policy to perform single-shot planning of individual transit routes, and then
incorporate it as one of several sub-heuristics in a modified Bee Colony
Optimization (BCO) metaheuristic algorithm. Our experimental results
demonstrate that this hybrid algorithm outperforms the learned policy alone by
up to 20% and the original BCO algorithm by up to 53% on realistic problem
instances. We perform a set of ablations to study the impact of each component
of the modified algorithm.

本文介绍了将元启发式算法和神经网络求解器相结合应用于组合优化中，以运输网络设计问题为例进行了探索，通过将单次规划的神经网络策略训练和纳入到修改后的蜜蜂群优化算法中，实验结果表明，这种混合算法在现实问题实例上的性能比单独的学习策略提高了多达 20％，比原始蜜蜂群优化算法提高了多达 53％，并对修改后的算法的每个组件进行了一系列削减实验以研究其影响。