许多应用程序中的数据遵循普通微分方程（ODE）的系统。本文提出了一种新型的算法和符号结构，用于高斯过程的协方差函数（GPS），其实现严格遵循具有恒定系数的线性均匀ODES系统，我们称之为lode-gps。将这种强的感应偏置引入GP，可以改善此类数据的建模。使用史密斯正常形式算法，一种符号技术，我们克服了技术状态中的两个当前限制：（1）在一组解决方案中需要某些唯一性条件的需求，通常在经典的ODE求解器及其概率求解器及其概率对应物中假定，以及（2）对可控系统的限制，通常在编码协方差函数中的微分方程时假设。我们显示了Lode-GP在许多实验中的有效性，例如通过最大化的可能性来学习物理解释的参数。

Partial differential equations (PDEs) are important tools to model physical systems, and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works like a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDE, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

计算机代数可以使用符号算法回答有关部分微分方程的各种问题。但是，在计算机代数中将数据包含在方程式中很少。因此，最近，计算机代数模型与高斯流程（机器学习中的回归模型）相结合，以描述数据下某些微分方程的行为。尽管可以在这种情况下描述多项式边界条件，但我们将这些模型扩展到分析边界条件。此外，我们描述了具有某些分析系数的Weyl代数的gr \ obner和Janet碱基的必要算法。使用这些算法，我们提供了由分析功能界定并适应观察结果的域中无差流流的示例。

物理建模对于许多现代科学和工程应用至关重要。从数据科学或机器学习的角度来看，更多的域 - 不可吻合，数据驱动的模型是普遍的，物理知识 - 通常表示为微分方程 - 很有价值，因为它与数据是互补的，并且可能有可能帮助克服问题例如数据稀疏性，噪音和不准确性。在这项工作中，我们提出了一个简单但功能强大且通用的框架 - 自动构建物理学，可以将各种微分方程集成到高斯流程（GPS）中，以增强预测准确性和不确定性量化。这些方程可以是线性或非线性，空间，时间或时空，与未知的源术语完全或不完整，等等。基于内核分化，我们在示例目标函数，方程相关的衍生物和潜在源函数之前构建了GP，这些函数全部来自多元高斯分布。采样值被馈送到两个可能性：一个以适合观测值，另一个符合方程式。我们使用美白方法来逃避采样函数值和内核参数之间的强依赖性，并开发出一种随机变分学习算法。在模拟和几个现实世界应用中，即使使用粗糙的，不完整的方程式，自动元素都显示出对香草GPS的改进。

本论文主要涉及解决深层（时间）高斯过程（DGP）回归问题的状态空间方法。更具体地，我们代表DGP作为分层组合的随机微分方程（SDES），并且我们通过使用状态空间过滤和平滑方法来解决DGP回归问题。由此产生的状态空间DGP（SS-DGP）模型生成丰富的电视等级，与建模许多不规则信号/功能兼容。此外，由于他们的马尔可道结构，通过使用贝叶斯滤波和平滑方法可以有效地解决SS-DGPS回归问题。本论文的第二次贡献是我们通过使用泰勒力矩膨胀（TME）方法来解决连续离散高斯滤波和平滑问题。这诱导了一类滤波器和SmooThers，其可以渐近地精确地预测随机微分方程（SDES）解决方案的平均值和协方差。此外，TME方法和TME过滤器和SmoOthers兼容模拟SS-DGP并解决其回归问题。最后，本文具有多种状态 - 空间（深）GPS的应用。这些应用主要包括（i）来自部分观察到的轨迹的SDES的未知漂移功能和信号的光谱 - 时间特征估计。

Machine learning models can be improved by adapting them to respect existing background knowledge. In this paper we consider multitask Gaussian processes, with background knowledge in the form of constraints that require a specific sum of the outputs to be constant. This is achieved by conditioning the prior distribution on the constraint fulfillment. The approach allows for both linear and nonlinear constraints. We demonstrate that the constraints are fulfilled with high precision and that the construction can improve the overall prediction accuracy as compared to the standard Gaussian process.

我们介绍了Hida-Mat'Ern内核的班级，这是整个固定式高斯 - 马尔可夫流程的整个空间的规范家庭协方差。它在垫子内核上延伸，通过允许灵活地构造具有振荡组件的过程。任何固定内核，包括广泛使用的平方指数和光谱混合核，要么直接在该类内，也是适当的渐近限制，展示了该类的一般性。利用其Markovian Nature，我们展示了如何仅使用内核及其衍生物来代表状态空间模型的过程。反过来，这使我们能够更有效地执行高斯工艺推论，并且侧面通常计算负担。我们还表明，除了进一步减少计算复杂性之外，我们还显示了如何利用状态空间表示的特殊属性。

We present the GPry algorithm for fast Bayesian inference of general (non-Gaussian) posteriors with a moderate number of parameters. GPry does not need any pre-training, special hardware such as GPUs, and is intended as a drop-in replacement for traditional Monte Carlo methods for Bayesian inference. Our algorithm is based on generating a Gaussian Process surrogate model of the log-posterior, aided by a Support Vector Machine classifier that excludes extreme or non-finite values. An active learning scheme allows us to reduce the number of required posterior evaluations by two orders of magnitude compared to traditional Monte Carlo inference. Our algorithm allows for parallel evaluations of the posterior at optimal locations, further reducing wall-clock times. We significantly improve performance using properties of the posterior in our active learning scheme and for the definition of the GP prior. In particular we account for the expected dynamical range of the posterior in different dimensionalities. We test our model against a number of synthetic and cosmological examples. GPry outperforms traditional Monte Carlo methods when the evaluation time of the likelihood (or the calculation of theoretical observables) is of the order of seconds; for evaluation times of over a minute it can perform inference in days that would take months using traditional methods. GPry is distributed as an open source Python package (pip install gpry) and can also be found at https://github.com/jonaselgammal/GPry.

在非参数回归中，落在欧几里德空间的限制子集中是常见的。基于典型的内核的方法，不考虑收集观察的域的内在几何学可能产生次优效果。在本文中，我们专注于在高斯过程（GP）模型的背景下解决这个问题，提出了一种新的基于Graplacian的GPS（GL-GPS），该GPS（GL-GPS），该GPS（GL-GPS）学习尊重输入域几何的协方差。随着热核的难以计算地，我们使用Prop Laplacian（GL）的有限许多特征方来近似协方差。 GL由内核构成，仅取决于输入的欧几里德坐标。因此，我们可以从关于内核的完整知识中受益，以通过NYSTR \“{o} M型扩展来将协方差结构扩展到新到达的样本。我们为GL-GP方法提供了实质性的理论支持，并说明了性能提升各种应用。

受到控制障碍功能（CBF）在解决安全性方面的成功以及数据驱动技术建模功能的兴起的启发，我们提出了一种使用高斯流程（GPS）在线合成CBF的非参数方法。 CBF等数学结构通过先验设计候选功能来实现安全性。但是，设计这样的候选功能可能具有挑战性。这种设置的一个实际示例是在需要确定安全且可导航区域的灾难恢复方案中设计CBF。在这样的示例中，安全性边界未知，不能先验设计。在我们的方法中，我们使用安全样本或观察结果来在线构建CBF，通过在这些样品上具有灵活的GP，并称我们为高斯CBF的配方。除非参数外，例如分析性障碍性和稳健的不确定性估计，GP具有有利的特性。这允许通过合并方差估计来实现具有高安全性保证的后部组件，同时还计算封闭形式中相关的部分导数以实现安全控制。此外，我们方法的合成安全函数允许根据数据任意更改相应的安全集，从而允许非Convex安全集。我们通过证明对固定但任意的安全集和避免碰撞的安全性在线构建安全集的安全控制，从而在四极管上验证了我们的方法。最后，我们将高斯CBF与常规的CBF并列，在嘈杂状态下，以突出其灵活性和对噪声的鲁棒性。实验视频可以在：https：//youtu.be/hx6uokvcigk上看到。

线性系统发生在整个工程和科学中，最著名的是差分方程。在许多情况下，系统的强迫函数尚不清楚，兴趣在于使用对系统的嘈杂观察来推断强迫以及其他未知参数。在微分方程中，强迫函数是自变量（通常是时间和空间）的未知函数，可以建模为高斯过程（GP）。在本文中，我们展示了如何使用GP内核的截断基础扩展，如何使用线性系统的伴随有效地推断成GP的功能。我们展示了如何实现截短的GP的确切共轭贝叶斯推断，在许多情况下，计算的计算大大低于使用MCMC方法所需的计算。我们证明了普通和部分微分方程系统的方法，并表明基础扩展方法与数量适中的基础向量相近。最后，我们展示了如何使用贝叶斯优化来推断非线性模型参数（例如内核长度尺度）的点估计值。

Partial differential equations (PDEs) are widely used for description of physical and engineering phenomena. Some key parameters involved in PDEs, which represents certain physical properties with important scientific interpretations, are difficult or even impossible to be measured directly. Estimation of these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations of numerical solution of PDE through algorithms such as finite element method, which can be time-consuming especially for nonlinear PDEs. In this paper, we propose a novel method for estimating unknown parameters in PDEs, called PDE-Informed Gaussian Process Inference (PIGPI). Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transfers the nonlinear PDE into an equivalent PDE system linear in all derivatives that our PIGPI can handle. PIGPI can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. The method completely bypasses the numerical solver for PDE, thus achieving drastic savings in computation time, especially for nonlinear PDEs. Moreover, the PIGPI method can give the uncertainty quantification for both the unknown parameters and the PDE solution. The proposed method is demonstrated by several application examples from different areas.

非线性动态系统的识别仍然是整个工程的重大挑战。这项工作提出了一种基于贝叶斯过滤的方法，以提取和确定系统中未知的非线性项的贡献，可以将其视为恢复力表面类型方法的替代观点。为了实现这种识别，最初将非线性恢复力的贡献作为高斯过程建模。该高斯过程将转换为状态空间模型，并与系统的线性动态组件结合使用。然后，通过推断过滤和平滑分布，可以提取系统的内部状态和非线性恢复力。在这些状态下，可以构建非线性模型。在模拟案例研究和实验基准数据集中，该方法被证明是有效的。

在科学的背景下，众所周知的格言“一张图片胜过千言万语”可能是“一个型号胜过一千个数据集”。在本手稿中，我们将Sciml软件生态系统介绍作为混合物理法律和科学模型的信息，并使用数据驱动的机器学习方法。我们描述了一个数学对象，我们表示通用微分方程（UDE），作为连接生态系统的统一框架。我们展示了各种各样的应用程序，从自动发现解决高维汉密尔顿 - Jacobi-Bellman方程的生物机制，可以通过UDE形式主义和工具进行措辞和有效地处理。我们展示了软件工具的一般性，以处理随机性，延迟和隐式约束。这使得各种SCIML应用程序变为核心训练机构的核心集，这些训练机构高度优化，稳定硬化方程，并与分布式并行性和GPU加速器兼容。

Interacting particle or agent systems that display a rich variety of swarming behaviours are ubiquitous in science and engineering. A fundamental and challenging goal is to understand the link between individual interaction rules and swarming. In this paper, we study the data-driven discovery of a second-order particle swarming model that describes the evolution of $N$ particles in $\mathbb{R}^d$ under radial interactions. We propose a learning approach that models the latent radial interaction function as Gaussian processes, which can simultaneously fulfill two inference goals: one is the nonparametric inference of {the} interaction function with pointwise uncertainty quantification, and the other one is the inference of unknown scalar parameters in the non-collective friction forces of the system. We formulate the learning problem as a statistical inverse problem and provide a detailed analysis of recoverability conditions, establishing that a coercivity condition is sufficient for recoverability. Given data collected from $M$ i.i.d trajectories with independent Gaussian observational noise, we provide a finite-sample analysis, showing that our posterior mean estimator converges in a Reproducing kernel Hilbert space norm, at an optimal rate in $M$ equal to the one in the classical 1-dimensional Kernel Ridge regression. As a byproduct, we show we can obtain a parametric learning rate in $M$ for the posterior marginal variance using $L^{\infty}$ norm, and the rate could also involve $N$ and $L$ (the number of observation time instances for each trajectory), depending on the condition number of the inverse problem. Numerical results on systems that exhibit different swarming behaviors demonstrate efficient learning of our approach from scarce noisy trajectory data.

高斯流程已成为各种安全至关重要环境的有前途的工具，因为后方差可用于直接估计模型误差并量化风险。但是，针对安全 - 关键环境的最新技术取决于核超参数是已知的，这通常不适用。为了减轻这种情况，我们在具有未知的超参数的设置中引入了强大的高斯过程统一误差界。我们的方法计算超参数空间中的一个置信区域，这使我们能够获得具有任意超参数的高斯过程模型误差的概率上限。我们不需要对超参数的任何界限，这是相关工作中常见的假设。相反，我们能够以直观的方式从数据中得出界限。我们还采用了建议的技术来为一类基于学习的控制问题提供绩效保证。实验表明，界限的性能明显优于香草和完全贝叶斯高斯工艺。

神经切线核是根据无限宽度神经网络的参数分布定义的内核函数。尽管该极限不切实际，但神经切线内核允许对神经网络进行更直接的研究，并凝视着黑匣子的面纱。最近，从理论上讲，Laplace内核和神经切线内核在$ \ Mathbb {S}}^{D-1} $中共享相同的复制核Hilbert空间，暗示了它们的等价。在这项工作中，我们分析了两个内核的实际等效性。我们首先是通过与核的准确匹配，然后通过与高斯过程的后代匹配来进行匹配。此外，我们分析了$ \ mathbb {r}^d $中的内核，并在回归任务中进行实验。

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics, and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than classical feed-forward neural networks, recurrent neural networks, and convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering, In this work, we review such methods and extend them for parametric studies as well as for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

我们开发了一个计算程序，以估计具有附加噪声的半摩托车高斯过程回归模型的协方差超参数。也就是说，提出的方法可用于有效估计相关误差的方差，以及基于最大化边际似然函数的噪声方差。我们的方法涉及适当地降低超参数空间的维度，以简化单变量的根发现问题的估计过程。此外，我们得出了边际似然函数及其衍生物的边界和渐近线，这对于缩小高参数搜索的初始范围很有用。使用数值示例，我们证明了与传统参数优化相比，提出方法的计算优势和鲁棒性。

In a fissile material, the inherent multiplicity of neutrons born through induced fissions leads to correlations in their detection statistics. The correlations between neutrons can be used to trace back some characteristics of the fissile material. This technique known as neutron noise analysis has applications in nuclear safeguards or waste identification. It provides a non-destructive examination method for an unknown fissile material. This is an example of an inverse problem where the cause is inferred from observations of the consequences. However, neutron correlation measurements are often noisy because of the stochastic nature of the underlying processes. This makes the resolution of the inverse problem more complex since the measurements are strongly dependent on the material characteristics. A minor change in the material properties can lead to very different outputs. Such an inverse problem is said to be ill-posed. For an ill-posed inverse problem the inverse uncertainty quantification is crucial. Indeed, seemingly low noise in the data can lead to strong uncertainties in the estimation of the material properties. Moreover, the analytical framework commonly used to describe neutron correlations relies on strong physical assumptions and is thus inherently biased. This paper addresses dual goals. Firstly, surrogate models are used to improve neutron correlations predictions and quantify the errors on those predictions. Then, the inverse uncertainty quantification is performed to include the impact of measurement error alongside the residual model bias.