Physics-Informed neural network solver for numerical analysis in geoengineering

Figures & data

Figure 1. Framework of a generic PINN (Karniadakis et al. 2021).

Figure 2. Illustration of implementing AD based on the chain rule (Rao, Sun, and Liu 2021).

Figure 3. Framework of PINNs for (a) forward problems; and (b) inverse problems.

Figure 4. Memory cell of an LSTM-based neural network (Yu et al. 2019).

Table 1. Various domain decomposition methods for PINNs.

PINN variants	Features	References
PPINN	Discrete the long-time domain	Meng et al. (2020)
cPINN	Decomposite the whole time-space domain; For nonlinear conservation laws	Jagtap, Kharazmi, and Karniadakis (2020)
XPINN	Discrete the domain arbitrarily; Applicable to any PDEs	Jagtap and Karniadakis (2021)
APINN	A soft-mannered decomposition method can be discovered through optimization	Hu et al. (2022)
hp-VPINN	The loss is presented in a variational form	Kharazmi, Zhang, and Karniadakis (2021); Kharazmi, Zhang, and Karniadakis (2019)

Figure 5. Framework of convolutional neural network (Zhang and Yin 2021).

Figure 6. Fixed sampling methods: (a) uniform; (b) normal sampling; (c) Latin hypercube sampling; (d) Sobol' sequence; (e) Halton sequence; (f) Hammersley sequence.

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Data availability statement

All data that support the findings of this study are available from the corresponding author upon reasonable request.