Artificial Neural Networks for Solving Ordinary and Partial Differential Equations

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ODE's, to systems of coupled ODE's and also to PDE's. In this article we illustrate the method by solving a variety of model problems and present comparisons with finite elements for several cases of partial differential equations.Comment: LAtex file, 26 pages, 21 figs, submitted to IEEE TN

An Improved Training Method For Wavelet Neural Networks For Solving Ordinary And Partial Differential Equations

Differential equations (DEs) have been widely used for modelling countless studies and applications in various disciplines. In recent years, artificial neural networks (ANNs) have gained attention in solving DEs in a more efficient way to approximate solutions of DEs. In this thesis, wavelet neural networks (WNNs) were employed and programmed to solve ordinary differential equations (ODEs) and partial differential equations (PDEs)

Fast Neural Network Predictions from Constrained Aerodynamics Datasets

Incorporating computational fluid dynamics in the design process of jets, spacecraft, or gas turbine engines is often challenged by the required computational resources and simulation time, which depend on the chosen physics-based computational models and grid resolutions. An ongoing problem in the field is how to simulate these systems faster but with sufficient accuracy. While many approaches involve simplified models of the underlying physics, others are model-free and make predictions based only on existing simulation data. We present a novel model-free approach in which we reformulate the simulation problem to effectively increase the size of constrained pre-computed datasets and introduce a novel neural network architecture (called a cluster network) with an inductive bias well-suited to highly nonlinear computational fluid dynamics solutions. Compared to the state-of-the-art in model-based approximations, we show that our approach is nearly as accurate, an order of magnitude faster, and easier to apply. Furthermore, we show that our method outperforms other model-free approaches

A machine learning framework for data driven acceleration of computations of differential equations

We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods