A.1 Shower shape variables452 We extend the list of shower shape variables described in Sec. Marginals of each point feature by considering the set all the points from all454 the point clouds together.455 Layer Energy Ei. Denotes the total energy deposited in layer i of the shower. Total energy across all layers of the shower. The layer lateral widths can be interpreted as the spread around the center of energy in the lateral462 plane in respective dimensions.

Physics-informed neural networks approach the approximation of differential equations by directly incorporating their structure and given conditions in a loss function. This enables conditions like, e.g., invariants to be easily added during the modelling phase. In addition, the approach can be considered as mesh free and can be utilised to compute solutions on arbitrary grids after the training phase. Therefore, physics-informed neural networks are emerging as a promising alternative to solving differential equations with methods from numerical mathematics. However, their performance highly depends on a large variety of factors. In this paper, we systematically investigate and evaluate a core component of the approach, namely the training point distribution. We test two ordinary and two partial differential equations with five strategies for training data generation and shallow network architectures, with one and two hidden layers. In addition to common distributions, we introduce sine-based training points, which are motivated by the construction of Chebyshev nodes. The results are challenged by using certain parameter combinations like, e.g., random and fixed-seed weight initialisation for reproducibility. The results show the impact of the training point distributions on the solution accuracy and we find evidence that they are connected to the characteristics of the differential equation.

Physics-informed neural networks (PINNs) is becoming a popular alternative method for solving partial differential equations (PDEs). However, they require dedicated manual modifications to the hyperparameters of the network, the sampling methods and loss function weights for different PDEs, which reduces the efficiency of the solvers. In this paper, we pro- pose a general multi-stage framework, i.e. BO-SA-PINNs to alleviate this issue. In the first stage, Bayesian optimization (BO) is used to select hyperparameters for the training process, and based on the results of the pre-training, the network architecture, learning rate, sampling points distribution and loss function weights suitable for the PDEs are automatically determined. The proposed hyperparameters search space based on experimental results can enhance the efficiency of BO in identifying optimal hyperparameters. After selecting the appropriate hyperparameters, we incorporate a global self-adaptive (SA) mechanism the second stage. Using the pre-trained model and loss information in the second-stage training, the exponential moving average (EMA) method is employed to optimize the loss function weights, and residual-based adaptive refinement with distribution (RAR-D) is used to optimize the sampling points distribution. In the third stage, L-BFGS is used for stable training. In addition, we introduce a new activation function that enables BO-SA-PINNs to achieve higher accuracy. In numerical experiments, we conduct comparative and ablation experiments to verify the performance of the model on Helmholtz, Maxwell, Burgers and high-dimensional Poisson equations. The comparative experiment results show that our model can achieve higher accuracy and fewer iterations in test cases, and the ablation experiments demonstrate the positive impact of every improvement.

Surface reconstruction from point clouds is a fundamental challenge in computer graphics and medical imaging. In this paper, we explore the application of advanced neural network architectures for the accurate and efficient reconstruction of surfaces from data points. We introduce a novel variant of the Highway network (Hw) called Square-Highway (SqrHw) within the context of multilayer perceptrons and investigate its performance alongside plain neural networks and a simplified Hw in various numerical examples. These examples include the reconstruction of simple and complex surfaces, such as spheres, human hands, and intricate models like the Stanford Bunny. We analyze the impact of factors such as the number of hidden layers, interior and exterior points, and data distribution on surface reconstruction quality. Our results show that the proposed SqrHw architecture outperforms other neural network configurations, achieving faster convergence and higher-quality surface reconstructions. Additionally, we demonstrate the SqrHw's ability to predict surfaces over missing data, a valuable feature for challenging applications like medical imaging. Furthermore, our study delves into further details, demonstrating that the proposed method based on highway networks yields more stable weight norms and backpropagation gradients compared to the Plain Network architecture. This research not only advances the field of computer graphics but also holds utility for other purposes such as function interpolation and physics-informed neural networks, which integrate multilayer perceptrons into their algorithms.

We introduce the Lennard-Jones layer (LJL) for the equalization of the density of 2D and 3D point clouds through systematically rearranging points without destroying their overall structure (distribution normalization). LJL simulates a dissipative process of repulsive and weakly attractive interactions between individual points by considering the nearest neighbor of each point at a given moment in time. This pushes the particles into a potential valley, reaching a well-defined stable configuration that approximates an equidistant sampling after the stabilization process. We apply LJLs to redistribute randomly generated point clouds into a randomized uniform distribution. Moreover, LJLs are embedded in the generation process of point cloud networks by adding them at later stages of the inference process. The improvements in 3D point cloud generation utilizing LJLs are evaluated qualitatively and quantitatively. Finally, we apply LJLs to improve the point distribution of a score-based 3D point cloud denoising network. In general, we demonstrate that LJLs are effective for distribution normalization which can be applied at negligible cost without retraining the given neural network.

The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. However, as FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. We address this issue by proposing a novel method that leverages batch matrix multiplications to efficiently construct Vandermonde-structured matrices and compute forward and inverse transforms, on arbitrarily distributed points. An efficient implementation of such structured matrix methods is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows one to extend neural operators to very general point distributions with significant gains in training speed over baselines, while retaining or improving accuracy.

Distribution-to-Distribution (D2D) point cloud registration algorithms are fast, interpretable, and perform well in unstructured environments. Unfortunately, existing strategies for predicting solution error for these methods are overly optimistic, particularly in regions containing large or extended physical objects. In this paper we introduce the Iterative Closest Ellipsoidal Transform (ICET), a novel 3D LIDAR scan-matching algorithm that re-envisions NDT in order to provide robust accuracy prediction from first principles. Like NDT, ICET subdivides a LIDAR scan into voxels in order to analyze complex scenes by considering many smaller local point distributions, however, ICET assesses the voxel distribution to distinguish random noise from deterministic structure. ICET then uses a weighted least-squares formulation to incorporate this noise/structure distinction into computing a localization solution and predicting the solution-error covariance. In order to demonstrate the reasonableness of our accuracy predictions, we verify 3D ICET in three LIDAR tests involving real-world automotive data, high-fidelity simulated trajectories, and simulated corner-case scenes. For each test, ICET consistently performs scan matching with sub-centimeter accuracy. This level of accuracy, combined with the fact that the algorithm is fully interpretable, make it well suited for safety-critical transportation applications. Code is available at https://github.com/mcdermatt/ICET

Traditionally, supervised machine learning algorithms are trained based on past data under the assumption that the past data is representative of the future. However, machine learning algorithms are increasingly being used in settings where the output of the algorithm changes the environment and hence, the data distribution. Indeed, online labor markets (Anagnostopoulos et al., 2018; Horton, 2010), predictive policing (Lum and Isaac, 2016), on-street parking (Dowling et al., 2020; Pierce and Shoup, 2018), and vehicle sharing markets (Banerjee et al., 2015) are all examples of real-world settings in which the algorithm's decisions change the underlying data distribution due to the fact that the algorithm interacts with strategic users. To address this problem, the machine learning community introduced the problem of performative prediction which models the data distribution as being decision-dependent thereby accounting for feedback induced distributional shift (Brown et al., 2020; Drusvyatskiy and Xiao, 2020; Mendler-Dünner et al., 2020; Miller et al., 2021; Perdomo et al., 2020). With the exception of (Brown et al., 2020), this work has focused on static environments. In many of the aforementioned application domains, however, the underlying data distribution also may have memory or even be changing dynamically in time. When a decision-making mechanism is announced it may take time to see the full effect of the decision as the environment and strategic data sources respond given their prior history or interactions. For example, many municipalities announce quarterly a new quasi-static set of prices for on-street parking. In this scenario, the institution may adjust parking rates for certain blocks in order to to achieve a desired occupancy range to reduce cruising phenomena and increase business district vitality (Dowling et al., 2017; Fiez et al., 2018; Pierce and Shoup, 2013; Shoup, 2006).

An important issue in clustering concerns the avoidance of false positives while searching for clusters. This work addressed this problem considering agglomerative methods, namely single, average, median, complete, centroid and Ward's approaches applied to unimodal and bimodal datasets obeying uniform, gaussian, exponential and power-law distributions. A model of clusters was also adopted, involving a higher density nucleus surrounded by a transition, followed by outliers. This paved the way to defining an objective means for identifying the clusters from dendrograms. The adopted model also allowed the relevance of the clusters to be quantified in terms of the height of their subtrees. The obtained results include the verification that many methods detect two clusters in unimodal data. The single-linkage method was found to be more resilient to false positives. Also, several methods detected clusters not corresponding directly to the nucleus. The possibility of identifying the type of distribution was also investigated.