A Data Driven Approach for Resolving Time-dependent Differential Equations with Noise

We propose data-driven surrogate models to solve systems of time-dependent differential equations coupled with noise. Using a feedforward neural network, we separately learn the noise and solution, tackling approximations across regimes with bifurcations and rare events. Focusing on irregular data generated by a stochastic noise model on a one-dimensional spatial lattice coupled to a differential equation, we examine two profiles: the periodic complex Ginzburg-Landau equation and a saddle bifurcation equation exhibiting rare events. This coupling introduces conditional data, enabling solutions to reach new states while posing challenges for accurately learning the underlying dynamics.