Learning Parameterized ODEs From Data

In contemporary research, neural networks are being used to derive Ordinary Differential Equations (ODEs) from observations. However, parameterized ODEs pose a more significant challenge than non-parameterized ODEs since the networks are required to understand the roles of the parameters, i.e., the structure of the equations. This paper proposes a novel approach by combining Symbolic Neural Network (S-Net) with ODE Solver to solve this issue. First, S-Net learns the structure of the parameterized ODEs and then predicts the dynamics based on the new parameters with the new initial states. To assess its performance, we compare our approach with a widely used Ordinary Neural Network (O-Net) that directly learns and predicts ODEs. Our numerical experiments demonstrate that our approach outperforms O-Net when applied to the Lotka-Volterra and Lorenz equations.