Neural network-enhanced integrators for simulating ordinary differential equations
This work addresses the need for efficient numerical integration in applications like engineering simulations, but it is incremental as it builds on classical methods with neural network enhancements.
The study tackled the problem of efficiently and accurately simulating ordinary differential equations across various initial conditions and parameters by proposing neural network-enhanced integrators that learn integration errors as additive corrections. The result demonstrated that these enhanced integrators perform at least as well as classical Runge-Kutta schemes, with effectiveness shown through numerical studies on a realistic wind turbine model using OpenFast parameters.
Numerous applications necessitate the computation of numerical solutions to differential equations across a wide range of initial conditions and system parameters, which feeds the demand for efficient yet accurate numerical integration methods.This study proposes a neural network (NN) enhancement of classical numerical integrators. NNs are trained to learn integration errors, which are then used as additive correction terms in numerical schemes. The performance of these enhanced integrators is compared with well-established methods through numerical studies, with a particular emphasis on computational efficiency. Analytical properties are examined in terms of local errors and backward error analysis. Embedded Runge-Kutta schemes are then employed to develop enhanced integrators that mitigate generalization risk, ensuring that the neural network's evaluation in previously unseen regions of the state space does not destabilize the integrator. It is guaranteed that the enhanced integrators perform at least as well as the desired classical Runge-Kutta schemes. The effectiveness of the proposed approaches is demonstrated through extensive numerical studies using a realistic model of a wind turbine, with parameters derived from the established simulation framework OpenFast.