Code: bloch_torrey_pinn.ipynb

References:

[1] Kenkre et. al. Simple Solutions of the Torrey–Bloch Equations in the NMR Study of Molecular Diffusion : https://physics.unm.edu/kenkre/papers/Art164.pdf

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Solving Torrey-Bloch Equations via Physics-Informed Neural Networks (PINNs)

Trying to reproduce the results in [1] using PINNs (deepxde library)

Bloch-Torrey equation for the magnetization density $M(\mathbf{r},t)$ with diffusion coefficient $D$ and arbitrary time-dependent liear gradient field is

$$M_t(\mathbf{r},t) = -igf(t)xM(\mathbf{r},t) + D\nabla^2M(\mathbf{r},t)$$

In 1-D

$$M_t(x,t) = -igf(t)xM(x,t) + D M_{xx}(x,t)$$

NMR signal is defined by

$$M(t) = \int_{-∞}^∞ M(x,t)dx$$

Denoting real and imaginary parts of $M(x,t) = u(x, t) + iv(x, t)$ we have

$$u_t - v(x, t) g f(t) x - Du_{xx}=0$$ $$v_t + u(x, t) g f(t) x - Dv_{xx}=0$$

where $f(x)$ is the envelope function of the external magnetic field.