Embedding stochastic differential equations into neural networks via dual processes
This method addresses the overfitting issue in neural network training for stochastic systems by eliminating dependence on training data, though it appears incremental as it builds on existing dual process theory.
The authors tackled the problem of predicting expectations of stochastic differential equations without requiring input-output datasets by embedding dual processes into neural networks, achieving high accuracy near the origin for systems like the Ornstein-Uhlenbeck process and noisy van der Pol system.
We propose a new approach to constructing a neural network for predicting expectations of stochastic differential equations. The proposed method does not need data sets of inputs and outputs; instead, the information obtained from the time-evolution equations, i.e., the corresponding dual process, is directly compared with the weights in the neural network. As a demonstration, we construct neural networks for the Ornstein-Uhlenbeck process and the noisy van der Pol system. The remarkable feature of learned networks with the proposed method is the accuracy of inputs near the origin. Hence, it would be possible to avoid the overfitting problem because the learned network does not depend on training data sets.