Probability density prediction for linear systems under fractional Gaussian noise excitation using physics-informed neural networks

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Abstract

This study centered on employing the physics-informed neural networks (PINNs) approach for resolving the time-dependent Fokker-Planck-Kolmogorov (FPK) equation for the first time, culminating in the derivation of the transient probability density function. First, we derived the FPK equation for a dynamical system driven by fractional Gaussian noise (FGN). Second, a deep learning method based on PINNs was introduced for resolving the corresponding time-dependent FPK equation. Finally, two examples under two different excitation conditions were discussed to determine the effectiveness and feasibility of the PINNs algorithm. The results show that the PINNs algorithm can get the transient solution of the system under additive and multiplicative FGN. Concurrently, the Monte Carlo approach was utilized to evaluate the precision and computational efficiency of the PINNs algorithm. We found that the different comparison results are in good consistency, which proves that the PINNs algorithm is not only efficient, but also effective and interpretable.

Keywords

Fokker-Planck-Kolmogorov equation / physics-informed neural networks algorithm / dynamical system / fractional Gaussian noise / Monte Carlo approach

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