Continuous dynamical systems, characterized by differential equations, are
ubiquitously used to model several important problems: plasma dynamics, flow
through porous media, weather forecasting, and epidemic dynamics. Recently, a
wide range of data-driven models has been used successfully to model these
systems. However, in contrast to established fields like computer vision,
limited studies are available analyzing the strengths and potential
applications of different classes of these models that could steer
decision-making in scientific machine learning. Here, we introduce CodBench, an
exhaustive benchmarking suite comprising 11 state-of-the-art data-driven models
for solving differential equations. Specifically, we comprehensively evaluate 4
distinct categories of models, viz., feed forward neural networks, deep
operator regression models, frequency-based neural operators, and transformer
architectures against 8 widely applicable benchmark datasets encompassing
challenges from fluid and solid mechanics. We conduct extensive experiments,
assessing the operators' capabilities in learning, zero-shot super-resolution,
data efficiency, robustness to noise, and computational efficiency.
Interestingly, our findings highlight that current operators struggle with the
newer mechanics datasets, motivating the need for more robust neural operators.
All the datasets and codes will be shared in an easy-to-use fashion for the
scientific community. We hope this resource will be an impetus for accelerated
progress and exploration in modeling dynamical systems.

连续动力系统建模方法研究了数据驱动模型在解决微分方程方面的表现，通过 CodBench 进行了四类不同模型的全面评估并揭示了神经算子对新型力学数据集的困难，为建模动力系统的加速进展和探索提供了动力学研究资源。

CoDBench：对连续动力系统的数据驱动模型进行的关键评估

CoDBench: A Critical Evaluation of Data-driven Models for Continuous  Dynamical Systems

Modeling spatiotemporal dynamical systems is a fundamental challenge in
machine learning. Transformer models have been very successful in NLP and
computer vision where they provide interpretable representations of data.
However, a limitation of transformers in modeling continuous dynamical systems
is that they are fundamentally discrete time and space models and thus have no
guarantees regarding continuous sampling. To address this challenge, we present
the Continuous Spatiotemporal Transformer (CST), a new transformer architecture
that is designed for the modeling of continuous systems. This new framework
guarantees a continuous and smooth output via optimization in Sobolev space. We
benchmark CST against traditional transformers as well as other spatiotemporal
dynamics modeling methods and achieve superior performance in a number of tasks
on synthetic and real systems, including learning brain dynamics from calcium
imaging data.

这篇论文介绍了一种新的基于 Continuous Spatiotemporal Transformer (CST) 的连续动态系统建模方法，能够通过 Sobolev 空间的优化保证连续且平滑输出，实现了对实验数据中脑电波等连续系统的建模，性能在各项任务中均优于传统的基于离散时间和空间的 Transformer 模型及其他动态系统建模方法。

连续时空转换器

Continuous Spatiotemporal Transformers

We present Barrier Certificate-based Simplex (BC-Simplex), a new, provably
correct design for runtime assurance of continuous dynamical systems.
BC-Simplex is centered around the Simplex Control Architecture, which consists
of a high-performance advanced controller which is not guaranteed to maintain
safety of the plant, a verified-safe baseline controller, and a decision module
that switches control of the plant between the two controllers to ensure safety
without sacrificing performance. In BC-Simplex, Barrier certificates are used
to prove that the baseline controller ensures safety. Furthermore, BC-Simplex
features a new automated method for deriving, from the barrier certificate, the
conditions for switching between the controllers. Our method is based on the
Taylor expansion of the barrier certificate and yields computationally
inexpensive switching conditions. We consider a significant application of
BC-Simplex to a microgrid featuring an advanced controller in the form of a
neural network trained using reinforcement learning. The microgrid is modeled
in RTDS, an industry-standard high-fidelity, real-time power systems simulator.
Our results demonstrate that BC-Simplex can automatically derive switching
conditions for complex systems, the switching conditions are not overly
conservative, and BC-Simplex ensures safety even in the presence of adversarial
attacks on the neural controller.

提出基于屏障证明的 Simplex（BC-Simplex）算法，旨在同时保证连续动态系统的安全和性能；算法采用屏障证明来证明基线控制器确保安全，基于屏障证明的泰勒展开式推导出了控制器切换的条件，并针对一个包含神经网络控制器的微电网进行了实验验证，结果表明该算法能够自动导出控制器切换条件，且还可以在神经控制器被攻击的情况下仍然保证系统安全。