Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical methods based on discretization are used to solve PDEs. They generally use an estimate of the unknown model parameters and, if available, physical measurements for initialization. Such solvers are often embedded into larger scientific models or analyses with a downstream application such that error quantification plays a key role. However, by entirely ignoring parameter and measurement uncertainty, classical PDE solvers may fail to produce consistent estimates of their inherent approximation error. In this work, we approach this problem in a principled fashion by interpreting solving linear PDEs as physics-informed Gaussian process (GP) regression. Our framework is based on a key generalization of a widely-applied theorem for conditioning GPs on a finite number of direct observations to observations made via an arbitrary bounded linear operator. Crucially, this probabilistic viewpoint allows to (1) quantify the inherent discretization error; (2) propagate uncertainty about the model parameters to the solution; and (3) condition on noisy measurements. Demonstrating the strength of this formulation, we prove that it strictly generalizes methods of weighted residuals, a central class of PDE solvers including collocation, finite volume, pseudospectral, and (generalized) Galerkin methods such as finite element and spectral methods. This class can thus be directly equipped with a structured error estimate and the capability to incorporate uncertain model parameters and observations. In summary, our results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models.

Gaussian processes scale prohibitively with the size of the dataset. In response, many approximation methods have been developed, which inevitably introduce approximation error. This additional source of uncertainty, due to limited computation, is entirely ignored when using the approximate posterior. Therefore in practice, GP models are often as much about the approximation method as they are about the data. Here, we develop a new class of methods that provides consistent estimation of the combined uncertainty arising from both the finite number of data observed and the finite amount of computation expended. The most common GP approximations map to an instance in this class, such as methods based on the Cholesky factorization, conjugate gradients, and inducing points. For any method in this class, we prove (i) convergence of its posterior mean in the associated RKHS, (ii) decomposability of its combined posterior covariance into mathematical and computational covariances, and (iii) that the combined variance is a tight worst-case bound for the squared error between the method's posterior mean and the latent function. Finally, we empirically demonstrate the consequences of ignoring computational uncertainty and show how implicitly modeling it improves generalization performance on benchmark datasets.

概率数值方法（PNMS）通过概率推断解决数值问题。它们已开发用于线性代数，优化，集成和微分方程模拟。PNMS自然地纳入了关于问题的先前信息，并通过有限计算资源以及随机输入来量化不确定性。在本文中，我们提出了probnum：提供最先进的概率数值求解器的Python库。Probnum通过模块化设计以及包装器，可以通过模块化设计来定制PNMS的定制组成，以供自卸使用。在线，在线，文档，开发人员指南和基准，请访问www.probnum.org。

For improving short-length codes, we demonstrate that classic decoders can also be used with real-valued, neural encoders, i.e., deep-learning based codeword sequence generators. Here, the classical decoder can be a valuable tool to gain insights into these neural codes and shed light on weaknesses. Specifically, the turbo-autoencoder is a recently developed channel coding scheme where both encoder and decoder are replaced by neural networks. We first show that the limited receptive field of convolutional neural network (CNN)-based codes enables the application of the BCJR algorithm to optimally decode them with feasible computational complexity. These maximum a posteriori (MAP) component decoders then are used to form classical (iterative) turbo decoders for parallel or serially concatenated CNN encoders, offering a close-to-maximum likelihood (ML) decoding of the learned codes. To the best of our knowledge, this is the first time that a classical decoding algorithm is applied to a non-trivial, real-valued neural code. Furthermore, as the BCJR algorithm is fully differentiable, it is possible to train, or fine-tune, the neural encoder in an end-to-end fashion.

Recently, there has been increasing interest in synthesizing data to improve downstream text-to-SQL tasks. In this paper, we first examined the existing synthesized datasets and discovered that state-of-the-art text-to-SQL algorithms did not further improve on popular benchmarks when trained with augmented synthetic data. We observed two shortcomings: illogical synthetic SQL queries from independent column sampling and arbitrary table joins. To address these issues, we propose a novel synthesis framework that incorporates key relationships from schema, imposes strong typing, and conducts schema-distance-weighted column sampling. We also adopt an intermediate representation (IR) for the SQL-to-text task to further improve the quality of the generated natural language questions. When existing powerful semantic parsers are pre-finetuned on our high-quality synthesized data, our experiments show that these models have significant accuracy boosts on popular benchmarks, including new state-of-the-art performance on Spider.

In this paper a global reactive motion planning framework for robotic manipulators in complex dynamic environments is presented. In particular, the circular field predictions (CFP) planner from Becker et al. (2021) is extended to ensure obstacle avoidance of the whole structure of a robotic manipulator. Towards this end, a motion planning framework is developed that leverages global information about promising avoidance directions from arbitrary configuration space motion planners, resulting in improved global trajectories while reactively avoiding dynamic obstacles and decreasing the required computational power. The resulting motion planning framework is tested in multiple simulations with complex and dynamic obstacles and demonstrates great potential compared to existing motion planning approaches.

Whole slide images (WSI) are microscopy images of stained tissue slides routinely prepared for diagnosis and treatment selection in medical practice. WSI are very large (gigapixel size) and complex (made of up to millions of cells). The current state-of-the-art (SoTA) approach to classify WSI subdivides them into tiles, encodes them by pre-trained networks and applies Multiple Instance Learning (MIL) to train for specific downstream tasks. However, annotated datasets are often small, typically a few hundred to a few thousand WSI, which may cause overfitting and underperforming models. Conversely, the number of unannotated WSI is ever increasing, with datasets of tens of thousands (soon to be millions) of images available. While it has been previously proposed to use these unannotated data to identify suitable tile representations by self-supervised learning (SSL), downstream classification tasks still require full supervision because parts of the MIL architecture is not trained during tile level SSL pre-training. Here, we propose a strategy of slide level SSL to leverage the large number of WSI without annotations to infer powerful slide representations. Applying our method to The Cancer-Genome Atlas, one of the most widely used data resources in cancer research (16 TB image data), we are able to downsize the dataset to 23 MB without any loss in predictive power: we show that a linear classifier trained on top of these embeddings maintains or improves previous SoTA performances on various benchmark WSI classification tasks. Finally, we observe that training a classifier on these representations with tiny datasets (e.g. 50 slides) improved performances over SoTA by an average of +6.3 AUC points over all downstream tasks.

We present G-MSM (Graph-based Multi-Shape Matching), a novel unsupervised learning approach for non-rigid shape correspondence. Rather than treating a collection of input poses as an unordered set of samples, we explicitly model the underlying shape data manifold. To this end, we propose an adaptive multi-shape matching architecture that constructs an affinity graph on a given set of training shapes in a self-supervised manner. The key idea is to combine putative, pairwise correspondences by propagating maps along shortest paths in the underlying shape graph. During training, we enforce cycle-consistency between such optimal paths and the pairwise matches which enables our model to learn topology-aware shape priors. We explore different classes of shape graphs and recover specific settings, like template-based matching (star graph) or learnable ranking/sorting (TSP graph), as special cases in our framework. Finally, we demonstrate state-of-the-art performance on several recent shape correspondence benchmarks, including real-world 3D scan meshes with topological noise and challenging inter-class pairs.

In this work we present a fast occupancy map building approach based on the VDB datastructure. Existing log-odds based occupancy mapping systems are often not able to keep up with the high point densities and framerates of modern sensors. Therefore, we suggest a highly optimized approach based on a modern datastructure coming from a computer graphic background. A multithreaded insertion scheme allows occupancy map building at unprecedented speed. Multiple optimizations allow for a customizable tradeoff between runtime and map quality. We first demonstrate the effectiveness of the approach quantitatively on a set of ablation studies and typical benchmark sets, before we practically demonstrate the system using a legged robot and a UAV.

使用深度学习技术，可以在MRI图像中自动检测到旁那鼻鼻窦系统中的异常，并可以根据其体积，形状和其他参数（例如局部对比度）进行进一步分析和分类。但是，由于培训数据有限，传统的监督学习方法通​​常无法概括。现有的旁那间异常分类中的深度学习方法最多可诊断出一种异常。在我们的工作中，我们考虑三个异常。具体而言，我们采用3D CNN来分离上颌鼻窦体积，而没有异常的鼻窦体积，并具有异常。为了从一个小标记的数据集中学习强大的表示形式，我们提出了一种新颖的学习范式，结合了对比损失和跨内向损失。特别是，我们使用有监督的对比损失，鼓励有或没有异常的上颌窦量的嵌入来形成两个不同的簇，而跨层损失则鼓励3D CNN保持其歧视能力。我们报告说，两种损失的优化是有利的，而不是仅通过一次损失而优化。我们还发现我们的培训策略会提高标签效率。使用我们的方法，3D CNN分类器的AUROC为0.85，而用横向渗透损失优化的3D CNN分类器可实现0.66的AUROC。