Solving Differential Equations Using Neural Network Solution Bundles
This method addresses efficiency challenges in dynamical systems analysis for researchers and engineers, though it is incremental as it builds on existing neural network approaches for ODEs.
The authors tackled the computational cost of solving ordinary differential equations (ODEs) for many varied initial conditions by proposing a neural network as a solution bundle, which enables fast, parallelizable evaluation and facilitates Bayesian inference for parameter estimation.
The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution whose values are computed at discrete times. When many varied solutions with different initial conditions to the ODE are required, the computational cost can become significant. We propose that a neural network be used as a solution bundle, a collection of solutions to an ODE for various initial states and system parameters. The neural network solution bundle is trained with an unsupervised loss that does not require any prior knowledge of the sought solutions, and the resulting object is differentiable in initial conditions and system parameters. The solution bundle exhibits fast, parallelizable evaluation of the system state, facilitating the use of Bayesian inference for parameter estimation in real dynamical systems.