Embedding stochastic differential equations into neural networks via dual processes

Abstract

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.

14 pages, 4 figures

References in corpus (4)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• Hamiltonian neural networks for solving equations of motion
• Duality in stochastic processes from the viewpoint of basis expansions
• Numerical methods to evaluate Koopman matrix from system equations