In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.
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機器學習（Machine Learning）是一個研究計算學習方法的國際論壇。該雜志發表文章，報告廣泛的學習方法應用于各種學習問題的實質性結果。該雜志的特色論文描述研究的問題和方法，應用研究和研究方法的問題。有關學習問題或方法的論文通過實證研究、理論分析或與心理現象的比較提供了堅實的支持。應用論文展示了如何應用學習方法來解決重要的應用問題。研究方法論文改進了機器學習的研究方法。所有的論文都以其他研究人員可以驗證或復制的方式描述了支持證據。論文還詳細說明了學習的組成部分，并討論了關于知識表示和性能任務的假設。
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The semantic segmentation of pelvic organs via MRI has important clinical significance. Recently, deep learning-enabled semantic segmentation has facilitated the three-dimensional geometric reconstruction of pelvic floor organs, providing clinicians with accurate and intuitive diagnostic results. However, the task of labeling pelvic floor MRI segmentation, typically performed by clinicians, is labor-intensive and costly, leading to a scarcity of labels. Insufficient segmentation labels limit the precise segmentation and reconstruction of pelvic floor organs. To address these issues, we propose a semi-supervised framework for pelvic organ segmentation. The implementation of this framework comprises two stages. In the first stage, it performs self-supervised pre-training using image restoration tasks. Subsequently, fine-tuning of the self-supervised model is performed, using labeled data to train the segmentation model. In the second stage, the self-supervised segmentation model is used to generate pseudo labels for unlabeled data. Ultimately, both labeled and unlabeled data are utilized in semi-supervised training. Upon evaluation, our method significantly enhances the performance in the semantic segmentation and geometric reconstruction of pelvic organs, Dice coefficient can increase by 2.65% averagely. Especially for organs that are difficult to segment, such as the uterus, the accuracy of semantic segmentation can be improved by up to 3.70%.
As a surrogate for computationally intensive meso-scale simulation of woven composites, this article presents Recurrent Neural Network (RNN) models. Leveraging the power of transfer learning, the initialization challenges and sparse data issues inherent in cyclic shear strain loads are addressed in the RNN models. A mean-field model generates a comprehensive data set representing elasto-plastic behavior. In simulations, arbitrary six-dimensional strain histories are used to predict stresses under random walking as the source task and cyclic loading conditions as the target task. Incorporating sub-scale properties enhances RNN versatility. In order to achieve accurate predictions, the model uses a grid search method to tune network architecture and hyper-parameter configurations. The results of this study demonstrate that transfer learning can be used to effectively adapt the RNN to varying strain conditions, which establishes its potential as a useful tool for modeling path-dependent responses in woven composites.
In this paper we introduce a novel semantics, called defense semantics, for Dung's abstract argumentation frameworks in terms of a notion of (partial) defence, which is a triple encoding that one argument is (partially) defended by another argument via attacking the attacker of the first argument. In terms of defense semantics, we show that defenses related to self-attacked arguments and arguments in 3-cycles are unsatifiable under any situation and therefore can be removed without affecting the defense semantics of an AF. Then, we introduce a new notion of defense equivalence of AFs, and compare defense equivalence with standard equivalence and strong equivalence, respectively. Finally, by exploiting defense semantics, we define two kinds of reasons for accepting arguments, i.e., direct reasons and root reasons, and a notion of root equivalence of AFs that can be used in argumentation summarization.
In this paper, we study the Radial Basis Function (RBF) approximation to differential operators on smooth tensor fields defined on closed Riemannian submanifolds of Euclidean space, identified by randomly sampled point cloud data. {The formulation in this paper leverages a fundamental fact that the covariant derivative on a submanifold is the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. To differentiate a test function (or vector field) on the submanifold with respect to the Euclidean metric, the RBF interpolation is applied to extend the function (or vector field) in the ambient Euclidean space. When the manifolds are unknown, we develop an improved second-order local SVD technique for estimating local tangent spaces on the manifold. When the classical pointwise non-symmetric RBF formulation is used to solve Laplacian eigenvalue problems, we found that while accurate estimation of the leading spectra can be obtained with large enough data, such an approximation often produces irrelevant complex-valued spectra (or pollution) as the true spectra are real-valued and positive. To avoid such an issue,} we introduce a symmetric RBF discrete approximation of the Laplacians induced by a weak formulation on appropriate Hilbert spaces. Unlike the non-symmetric approximation, this formulation guarantees non-negative real-valued spectra and the orthogonality of the eigenvectors. Theoretically, we establish the convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner Laplacian {for the symmetric formulation} in the limit of large data with convergence rates. Numerically, we provide supporting examples for approximations of the Laplace-Beltrami operator and various vector Laplacians, including the Bochner, Hodge, and Lichnerowicz Laplacians.
This article proposes a highly accurate and conservative method for hyperbolic systems using the finite volume approach. This innovative scheme constructs the intermediate states at the interfaces of the control volumes using the method of characteristics. The approach is simple to implement, generates entropic solutions, and avoids solving Riemann problems. A diffusion control parameter is introduced to increase the accuracy of the scheme. Numerical examples are presented for the Euler equation for an ideal gas. The results demonstrate the method's ability to capture contact discontinuity and shock wave profiles with high accuracy and low cost as well as its robustness.
We propose Diffusion Model Variational Inference (DMVI), a novel method for automated approximate inference in probabilistic programming languages (PPLs). DMVI utilizes diffusion models as variational approximations to the true posterior distribution by deriving a novel bound to the marginal likelihood objective used in Bayesian modelling. DMVI is easy to implement, allows hassle-free inference in PPLs without the drawbacks of, e.g., variational inference using normalizing flows, and does not make any constraints on the underlying neural network model. We evaluate DMVI on a set of common Bayesian models and show that its posterior inferences are in general more accurate than those of contemporary methods used in PPLs while having a similar computational cost and requiring less manual tuning.
Most state-of-the-art machine learning techniques revolve around the optimisation of loss functions. Defining appropriate loss functions is therefore critical to successfully solving problems in this field. We present a survey of the most commonly used loss functions for a wide range of different applications, divided into classification, regression, ranking, sample generation and energy based modelling. Overall, we introduce 33 different loss functions and we organise them into an intuitive taxonomy. Each loss function is given a theoretical backing and we describe where it is best used. This survey aims to provide a reference of the most essential loss functions for both beginner and advanced machine learning practitioners.
We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.
In this paper we develop a novel neural network model for predicting implied volatility surface. Prior financial domain knowledge is taken into account. A new activation function that incorporates volatility smile is proposed, which is used for the hidden nodes that process the underlying asset price. In addition, financial conditions, such as the absence of arbitrage, the boundaries and the asymptotic slope, are embedded into the loss function. This is one of the very first studies which discuss a methodological framework that incorporates prior financial domain knowledge into neural network architecture design and model training. The proposed model outperforms the benchmarked models with the option data on the S&P 500 index over 20 years. More importantly, the domain knowledge is satisfied empirically, showing the model is consistent with the existing financial theories and conditions related to implied volatility surface.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
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