Solve Systems of Ordinary Differential Equations Using Deep Neural Networks

The systems of ordinary differential equations have been ubiquitously investigated and had many applications for various areas in real life. This paper investigates a deep learning method to solve the systems of ordinary differential equations (ODEs). We formulate the original problem with the initial conditions as an optimization problem. By minimizing a loss function associated with the optimization problem, we can construct an appropriate neural network to estimate the exact solutions of the systems of equations. We do experiments by considering two types of ODEs, Lotka-Volterra and Biochemical Oscillator equations. The experimental results show that we can obtain accurate results in solving these two systems of ODEs, where the numerical errors (mean square errors) vary from 10
⁻⁶
to 10
⁻¹¹
for different neural networks, compared to the traditional approaches.