Identification of Ordinary Differential Equation Surrogates from Numerical Integration with Adaptive Time-Step Schemes

Abstract

The formulation of an ordinary differential equation (ODE) to describe a dynamic system is generally the first and most crucial step before numerical investigations may be performed. However, this process step will fail if the underlying physics is not fully understood or cannot be described. In such cases, running real-world experiments may be the only possibility to get some insight into the system behavior. Based on the collected data, a surrogate for the ODE may be adjusted until it behaves like the original system. Such a substitution is also recommended if time-intensive simulations need to be combined e.g. with global optimization algorithms. Here, an approach is presented where a neural network surrogate for the state function of the system is trained while being embedded into a numerical integration scheme with step-size control. During forward propagation, an initial value problem is solved while the embedded neural network is treated as state function. To update the adjustable weights of the neural network, the output system states are compared to target system states and the corresponding loss is propagated backward using error backpropagation methods with the goal to minimize the discrepancy against measured data. The surrogate state function optimized in this way may be finally used as a substitute for the unknown ODE under investigation.