Numerical framework demonstrates accurate source identification in fractional Poisson equations up to ten dimensions, suggesting robust utility for high-dimensional physical systems.
In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using MC-fPINNs. We construct two neural networks uNN(x;θ ) and fNN(x;ψ) to approximate the solution u*(x) and the forcing term f*(x) of the fractional Poisson equation. To optimize these two neural networks, we use the Monte Carlo sampling method mentioned in MC-fPINNs and define a new loss function combining measurement data and the underlying physical model. Meanwhile, we present a comprehensive error analysis for this method, along with a prior rule to select the appropriate parameters of neural networks. Several numerical examples are given to demonstrate the great precision and robustness of this method in solving high-dimensional problems up to 10D, with various fractional order α and different noise levels of the measurement data ranging from 1$%$ to 10$%$.
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Sheng et al. (2024) studied this question.
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