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Neural Networks (NNs) are the method of choice for building learning algorithms. Their popularity stems from their empirical success on several challenging learning problems. However, most scholars agree that a convincing theoretical…
Recent theoretical studies proved that deep neural network (DNN) estimators obtained by minimizing empirical risk with a certain sparsity constraint can attain optimal convergence rates for regression and classification problems. However,…
Chance constrained program where one seeks to minimize an objective over decisions which satisfy randomly disturbed constraints with a given probability is computationally intractable. This paper proposes an approximate approach to address…
To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness…
Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on…
Neural networks are widely used to approximate unknown functions in control. A common neural network architecture uses a single hidden layer (i.e. a shallow network), in which the input parameters are fixed in advance and only the output…
This paper proposes an Adaptive-Growth Randomized Neural Network (AG-RaNN) method for computing multivalued solutions of nonlinear first-order PDEs with hyperbolic characteristics, including quasilinear hyperbolic balance laws and…
Physics-informed neural networks (PINNs) and their variants have recently emerged as alternatives to traditional partial differential equation (PDE) solvers, but little literature has focused on devising accurate numerical integration…
Recently, the proximal Newton-type method and its variants have been generalized to solve composite optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. In…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…
Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…
We study the problem of approximate near neighbor (ANN) search and show the following results: - An improved framework for solving the ANN problem using locality-sensitive hashing, reducing the number of evaluations of locality-sensitive…
Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H\"older spaces by these networks is crucial for understanding the…
Physics-informed neural networks (PiNNs) recently emerged as a powerful solver for a large class of partial differential equations under various initial and boundary conditions. In this paper, we propose trapz-PiNNs, physics-informed neural…
Randomized methods of neural network learning suffer from a problem with the generation of random parameters as they are difficult to set optimally to obtain a good projection space. The standard method draws the parameters from a fixed…
Although recent provable methods have been developed to compute preimage bounds for neural networks, their scalability is fundamentally limited by the #P-hardness of the problem. In this work, we adopt a novel probabilistic perspective,…