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We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising $p$-spin glass model, and (b)…
Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher-order…
A large number of engineering, science and computational problems have yet to be solved in a computationally efficient way. One of the emerging challenges is how evolving technologies grow towards autonomy and intelligent decision making.…
Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions…
We study the problem of learning similarity by using nonlinear embedding models (e.g., neural networks) from all possible pairs. This problem is well-known for its difficulty of training with the extreme number of pairs. For the special…
We give a $\mathrm{poly}(\log n, 1/\epsilon)$-query adaptive algorithm for testing whether an unknown Boolean function $f: \{-1,1\}^n \to \{-1,1\}$, which is promised to be a halfspace, is monotone versus $\epsilon$-far from monotone. Since…
Decision Diagrams(DDs) are one of the most popular representations for boolean functions. They are widely used in the design and verification of circuits. Different types of DDs have been proven to represent important functions in…
Understanding the characteristics of neural networks is important but difficult due to their complex structures and behaviors. Some previous work proposes to transform neural networks into equivalent Boolean expressions and apply…
Anomaly detection has many applications ranging from bank-fraud detection and cyber-threat detection to equipment maintenance and health monitoring. However, choosing a suitable algorithm for a given application remains a challenging design…
Optimization can be found in many real-life applications. Designing an effective algorithm for a specific optimization problem typically requires a tedious amount of effort from human experts with domain knowledge and algorithm design…
This paper considers the problem of approximating a Boolean function $f$ using another Boolean function from a specified class. Two classes of approximating functions are considered: $k$-juntas, and linear Boolean functions. The $n$ input…
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
High-dimensional problems have long been considered the Achilles' heel of Bayesian optimization algorithms. Spurred by the curse of dimensionality, a large collection of algorithms aim to make it more performant in this setting, commonly by…
This paper presents an application of evolutionary search procedures to artificial neural networks. Here, we can distinguish among three kinds of evolution in artificial neural networks, i.e. the evolution of connection weights, of…
A nonlinear inequality is formulated in the paper. An estimate of the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be…
Recent advances in nonlinear dynamical systems theory provide a new insight into numerical properties of discrete algorithms developed to solve nonlinear initial value problems. Basic features like accuracy and stability are well pointed…
Successful application of Adomian decomposition method (ADM) in solving problems in nonlinear ordinary and partial differential equations depend strictly on the Adomian polynomial. In this paper, we present a simple modified known Adomian…
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem…
Boolean networks are special types of finite state time-discrete dynamical systems. A Boolean network can be described by a function from an n-dimensional vector space over the field of two elements to itself. A fundamental problem in…
The lower and upper bound of any given algorithm is one of the most crucial pieces of information needed when evaluating the computational effectiveness for said algorithm. Here a novel method of Boolean Algebraic Programming for symbolic…