Deep learning has gained attention for solving PDEs, but the black-box nature of neural networks hinders precise enforcement of boundary conditions. To address this, we propose a boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (KAN) with radial basis functions (BEKAN). In BEKAN, we propose three distinct and combinable approaches for incorporating Dirichlet, periodic, and Neumann boundary conditions into the network. For Dirichlet problem, we use smooth and global Gaussian RBFs to construct univariate basis functions for approximating the solution and to encode boundary information at the activation level of the network. To handle periodic problems, we employ a periodic layer constructed from a set of sinusoidal functions to enforce the boundary conditions exactly. For a Neumann problem, we devise a least-squares formulation to guide the parameter evolution toward satisfying the Neumann condition. By virtue of the boundary-embedded RBFs, the periodic layer, and the evolutionary framework, we can perform accurate PDE simulations while rigorously enforcing boundary conditions. For demonstration, we conducted extensive numerical experiments on Dirichlet, Neumann, periodic, and mixed boundary value problems. The results indicate that BEKAN outperforms both multilayer perceptron (MLP) and B-splines KAN in terms of accuracy. In conclusion, the proposed approach enhances the capability of KANs in solving PDE problems while satisfying boundary conditions, thereby facilitating advancements in scientific computing and engineering applications.

The rapid growth of deploying machine learning (ML) models within embedded systems on a chip (SoCs) has led to transformative shifts in fields like healthcare and autonomous vehicles. One of the primary challenges for training such embedded ML models is the lack of publicly available high-quality training data. Transfer learning approaches address this challenge by utilizing the knowledge encapsulated in an existing ML model as a starting point for training a new ML model. However, existing transfer learning approaches require direct access to the existing model which is not always feasible, especially for ML models deployed on embedded SoCs. Therefore, in this paper, we introduce a novel unconventional transfer learning approach to train a new ML model by extracting and using weights from an existing ML model running on an embedded SoC without having access to the model within the SoC. Our approach captures power consumption measurements from the SoC while it is executing the ML model and translates them to an approximated weights matrix used to initialize the new ML model. This improves the learning efficiency and predictive performance of the new model, especially in scenarios with limited data available to train the model. Our novel approach can effectively increase the accuracy of the new ML model up to 3 times compared to classical training methods using the same amount of limited training data.

Evolutional deep neural networks (EDNN) solve partial differential equations (PDEs) by marching the network representation of the solution fields, using the governing equations. Use of a single network to solve coupled PDEs on large domains requires a large number of network parameters and incurs a significant computational cost. We introduce coupled EDNN (C-EDNN) to solve systems of PDEs by using independent networks for each state variable, which are only coupled through the governing equations. We also introduce distributed EDNN (D-EDNN) by spatially partitioning the global domain into several elements and assigning individual EDNNs to each element to solve the local evolution of the PDE. The networks then exchange the solution and fluxes at their interfaces, similar to flux-reconstruction methods, and ensure that the PDE dynamics are accurately preserved between neighboring elements. Together C-EDNN and D-EDNN form the general class of Multi-EDNN methods. We demonstrate these methods with aid of canonical problems including linear advection, the heat equation, and the compressible Navier-Stokes equations in Couette and Taylor-Green flows.

Encoder-decoder neural networks (EDNN) condense information most relevant to the output of the feedforward network to activation values at a bottleneck layer. We study the use of this architecture in emulation and interpretation of simulated X-ray spectroscopic data with the aim to identify key structural characteristics for the spectra, previously studied using emulator-based component analysis (ECA). We find an EDNN to outperform ECA in covered target variable variance, but also discover complications in interpreting the latent variables in physical terms. As a compromise of the benefits of these two approaches, we develop a network where the linear projection of ECA is used, thus maintaining the beneficial characteristics of vector expansion from the latent variables for their interpretation. These results underline the necessity of information recovery after its condensation and identification of decisive structural degrees for the output spectra for a justified interpretation.

We propose extrinsic and intrinsic deep neural network architectures as general frameworks for deep learning on manifolds. Specifically, extrinsic deep neural networks (eDNNs) preserve geometric features on manifolds by utilizing an equivariant embedding from the manifold to its image in the Euclidean space. Moreover, intrinsic deep neural networks (iDNNs) incorporate the underlying intrinsic geometry of manifolds via exponential and log maps with respect to a Riemannian structure. Consequently, we prove that the empirical risk of the empirical risk minimizers (ERM) of eDNNs and iDNNs converge in optimal rates. Overall, The eDNNs framework is simple and easy to compute, while the iDNNs framework is accurate and fast converging. To demonstrate the utilities of our framework, various simulation studies, and real data analyses are presented with eDNNs and iDNNs.

We demonstrate the use of an extensive deep neural network to localize instances of objects in images. The EDNN is naturally able to accurately perform multi-class counting using only ground truth count values as labels. Without providing any conceptual information, object annotations, or pixel segmentation information, the neural network is able to formulate its own conceptual representation of the items in the image. Using images labelled with only the counts of the objects present, the structure of the extensive deep neural network can be exploited to perform localization of the objects within the visual field. We demonstrate that a trained EDNN can be used to count objects in images much larger than those on which it was trained. In order to demonstrate our technique, we introduce seven new datasets: five progressively harder MNIST digit-counting data sets, and two data sets of 3d-rendered rubber ducks in various situations. On most of these datasets, the EDNN achieves greater than 99% test set accuracy in counting objects.