Related papers: Computing Anti-Derivatives using Deep Neural Netwo…
We propose a simple recursive algorithm that allows the computation of the first- and second-order derivatives with respect to the inputs of an arbitrary deep feed forward neural network (DFNN). The algorithm naturally incorporates the…
The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…
We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
We present a method for the numerical calculation of derivatives of functions of general complex matrices. The method can be used in combination with any algorithm that evaluates or approximates the desired matrix function, in particular…
In insurance mathematics optimal control problems over an infinite time horizon arise when computing risk measures. Their solutions correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In…
We investigate the potential of applying (D)NN ((deep) neural networks) for approximating nonlinear mappings arising in the finite element discretization of nonlinear PDEs (partial differential equations). As an application, we apply the…
Automatic differentiation is everywhere, but there exists only minimal documentation of how it works in complex arithmetic beyond stating "derivatives in $\mathbb{C}^d$" $\cong$ "derivatives in $\mathbb{R}^{2d}$" and, at best, shallow…
A novel class of derivative-free optimization algorithms is developed. The main idea is to utilize certain non-commutative maps in order to approximate the gradient of the objective function. Convergence properties of the novel algorithms…
In this work, multi-variable derivative-free optimization algorithms for unconstrained optimization problems are developed. A novel procedure for approximating the gradient of multi-variable objective functions based on non-commutative maps…
This paper discusses a new method to solve definite integrals using artificial neural networks. The objective is to build a neural network that would be a novel alternative to pre-established numerical methods and with the help of a…
Projection-based model reduction has become a popular approach to reduce the cost associated with integrating large-scale dynamical systems so they can be used in many-query settings such as optimization and uncertainty quantification. For…
Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering…
Neural fields offer continuous, learnable representations that extend beyond traditional discrete formats in visual computing. We study the problem of learning neural representations of repeated antiderivatives directly from a function, a…
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral…
Gradient-based techniques are becoming increasingly critical in quantitative fields, notably in statistics and computer science. The utility of these techniques, however, ultimately depends on how efficiently we can evaluate the derivatives…
We propose a learning paradigm for the numerical approximation of differential invariants of planar curves. Deep neural-networks' (DNNs) universal approximation properties are utilized to estimate geometric measures. The proposed framework…
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…
We consider least squares approximation of a function of one variable by a continuous, piecewise-linear approximand that has a small number of breakpoints. This problem was notably considered by Bellman who proposed an approximate algorithm…