Deep learning and inverse discovery of polymer self-consistent field theory inspired by physics-informed neural networks

We devise a deep learning solver inspired by physics-informed neural networks (PINNs) to tackle the polymer self-consistent field theory (SCFT) equations for one-dimensional AB-diblock copolymers. The PINNs framework comprises two parallel feedforward neural networks that separately represent the segmental partition functions and self-consistent chemical potential fields. The two networks are coupled through a loss function incorporating the governing equation, initial and boundary conditions, and the incompressibility constraint. To avoid the metastable homogeneous solution, the network parameters are initialized based on known self-consistent fields obtained from the numerical pseudospectral method. For copolymers of length $N$ at a given volume fraction of A block ($f$) and the reduced Flory-Huggins interaction parameter ($\ensuremath{\chi}N$), the minimization of the loss function leads to the converged network parameters that successfully capture the stable lamellar phase. The periodicity of the lamellar structure is correctly reproduced for the explored sets of $[f,\ensuremath{\chi}N]$, irrespective of the presumed computational domain size for initialization. Moreover, the proposed PINNs are applicable to the inverse discovery of the interaction parameter and the embedded chemical potential fields for an observed structure. This capability of solving the inverse SCFT problem demonstrates the potential of using PINNs to accelerate the exploration of new polymeric materials.