Learning the Physics of Pattern Formation from Images

Using a framework of partial differential equation-constrained optimization, we demonstrate that multiple constitutive relations can be extracted simultaneously from a small set of images of pattern formation. Examples include state-dependent properties in phase-field models, such as the diffusivity, kinetic prefactor, free energy, and direct correlation function, given only the general form of the Cahn-Hilliard equation, Allen-Cahn equation, or dynamical density functional theory (phase-field crystal model). Constraints can be added based on physical arguments to accelerate convergence and avoid spurious results. Reconstruction of the free energy functional, which contains nonlinear dependence on the state variable and differential or convolutional operators, opens the possibility of learning nonequilibrium thermodynamics from only a few snapshots of the dynamics.

Figure 1

Learning phase-field models from images. (a) Inversion of chemical potential ${\mu }_h(c)$ and diffusivity $D(c)$ for the Cahn-Hilliard equation. At the top row are the 5 images provided for training. The following rows show ${\mu }_h(c)$ and $D(c)$ and the model prediction at each iteration of the optimization. Iteration 0 is the initial guess. (b) Inversion of ${\mu }_h(c)$ and $D(c)$ for the Cahn-Hilliard equation using 4 noisy images shown. (c) Inversion of chemical potential and kinetic prefactor $R_0(c)$ for the Allen-Cahn equation. (c) Inversion of ${\mu }_h(c)$ and $D(c)$ for the Cahn-Hilliard equation based on images taken from a part of a larger domain. Except for iteration 0, the functions shown at other iterations are scaled such that $\kappa$ is the same as the known truth for comparison.

Figure 2

Learning dynamic density functional theory (DDFT) from images. (a) Five images of the nucleation event used as training data. (b) Inversion of ${\mu }_h(\eta )$ and $\widehat{C_2}(|\mathbf{k}|)$. $\widehat{C_2}(|\mathbf{k}|)$ is represented by a quartic approximation [Eq. (8)]. ${\mu }_h(\eta )$ is represented by up to third order polynomials. The linear term in ${\mu }_h(\eta )$ and the constant term in $\widehat{C_2}(|\mathbf{k}|)$ are indistinguishable. Hence, they are adjusted such that ${\mu }_h^{\prime }(0)=1$ for comparison with the known truth. (c) The model prediction at the final iteration in (b) using the quartic approximation. The last two images are anaglyphs that highlight their difference with the data. Magenta indicates the model value is greater than data while green means the opposite. (d) Inversion of $\widehat{C_2}(|\mathbf{k}|)$ using Eq. (9). The shaded region in the last image shows the estimated error of $\widehat{C_2}(|\mathbf{k}|)$ inferred by adding $\epsilon {\sum }_i{d}_i^2$ to the objective function normalize by area, where $\epsilon =0.01$, using 100 Hermite functions.