Machine learning-based modeling of physical systems has experienced increased interest in recent years. Despite some impressive progress, there is still a lack of benchmarks for Scientific ML that are easy to use but still challenging and representative of a wide range of problems. We introduce PDEBench, a benchmark suite of time-dependent simulation tasks based on Partial Differential Equations (PDEs). PDEBench comprises both code and data to benchmark the performance of novel machine learning models against both classical numerical simulations and machine learning baselines. Our proposed set of benchmark problems contribute the following unique features: (1) A much wider range of PDEs compared to existing benchmarks, ranging from relatively common examples to more realistic and difficult problems; (2) much larger ready-to-use datasets compared to prior work, comprising multiple simulation runs across a larger number of initial and boundary conditions and PDE parameters; (3) more extensible source codes with user-friendly APIs for data generation and baseline results with popular machine learning models (FNO, U-Net, PINN, Gradient-Based Inverse Method). PDEBench allows researchers to extend the benchmark freely for their own purposes using a standardized API and to compare the performance of new models to existing baseline methods. We also propose new evaluation metrics with the aim to provide a more holistic understanding of learning methods in the context of Scientific ML. With those metrics we identify tasks which are challenging for recent ML methods and propose these tasks as future challenges for the community. The code is available at https://github.com/pdebench/PDEBench.

机器学习算法必须能够有效地应对大量数据集。因此，他们必须在任何现代系统上进行良好的扩展，并能够利用独立于供应商的加速器的计算能力。在监督学习领域，支持向量机（SVM）被广泛使用。但是，即使是现代化和优化的实现，例如LIBSVM或ThunderSVM对于尖端硬件的大型非平凡的密集数据集也不能很好地扩展：大多数SVM实现基于顺序最小优化，这是一种优化的固有顺序算法。因此，它们不适合高度平行的GPU。此外，我们不知道支持不同供应商的CPU和GPU的性能便携式实现。我们已经开发了PLSSVM库来解决这两个问题。首先，我们将SVM的配方作为最小二乘问题。然后训练SVM沸腾以求解已知高度平行算法的线性方程系统。其次，我们提供了一个独立但高效的实现：PLSSVM使用不同的可互换后端 - openmp，cuda，opencl，sycl-支持来自多个GPU的NVIDIA，AMD或INTEL等各种供应商的现代硬件。 PLSSVM可以用作LIBSVM的倒入替换。与LIBSVM相比，与ThunderSVM相比，我们观察到高达10的CPU和GPU的加速度。我们的实施量表在多核CPU上缩放，并在多达256个CPU线程和多个GPU上平行加速为74.7，在四个GPU上的并行加速为3.71。代码，实用程序脚本和文档都可以在GitHub上获得：https：//github.com/sc-sgs/plssvm。

科学机器学习（Sciml）的出现在思路科学领域开辟了一个新的领域，通过在基于物理和数据建模的界面的界面中开发方法。为此，近年来介绍了物理知识的神经网络（Pinns），通过在所谓的焊点上纳入物理知识来应对培训数据的稀缺。在这项工作中，我们研究了Pinns关于用于强制基于物理惩罚术语的配偶数量的预测性能。我们表明Pinns可能会失败，学习通过定义来满足物理惩罚术语的琐碎解决方案。我们制定了一种替代的采样方法和新的惩罚术语，使我们能够在具有竞争性结果的数据稀缺设置中纠正Pinns中的核心问题，同时减少最多80 \％的基准问题所需的搭配数量。

神经网络可用作PDE模型的代理。它们可以通过惩罚潜在方程或在训练期间损失函数中的物理性质保护来进行物理意识。电流方法允许另外尊重来自培训过程中的数值模拟或实验的数据。然而，该数据经常昂贵，因此只能用于复杂模型。在这项工作中，我们调查了物理感知模型如何富有计算方式，而是来自其他代理模型的数据，如减少阶模型（ROM）。为了避免相信过于低保的代理解决方案，我们开发一种对不精确数据中的错误敏感的方法。作为概念证明，我们考虑一维波浪方程，并表明，当纳入来自ROM的不精确数据时，训练精度增加了两个数量级。

数据驱动学习方法与经典仿真之间的接口造成了一个有趣的字段，提供了多种新应用。在这项工作中，我们建立了物理知识的神经网络（Pinns）的概念，并在浅水方程（SWE）模型中采用它们。这些模型在建模和模拟自由表面流程中起重要作用，例如洪波传播或海啸波。彼此比较Pinn残差的不同配方，并评估多种优化以加速收敛速率。我们用不同的1-D和2-D实验测试这些并最终证明关于具有不同沐浴浴的SWE场景，该方法能够与直接数值模拟相比，具有8.9美元的总相对$ L_2 $误差的直接数值模拟。e-3 $。

近年来，由于增加了计算能力，允许在合理的时间框架中培训大型集合的培训，所应用的集合学习的使用已经显着增加。许多应用，例如恶意软件检测，面部识别或财务决策，使用有限的学习算法，并以比任何其他单独的学习算法获得更好的预测性能的方式聚合它们。在半导体器件（PSV）的硅后验证领域中，通常提供数据集，其包括各种装置，例如，例如不同的制造线的芯片。在PSV中，任务是近似于多学习算法的数据的基础功能，每个数据在设备特定的子集上训练，而不是提高整个数据集上任意分类器的性能。此外，期望是，未知数量的子集描述了显示非常不同特征的函数。相应的集合成员称为异常值，可以严重影响近似值。我们的方法旨在找到对异常值强大的合适近似，并且代表了适用于尽可能多的类型的方式最佳或最坏的情况。使用“软最大”或“软MIN”功能代替最大或最小操作员。培训神经网络（NN）以在两阶段过程中学习此“软功能”。首先，我们选择代表最佳或最坏情况的集合成员的子集。其次，我们组合这些成员并定义使用本地异常因素系数（LOF）属性的加权来增加非异常值的影响并减少异常值。加权可确保对异常值的鲁棒性，并确保近似适用于大多数类型。

越来越多的现代芯片复杂性使设计验证更加困难。现有方法不再能够应对硅后验证中稳健性能调整等任务的复杂性。因此，我们提出了一种基于学习优化和加强学习的新方法，以便以高效且稳健的方式解决复杂和混合式调整任务。

Increasingly taking place in online spaces, modern political conversations are typically perceived to be unproductively affirming -- siloed in so called ``echo chambers'' of exclusively like-minded discussants. Yet, to date we lack sufficient means to measure viewpoint diversity in conversations. To this end, in this paper, we operationalize two viewpoint metrics proposed for recommender systems and adapt them to the context of social media conversations. This is the first study to apply these two metrics (Representation and Fragmentation) to real world data and to consider the implications for online conversations specifically. We apply these measures to two topics -- daylight savings time (DST), which serves as a control, and the more politically polarized topic of immigration. We find that the diversity scores for both Fragmentation and Representation are lower for immigration than for DST. Further, we find that while pro-immigrant views receive consistent pushback on the platform, anti-immigrant views largely operate within echo chambers. We observe less severe yet similar patterns for DST. Taken together, Representation and Fragmentation paint a meaningful and important new picture of viewpoint diversity.

Electricity prices in liberalized markets are determined by the supply and demand for electric power, which are in turn driven by various external influences that vary strongly in time. In perfect competition, the merit order principle describes that dispatchable power plants enter the market in the order of their marginal costs to meet the residual load, i.e. the difference of load and renewable generation. Many market models implement this principle to predict electricity prices but typically require certain assumptions and simplifications. In this article, we present an explainable machine learning model for the prices on the German day-ahead market, which substantially outperforms a benchmark model based on the merit order principle. Our model is designed for the ex-post analysis of prices and thus builds on various external features. Using Shapley Additive exPlanation (SHAP) values, we can disentangle the role of the different features and quantify their importance from empiric data. Load, wind and solar generation are most important, as expected, but wind power appears to affect prices stronger than solar power does. Fuel prices also rank highly and show nontrivial dependencies, including strong interactions with other features revealed by a SHAP interaction analysis. Large generation ramps are correlated with high prices, again with strong feature interactions, due to the limited flexibility of nuclear and lignite plants. Our results further contribute to model development by providing quantitative insights directly from data.

Data-driven modeling has become a key building block in computational science and engineering. However, data that are available in science and engineering are typically scarce, often polluted with noise and affected by measurement errors and other perturbations, which makes learning the dynamics of systems challenging. In this work, we propose to combine data-driven modeling via operator inference with the dynamic training via roll outs of neural ordinary differential equations. Operator inference with roll outs inherits interpretability, scalability, and structure preservation of traditional operator inference while leveraging the dynamic training via roll outs over multiple time steps to increase stability and robustness for learning from low-quality and noisy data. Numerical experiments with data describing shallow water waves and surface quasi-geostrophic dynamics demonstrate that operator inference with roll outs provides predictive models from training trajectories even if data are sampled sparsely in time and polluted with noise of up to 10%.