In this article, we focus on the opacity issue of sub-symbolic machine learning predictors by promoting two complementary activities—symbolic knowledge extraction (SKE) and symbolic knowledge ...injection (SKI)—from and into sub-symbolic predictors. We consider as symbolic any language being intelligible and interpretable for both humans and computers. Accordingly, we propose general meta-models for both SKE and SKI, along with two taxonomies for the classification of SKE and SKI methods. By adopting an explainable artificial intelligence (XAI) perspective, we highlight how such methods can be exploited to mitigate the aforementioned opacity issue. Our taxonomies are attained by surveying and classifying existing methods from the literature, following a systematic approach, and by generalising the results of previous surveys targeting specific sub-topics of either SKE or SKI alone. More precisely, we analyse 132 methods for SKE and 117 methods for SKI, and we categorise them according to their purpose, operation, expected input/output data and predictor types. For each method, we also indicate the presence/lack of runnable software implementations. Our work may be of interest for data scientists aiming at selecting the most adequate SKE/SKI method for their needs, and may also work as suggestions for researchers interested in filling the gaps of the current state-of-the-art as well as for developers willing to implement SKE/SKI-based technologies.
Performance limitations for implanted antennas, taking radiation efficiency as the metric, are presented. The performance limitations use a convex optimization procedure with the current density ...inside the implant acting as its degree of freedom. The knowledge of the limitations provides useful information in design procedure and physical insight. Ohmic losses in the antenna and surrounding tissue are both considered and quantitatively compared. The interaction of all parts of the system is taken into account in a full-wave manner via the hybrid computation method. The optimization framework is thoroughly tested on a realistic implanted antenna design that is treated both experimentally and as a model in a commercial electromagnetic solver. Good agreement is reported. To demonstrate the feasibility of developed performance limitations, they are compared to the performance of a loop and a dipole antenna showing the importance of various loss mechanisms during the design process. The trade-off between tissue loss and antenna ohmic loss indicates critical points at which the optimal solution drastically changes and the chosen topology for a specific design should be changed.
Computing the intersection between two parametric surfaces (SSI) is one of the most fundamental problems in geometric and solid modeling. Maintaining the SSI topology is critical to its computation ...robustness. We propose a topology-driven hybrid symbolic-numeric framework to approximate rational parametric surface-surface intersection (SSI) based on a concept of interval algebraic topology analysis (IATA), which configures within a 4D interval box the SSI topology. We map the SSI topology to an algebraic system's solutions within the framework, classify and enumerate all topological cases as a mixture of four fundamental cases (or their specific sub-cases). Various complicated topological situations are covered, such as cusp points or curves, tangent points (isolated or not) or curves, tiny loops, self-intersections, or their mixtures. The theoretical formulation is also implemented numerically using advanced real solution isolation techniques, and computed within a topology-driven framework which maximally utilizes the advantages of the topology maintenance of algebraic analysis, the robustness of iterative subdivision, and the efficiency of forward marching. The approach demonstrates improved robustness under benchmark topological cases when compared with available open-source and commercial solutions, including IRIT, SISL, and Parasolid.
Automatic cubatures approximate integrals to user-specified error tolerances. For high-dimensional problems, it is difficult to adaptively change the sampling pattern, but one can automatically ...determine the sample size,
n
, given a reasonable, fixed sampling pattern. We take this approach here using a Bayesian perspective. We postulate that the integrand is an instance of a Gaussian stochastic process parameterized by a constant mean and a covariance kernel defined by a scale parameter times a parameterized function specifying how the integrand values at two different points in the domain are related. These hyperparameters are inferred or integrated out using integrand values via one of three techniques: empirical Bayes, full Bayes, or generalized cross-validation. The sample size,
n
, is increased until the half-width of the credible interval for the Bayesian posterior mean is no greater than the error tolerance. The process outlined above typically requires a computational cost of
O
(
N
opt
n
3
)
, where
N
opt
is the number of optimization steps required to identify the hyperparameters. Our innovation is to pair low discrepancy nodes with matching covariance kernels to lower the computational cost to
O
(
N
opt
n
log
n
)
. This approach is demonstrated explicitly with rank-1 lattice sequences and shift-invariant kernels. Our algorithm is implemented in the Guaranteed Automatic Integration Library (GAIL).
Load flow (LF) is an extensively used tool in planning and operation of power systems. Formulation of LF problem can be assimilated as a set of autonomous ordinary differential equations, therefore, ...many numeric methods can be used to solve this problem. However, LF methods often need to compute one or more Jacobian matrix inversions in each iteration. Owing to this fact, these methods might not be computationally efficient. In this study, the authors propose combined Runge–Kutta Broyden's LF (RK4B) method in order to reduce the required Jacobian matrix inversion to only one in the first iteration. In this proposed method, Broyden's approach is employed in fourth-order Runge–Kutta method. In addition, two modifications of the proposed method are presented to reduce the number of iterations and improve the computational performance. The proposed method and the two modifications are validated using several well- and ill-conditioned cases. Results show that the combined approach has better computational performance than the classical multistage numeric methods, besides it preserves the robustness features of fourth-order Runge–Kutta method.
Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all ...roots simultaneously. In this form, the process yields better circuit complexity in the case when the number of roots r is small but the multiplicities are exponentially large. Our method sets up a linear system in r unknowns and iteratively builds the roots as formal power series. For an algebraic circuit \( f(x_1,\ldots ,x_n) \) of size s, we prove that each factor has size at most a polynomial ins and the degree of the squarefree part of f. Consequently, if \( f_1 \) is a \( 2^{\Omega (n)} \) -hard polynomial, then any nonzero multiple \( \prod _{i} f_i^{e_i} \) is equally hard for arbitrary positive \( e_i \) ’s, assuming that \( \sum _i\deg (f_i) \) is at most \( 2^{O(n)} \) .It is an old open question whether the class of poly(n) size formulas (respectively, algebraic branching programs) is closed under factoring. We show that given a polynomial f of degree \( n^{O(1)} \) and formula (respectively, algebraic branching program) size \( n^{O(\log n)} \) , we can find a similar-size formula (respectively, algebraic branching program) factor in randomized poly( \( n^{\log n} \) ) time. Consequently, if the determinant requires an \( n^{\Omega (\log n)} \) size formula, then the same can be said about any of its nonzero multiples.In all of our proofs, we exploit the following property of multivariate polynomial factorization. Under a random linear transformation \( \tau \) , the polynomial \( f(\tau \overline{x}) \) completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. Therefore, with the help of the strong mathematical characterizations and the ‘allRootsNI’ technique, we make significant progress towards the old open problems; supplementing the vast body of classical results and concepts in algebraic circuit factorization (e.g., 17, 51, 54, 111).
This article presents a mathematical model and theoretical analysis of coating of a thin film of non-Newtonian polymers as they travel through a small space between two reverse-rotating rolls. The ...dimensionless forms of the governing equations are simplified with the help of the lubrication approximation theory (LAT). By using the perturbation technique, the analytical solutions for velocity, flow rate and pressure gradient were obtained. From an engineering point of view, some significant results such as thickness of the coated web, pressure distribution, separation points, separation force and power input were computed numerically. The effect of velocities ratio k and Weissenberg number We on these physical quantities is presented graphically; others are shown in tabular form. It is noted that the involved material parameters provide a mechanism to control the flow rate, pressure distribution, the thickness of coating, separation force and power input. Moreover, the separation point is shifted toward the nip region by increasing velocities ratio k.
The paper describes the methods of queuing theory to solve the problem of optimizing traffic light phases on signal-controlled road intersections. The flow of vehicles on multi-lane roads is ...described by Poisson processes. In this paper the concept of the effective number of lanes is used which indicates the maximum flow of cars with different modes of traffic lights. Methods of queuing theory helped to obtain explicit solutions of the problem of minimizing delays at signal-controlled road intersection.
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