A critical issue in approximating solutions of ordinary differential equations using neural networks is the exact satisfaction of the boundary or initial conditions. For this purpose, neural forms have been introduced, i.e., functional expressions that depend on neural networks which, by design, satisfy the prescribed conditions exactly. Expanding upon prior progress, the present work contributes in three distinct aspects. First, it presents a novel formalism for crafting optimized neural forms. Second, it outlines a method for establishing an upper bound on the absolute deviation from the exact solution. Third, it introduces a technique for converting problems with Neumann or Robin conditions into equivalent problems with parametric Dirichlet conditions. The proposed optimized neural forms were numerically tested on a set of diverse problems, encompassing first-order and second-order ordinary differential equations, as well as first-order systems. Stiff and delay differential equations were also considered. The obtained solutions were compared against solutions obtained via Runge-Kutta methods and exact solutions wherever available. The reported results and analysis verify that in addition to the exact satisfaction of the boundary or initial conditions, optimized neural forms provide closed-form solutions of superior interpolation capability and controllable overall accuracy.

Classical Physics-informed neural networks (PINNs) approximate solutions to PDEs with the help of deep neural networks trained to satisfy the differential operator and the relevant boundary conditions. We revisit this idea in the quantum computing realm, using parameterised random quantum circuits as trial solutions. We further adapt recent PINN-based techniques to our quantum setting, in particular Gaussian smoothing. Our analysis concentrates on the Poisson, the Heat and the Hamilton-Jacobi-Bellman equations, which are ubiquitous in most areas of science. On the theoretical side, we develop a complexity analysis of this approach, and show numerically that random quantum networks can outperform more traditional quantum networks as well as random classical networks.

Physics informed neural networks (PINNs), a type of machine learning approach, can be used to find the solution of differential equations by including all of the physics into the loss function and building a neural network that approximates the solution. In PINN, the neural network is optimized in such a way that the loss function is taken as a residual of the governing differential equation, boundary conditions, and initial conditions. The fundamental idea of PINN is that the neural network approximates the solution of a differential equation and satisfies any given constraints such that the loss function is minimized. A few of the earliest examples of using artificial neural networks to determine the solution of differential equations are in the work of Dissanayake et al.1 and I.E. The differential equation is presumed to be satisfied by a trial solution in the approach suggested by Lagaris et al..

Hey! Welcome to an other math article! This topic is under investigation and interesting things are occurring while researching! One thing that I use, that is very interesting, is the solution of an ode using a neural network. The method was firstly proposed in the paper https://www.cs.uoi.gr/ Especially now that the topics about scientific machine learning are very "hot" and we have in hands powerful tools to implement and test, this paper is must read. "The best model is the most simple one that does great job" .

In this paper, we propose a surrogate-assisted evolutionary algorithm (EA) for hyperparameter optimization of machine learning (ML) models. The proposed STEADE model initially estimates the objective function landscape using RadialBasis Function interpolation, and then transfers the knowledge to an EA technique called Differential Evolution that is used to evolve new solutions guided by a Bayesian optimization framework. We empirically evaluate our model on the hyperparameter optimization problems as a part of the black box optimization challenge at NeurIPS 2020 and demonstrate the improvement brought about by STEADE over the vanilla EA.