The past decade has seen increasing interest in applying Deep Learning (DL) to Computational Science and Engineering (CSE). Driven by impressive results in applications such as computer vision, Uncertainty Quantification (UQ), genetics, simulations and image processing, DL is increasingly supplanting classical algorithms, and seems poised to revolutionize scientific computing. However, DL is not yet well-understood from the standpoint of numerical analysis. Little is known about the efficiency and reliability of DL from the perspectives of stability, robustness, accuracy, and sample complexity. In particular, approximating solutions to parametric PDEs is an objective of UQ for CSE. Training data for such problems is often scarce and corrupted by errors. Moreover, the target function is a possibly infinite-dimensional smooth function taking values in the PDE solution space, generally an infinite-dimensional Banach space. This paper provides arguments for Deep Neural Network (DNN) approximation of such functions, with both known and unknown parametric dependence, that overcome the curse of dimensionality. We establish practical existence theorems that describe classes of DNNs with dimension-independent architecture size and training procedures based on minimizing the (regularized) $\ell^2$-loss which achieve near-optimal algebraic rates of convergence. These results involve key extensions of compressed sensing for Banach-valued recovery and polynomial emulation with DNNs. When approximating solutions of parametric PDEs, our results account for all sources of error, i.e., sampling, optimization, approximation and physical discretization, and allow for training high-fidelity DNN approximations from coarse-grained sample data. Our theoretical results fall into the category of non-intrusive methods, providing a theoretical alternative to classical methods for high-dimensional approximation.
相關內容
We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with a stability result we have estimates for the subsequent numerical analysis. With classical layer adapted meshes we present a numerical method, that achieves supercloseness and optimal convergence orders in the associated energy norm. We also consider coarser meshes in view of the weak layers. Some numerical examples conclude the paper and support the theory.
We present a new approach to stabilizing high-order Runge-Kutta discontinuous Galerkin (RKDG) schemes using weighted essentially non-oscillatory (WENO) reconstructions in the context of hyperbolic conservation laws. In contrast to RKDG schemes that overwrite finite element solutions with WENO reconstructions, our approach employs the reconstruction-based smoothness sensor presented by Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) to control the amount of added numerical dissipation. Incorporating a dissipation-based WENO stabilization term into a discontinuous Galerkin (DG) discretization, the proposed methodology achieves high-order accuracy while effectively capturing discontinuities in the solution. As such, our approach offers an attractive alternative to WENO-based slope limiters for DG schemes. The reconstruction procedure that we use performs Hermite interpolation on stencils composed of a mesh cell and its neighboring cells. The amount of numerical dissipation is determined by the relative differences between the partial derivatives of reconstructed candidate polynomials and those of the underlying finite element approximation. The employed smoothness sensor takes all derivatives into account to properly assess the local smoothness of a high-order DG solution. Numerical experiments demonstrate the ability of our scheme to capture discontinuities sharply. Optimal convergence rates are obtained for all polynomial degrees.
Pneumonia is the leading infectious cause of infant death in the world. When identified early, it is possible to alter the prognosis of the patient, one could use imaging exams to help in the diagnostic confirmation. Performing and interpreting the exams as soon as possible is vital for a good treatment, with the most common exam for this pathology being chest X-ray. The objective of this study was to develop a software that identify the presence or absence of pneumonia in chest radiographs. The software was developed as a computational model based on machine learning using transfer learning technique. For the training process, images were collected from a database available online with children's chest X-rays images taken at a hospital in China. After training, the model was then exposed to new images, achieving relevant results on identifying such pathology, reaching 98% sensitivity and 97.3% specificity for the sample used for testing. It can be concluded that it is possible to develop a software that identifies pneumonia in chest X-ray images.
We consider a sharp interface formulation for the multi-phase Mullins-Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins-Sekerka flow, demonstrate the capabilities of the introduced method.
An unconventional approach is applied to solve the one-dimensional Burgers' equation. It is based on spline polynomial interpolations and Hopf-Cole transformation. Taylor expansion is used to approximate the exponential term in the transformation, then the analytical solution of the simplified equation is discretized to form a numerical scheme, involving various special functions. The derived scheme is explicit and adaptable for parallel computing. However, some types of boundary condition cannot be specified straightforwardly. Three test cases were employed to examine its accuracy, stability, and parallel scalability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation performs equally well, managing to reduce the $\ell_{1}$, $\ell_{2}$ and $\ell_{\infty}$ error norms down to the order of $10^{-4}$. Due to the transformation, their stability condition $\nu \Delta t/\Delta x^2 > 0.02$ includes the viscosity/diffusion coefficient $\nu$. From the condition, the schemes can run at a large time step size $\Delta t$ even when grid spacing $\Delta x$ is small. These characteristics suggest that the method is more suitable for operational use than for research purposes.
The Boolean Satisfiability problem (SAT) is the most prototypical NP-complete problem and of great practical relevance. One important class of solvers for this problem are stochastic local search (SLS) algorithms that iteratively and randomly update a candidate assignment. Recent breakthrough results in theoretical computer science have established sufficient conditions under which SLS solvers are guaranteed to efficiently solve a SAT instance, provided they have access to suitable "oracles" that provide samples from an instance-specific distribution, exploiting an instance's local structure. Motivated by these results and the well established ability of neural networks to learn common structure in large datasets, in this work, we train oracles using Graph Neural Networks and evaluate them on two SLS solvers on random SAT instances of varying difficulty. We find that access to GNN-based oracles significantly boosts the performance of both solvers, allowing them, on average, to solve 17% more difficult instances (as measured by the ratio between clauses and variables), and to do so in 35% fewer steps, with improvements in the median number of steps of up to a factor of 8. As such, this work bridges formal results from theoretical computer science and practically motivated research on deep learning for constraint satisfaction problems and establishes the promise of purpose-trained SAT solvers with performance guarantees.
In the last ongoing years, there has been a significant ascending on the field of Natural Language Processing (NLP) for performing multiple tasks including English Language Teaching (ELT). An effective strategy to favor the learning process uses interactive devices to engage learners in their self-learning process. In this work, we present a working prototype of a humanoid robotic system to assist English language self-learners through text generation using Long Short Term Memory (LSTM) Neural Networks. The learners interact with the system using a Graphic User Interface that generates text according to the English level of the user. The experimentation was conducted using English learners and the results were measured accordingly to International English Language Testing System (IELTS) rubric. Preliminary results show an increment in the Grammatical Range of learners who interacted with the system.
Grid sentence is commonly used for studying the Lombard effect and Normal-to-Lombard conversion. However, it's unclear if Normal-to-Lombard models trained on grid sentences are sufficient for improving natural speech intelligibility in real-world applications. This paper presents the recording of a parallel Lombard corpus (called Lombard Chinese TIMIT, LCT) extracting natural sentences from Chinese TIMIT. Then We compare natural and grid sentences in terms of Lombard effect and Normal-to-Lombard conversion using LCT and Enhanced MAndarin Lombard Grid corpus (EMALG). Through a parametric analysis of the Lombard effect, We find that as the noise level increases, both natural sentences and grid sentences exhibit similar changes in parameters, but in terms of the increase of the alpha ratio, grid sentences show a greater increase. Following a subjective intelligibility assessment across genders and Signal-to-Noise Ratios, the StarGAN model trained on EMALG consistently outperforms the model trained on LCT in terms of improving intelligibility. This superior performance may be attributed to EMALG's larger alpha ratio increase from normal to Lombard speech.
This work proposes to measure the scope of a patent claim as the reciprocal of the self-information contained in this claim. Grounded in information theory, this approach is based on the assumption that a rare concept is more informative than a usual concept, inasmuch as it is more surprising. The self-information is calculated from the probability of occurrence of that claim, where the probability is calculated in accordance with a language model. Five language models are considered, ranging from the simplest models (each word or character is drawn from a uniform distribution) to intermediate models (using average word or character frequencies), to a large language model (GPT2). Interestingly, the simplest language models reduce the scope measure to the reciprocal of the word or character count, a metric already used in previous works. Application is made to nine series of patent claims directed to distinct inventions, where the claims in each series have a gradually decreasing scope. The performance of the language models is then assessed with respect to several ad hoc tests. The more sophisticated the model, the better the results. The GPT2 model outperforms models based on word and character frequencies, which are themselves ahead of models based on word and character counts.
In recent years, a certain type of problems have become of interest where one wants to query a trained classifier. Specifically, one wants to find the closest instance to a given input instance such that the classifier's predicted label is changed in a desired way. Examples of these ``inverse classification'' problems are counterfactual explanations, adversarial examples and model inversion. All of them are fundamentally optimization problems over the input instance vector involving a fixed classifier, and it is of interest to achieve a fast solution for interactive or real-time applications. We focus on solving this problem efficiently for two of the most widely used classifiers: logistic regression and softmax classifiers. Owing to special properties of these models, we show that the optimization can be solved in closed form for logistic regression, and iteratively but extremely fast for the softmax classifier. This allows us to solve either case exactly (to nearly machine precision) in a runtime of milliseconds to around a second even for very high-dimensional instances and many classes.
小貼士
登錄享
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191