Neural solver for sixth-order ordinary differential equations

A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning method is demonstrated through the solution of two distinct boundary value problems (BVPs), with the method being extended to include the solution of a sixth-order ordinary differential equation (ODE). The proposed mean squared loss function is comprised of two terms: the differential equation is satisfied by the first term, while the initial or boundary conditions are satisfied by the second. The total loss function is minimized using a quasi-Newton optimization method to obtain the desired network output. The approximation capability of the proposed method is verified for sixth-order ODEs. Point-wise comparisons of the approximations show strong agreement with available exact solutions. The proposed algorithm minimizes the overall learnable network hyperparameters in a given BVP. Simple minimization of the total loss function yields highly accurate results even with a low number of epochs. Therefore, the proposed framework offers an attractive setting for the computational mathematics community.