We propose to enhance the training of physics-informed neural networks (PINNs). To this aim, we introduce nonlinear additive and multiplicative preconditioning strategies for the widely used L-BFGS optimizer. The nonlinear preconditioners are constructed by utilizing the Schwarz domain-decomposition framework, where the parameters of the network are decomposed in a layer-wise manner. Through a series of numerical experiments, we demonstrate that both, additive and multiplicative preconditioners significantly improve the convergence of the standard L-BFGS optimizer, while providing more accurate solutions of the underlying partial differential equations. Moreover, the additive preconditioner is inherently parallel, thus giving rise to a novel approach to model parallelism.
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The exponential growth in scientific publications poses a severe challenge for human researchers. It forces attention to more narrow sub-fields, which makes it challenging to discover new impactful research ideas and collaborations outside one's own field. While there are ways to predict a scientific paper's future citation counts, they need the research to be finished and the paper written, usually assessing impact long after the idea was conceived. Here we show how to predict the impact of onsets of ideas that have never been published by researchers. For that, we developed a large evolving knowledge graph built from more than 21 million scientific papers. It combines a semantic network created from the content of the papers and an impact network created from the historic citations of papers. Using machine learning, we can predict the dynamic of the evolving network into the future with high accuracy, and thereby the impact of new research directions. We envision that the ability to predict the impact of new ideas will be a crucial component of future artificial muses that can inspire new impactful and interesting scientific ideas.
Quantum machine learning models have shown successful generalization performance even when trained with few data. In this work, through systematic randomization experiments, we show that traditional approaches to understanding generalization fail to explain the behavior of such quantum models. Our experiments reveal that state-of-the-art quantum neural networks accurately fit random states and random labeling of training data. This ability to memorize random data defies current notions of small generalization error, problematizing approaches that build on complexity measures such as the VC dimension, the Rademacher complexity, and all their uniform relatives. We complement our empirical results with a theoretical construction showing that quantum neural networks can fit arbitrary labels to quantum states, hinting at their memorization ability. Our results do not preclude the possibility of good generalization with few training data but rather rule out any possible guarantees based only on the properties of the model family. These findings expose a fundamental challenge in the conventional understanding of generalization in quantum machine learning and highlight the need for a paradigm shift in the study of quantum models for machine learning tasks.
The use of machine-learning techniques has grown in numerous research areas. Currently, it is also widely used in statistics, including the official statistics for data collection (e.g. satellite imagery, web scraping and text mining, data cleaning, integration and imputation) but also for data analysis. However, the usage of these methods in survey sampling including small area estimation is still very limited. Therefore, we propose a predictor supported by these algorithms which can be used to predict any population or subpopulation characteristics based on cross-sectional and longitudinal data. Machine learning methods have already been shown to be very powerful in identifying and modelling complex and nonlinear relationships between the variables, which means that they have very good properties in case of strong departures from the classic assumptions. Therefore, we analyse the performance of our proposal under a different set-up, in our opinion of greater importance in real-life surveys. We study only small departures from the assumed model, to show that our proposal is a good alternative in this case as well, even in comparison with optimal methods under the model. What is more, we propose the method of the accuracy estimation of machine learning predictors, giving the possibility of the accuracy comparison with classic methods, where the accuracy is measured as in survey sampling practice. The solution of this problem is indicated in the literature as one of the key issues in integration of these approaches. The simulation studies are based on a real, longitudinal dataset, freely available from the Polish Local Data Bank, where the prediction problem of subpopulation characteristics in the last period, with "borrowing strength" from other subpopulations and time periods, is considered.
Context: Experiment replications play a central role in the scientific method. Although software engineering experimentation has matured a great deal, the number of experiment replications is still relatively small. Software engineering experiments are composed of complex concepts, procedures and artefacts. Laboratory packages are a means of transfer-ring knowledge among researchers to facilitate experiment replications. Objective: This paper investigates the experiment replication process to find out what information is needed to successfully replicate an experiment. Our objective is to propose the content and structure of laboratory packages for software engineering experiments. Method: We evaluated seven replications of three different families of experiments. Each replication had a different experimenter who was, at the time, unfamiliar with the experi-ment. During the first iterations of the study, we identified experimental incidents and then proposed a laboratory package structure that addressed these incidents, including docu-ment usability improvements. We used the later iterations to validate and generalize the laboratory package structure for use in all software engineering experiments. We aimed to solve a specific problem, while at the same time looking at how to contribute to the body of knowledge on laboratory packages. Results: We generated a laboratory package for three different experiments. These packages eased the replication of the respective experiments. The evaluation that we conducted shows that the laboratory package proposal is acceptable and reduces the effort currently required to replicate experiments in software engineering. Conclusion: We think that the content and structure that we propose for laboratory pack-ages can be useful for other software engineering experiments.
In this study, we explore data assimilation for the Stochastic Camassa-Holm equation through the application of the particle filtering framework. Specifically, our approach integrates adaptive tempering, jittering, and nudging techniques to construct an advanced particle filtering system. All filtering processes are executed utilizing ensemble parallelism. We conduct extensive numerical experiments across various scenarios of the Stochastic Camassa-Holm model with transport noise and viscosity to examine the impact of different filtering procedures on the performance of the data assimilation process. Our analysis focuses on how observational data and the data assimilation step influence the accuracy and uncertainty of the obtained results.
In this paper we formally define the hierarchical clustering network problem (HCNP) as the problem to find a good hierarchical partition of a network. This new problem focuses on the dynamic process of the clustering rather than on the final picture of the clustering process. To address it, we introduce a new ierarchical clustering algorithm in networks, based on a new shortest path betweenness measure. To calculate it, the communication between each pair of nodes is weighed by he importance of the nodes that establish this communication. The weights or importance associated to each pair of nodes are calculated as the Shapley value of a game, named as the linear modularity game. This new measure, (the node-game shortest path betweenness measure), is used to obtain a hierarchical partition of the network by eliminating the link with the highest value. To evaluate the performance of our algorithm, we introduce several criteria that allow us to compare different dendrograms of a network from two point of view: modularity and homogeneity. Finally, we propose a faster algorithm based on a simplification of the node-game shortest path betweenness measure, whose order is quadratic on sparse networks. This fast version is competitive from a computational point of view with other hierarchical fast algorithms, and, in general, it provides better results.
Physics-informed neural networks (PINNs) are neural networks (NNs) that directly encode model equations, like Partial Differential Equations (PDEs), in the network itself. While most of the PINN algorithms in the literature minimize the local residual of the governing equations, there are energy-based approaches that take a different path by minimizing the variational energy of the model. We show that in the case of the steady thermal equation weakly coupled to magnetic equation, the energy-based approach displays multiple advantages compared to the standard residual-based PINN: it is more computationally efficient, it requires a lower order of derivatives to compute, and it involves less hyperparameters. The analyzed benchmark problem is the optimal design of an inductor for the controlled heating of a graphite plate. The optimized device is designed involving a multi-physics problem: a time-harmonic magnetic problem and a steady thermal problem. For the former, a deep neural network solving the direct problem is supervisedly trained on Finite Element Analysis (FEA) data. In turn, the solution of the latter relies on a hypernetwork that takes as input the inductor geometry parameters and outputs the model weights of an energy-based PINN (or ePINN). Eventually, the ePINN predicts the temperature field within the graphite plate.
In this paper, we combine the Smolyak technique for multi-dimensional interpolation with the Filon-Clenshaw-Curtis (FCC) rule for one-dimensional oscillatory integration, to obtain a new Filon-Clenshaw-Curtis-Smolyak (FCCS) rule for oscillatory integrals with linear phase over the $d-$dimensional cube $[-1,1]^d$. By combining stability and convergence estimates for the FCC rule with error estimates for the Smolyak interpolation operator, we obtain an error estimate for the FCCS rule, consisting of the product of a Smolyak-type error estimate multiplied by a term that decreases with $\mathcal{O}(k^{-\tilde{d}})$, where $k$ is the wavenumber and $\tilde{d}$ is the number of oscillatory dimensions. If all dimensions are oscillatory, a higher negative power of $k$ appears in the estimate. As an application, we consider the forward problem of uncertainty quantification (UQ) for a one-space-dimensional Helmholtz problem with wavenumber $k$ and a random heterogeneous refractive index, depending in an affine way on $d$ i.i.d. uniform random variables. After applying a classical hybrid numerical-asymptotic approximation, expectations of functionals of the solution of this problem can be formulated as a sum of oscillatory integrals over $[-1,1]^d$, which we compute using the FCCS rule. We give numerical results for the FCCS rule and the UQ algorithm showing that accuracy improves when both $k$ and the order of the rule increase. We also give results for dimension-adaptive sparse grid FCCS quadrature showing its efficiency as dimension increases.
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.
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