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In scientific computing, neural networks have been widely used to solve partial differential equations (PDEs). In this paper, we propose a novel RBF-assisted hybrid neural network for approximating solutions to PDEs. Inspired by the tendency of physics-informed neural networks (PINNs) to become local approximations after training, the proposed method utilizes a radial basis function (RBF) to provide the normalization and localization properties to the input data. The objective of this strategy is to assist the network in solving PDEs more effectively. During the RBF-assisted processing part, the method selects the center points and collocation points separately to effectively manage data size and computational complexity. Subsequently, the RBF processed data are put into the network for predicting the solutions to PDEs. Finally, a series of experiments are conducted to evaluate the novel method. The numerical results confirm that the proposed method can accelerate the convergence speed of the loss function and improve predictive accuracy. [ABSTRACT FROM AUTHOR]

The relationship between philosophy and science has always been complementary. Today, while science moves increasingly fast and philosophy shows some problems in catching up with it, it is not always possible to ignore such relationships, especially in some disciplines such as philosophy of mind, cognitive science, and neuroscience. However, the methodological procedures used to analyze these data are based on principles and assumptions that require a profound dialogue between philosophy and science. Following these ideas, this work aims to raise the problems that a classical connectionist theory can cause and problematize them in a cognitive framework, considering both philosophy and cognitive sciences but also the disciplines that are near to them, such as AI, computer sciences, and linguistics. For this reason, we embarked on an analysis of both the computational and theoretical problems that connectionism currently has. The second aim of this work is to advocate for collaboration between neuroscience and philosophy of mind because the promotion of deeper multidisciplinarity seems necessary in order to solve connectionism's problems. In fact, we believe that the problems that we detected can be solved by a thorough investigation at both a theoretical and an empirical level, and they do not represent an impasse but rather a starting point from which connectionism should learn and be updated while keeping its original and profoundly convincing core. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new class of micro-electromechanical oscillators. Some investigations based on Melnikov's approach are applied for identifying some chaotic possibilities. We demonstrate also some specialized modules for investigating the dynamics of these oscillators. This will be included as an integral part of a planned much more general Web-based application for scientific computing. It turns out that the theoretical apparatus for studying the circuit implementation (design, fabricating, etc.) of the considered differential model for large parameter values is extremely complex and requires a serious investigation. This is the reason to offer this model to the attention of specialists working in this scientific direction. Some open problems related to the use of existing computer algebraic systems for the study of this class of oscillators for large values of n , m and N are also posed. In general, the entire article is subordinated to this frank conversation with the readers with the sole purpose being the professional upgrading of the specialized modules provided for this purpose in subsequent licensed versions of CAS. [ABSTRACT FROM AUTHOR]

Matrix–matrix multiplication is of singular importance in linear algebra operations with a multitude of applications in scientific and engineering computing. Data structures for storing matrix elements are designed to minimize overhead information as well as to optimize the operation count. In this study, we utilize the notion of the compact diagonal storage method (CDM), which builds upon the previously developed diagonal storage—an orientation-independent uniform scheme to store the nonzero elements of a range of matrices. This study exploits both these storage schemes and presents efficient GPU-accelerated parallel implementations of matrix multiplication when the input matrices are banded and/or structured sparse. We exploit the data layouts in the diagonal storage schemes to expose a substantial amount of fine-grained parallelism and effectively utilize the GPU shared memory to improve the locality of data access for numerical calculations. Results from an extensive set of numerical experiments with the aforementioned types of matrices demonstrate orders-of-magnitude speedups compared with the sequential performance. [ABSTRACT FROM AUTHOR]

Simulating solute transport in heterogeneous porous media poses computational challenges due to the high-resolution meshing required for traditional solvers. To overcome these challenges, this study explores a mesh-free method based on deep learning to accelerate solute transport simulation. We employ Physics-informed Neural Networks (PiNN) with a periodic activation function to solve solute transport problems in both homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that rely on large training datasets, PiNNs use strong-form mathematical models to constrain the network in the training phase and simultaneously solve for multiple dependent or independent field variables, such as pressure and solute concentration fields. To demonstrate the effectiveness of using PiNNs with a periodic activation function to resolve solute transport in porous media, we construct PiNNs using two activation functions, sin and tanh, for seven case studies, including 1D and 2D scenarios. The accuracy of the PiNNs' predictions is then evaluated using absolute point error and mean square error metrics and compared to the ground truth solutions obtained analytically or numerically. Our results demonstrate that the PiNN with sin activation function, compared to tanh activation function, is up to two orders of magnitude more accurate and up to two times faster to train, especially in heterogeneous porous media. Moreover, PiNN's simultaneous predictions of pressure and concentration fields can reduce computational expenses in terms of inference time by three orders of magnitude compared to FEM simulations for two-dimensional cases. [ABSTRACT FROM AUTHOR]