A deep learning approach for the solution of probability density evolution of stochastic systems

Derivation of the probability density evolution provides invaluable insight into the behavior of many stochastic systems and their performance. However, for most real-time applications, numerical determination of the probability density evolution is a formidable task. The latter is due to the required temporal and spatial dis-cretization schemes that render most computational solutions prohibitive and impractical. In this respect, the development of an efficient computational surrogate model is of paramount importance. Recent studies on the physics-constrained networks show that a suitable surrogate can be achieved by encoding the physical insight into a deep neural network. To this aim, the present work introduces DeepPDEM which utilizes the concept of physics-informed networks to solve the evolution of the probability density via proposing a deep learning method. DeepPDEM learns the General Density Evolution Equation (GDEE) of stochastic structures. This approach paves the way for a mesh-free learning method that can solve the density evolution problem without prior simulation data. Moreover, it can also serve as an efficient surrogate for the solution at any other spatio-temporal points within optimization schemes or real-time applications. To demonstrate the potential applica-bility of the proposed framework, two network architectures with different activation functions as well as two optimizers are investigated. Numerical implementation on three different problems verifies the accuracy and efficacy of the proposed method.

Automated summary

The paper introduces a new deep learning method called DeepPDEM for solving the evolution of probability density. By utilizing the concept of physics-constrained networks, DeepPDEM learns the General Density Evolution Equation of stochastic structures. This method can solve the density evolution problem without prior simulation data and can serve as an efficient surrogate in optimization schemes or real-time applications.