A Neural-network-based Solver for the Three Dimensional Shape of Vesicle Membranes

Abstract

A neural-network-based numerical solver is developed for computing three-dimensional (3D) equilibrium shapes of deformable biomembranes, specifically phospholipid vesicles modeled by Helfrich's curvature elasticity theory. The solver represents vesicle morphology using a phase-field formulation, in which a scalar field distinguishes the interior and exterior of the vesicle through a diffuse interface. The phase field is parameterized by a compact feedforward neural network, and the equilibrium shape is obtained by direct minimization of the Helfrich bending energy subject to global surface-area and volume constraints, enforced via Lagrange multipliers. Automatic differentiation is used to evaluate all spatial derivatives, thereby avoiding finite-difference truncation errors and explicit surface discretization.

This framework produces both axisymmetric and fully non-axisymmetric vesicle shapes without imposing symmetry assumptions. Canonical free-space branches, namely prolates, oblates, and stomatocytes, are reproduced, and the classical bending-energy–reduced-volume diagram is recovered in close quantitative agreement with established results in the literature. In addition, a phase-field expression for the bilayer area-difference constraint is derived and incorporated into the solver, providing a numerical setting for the computation of non-axisymmetric equilibrium morphologies in free space.

A major contribution of this work is a systematic investigation of vesicle morphology under confinement. Vesicles are studied within a range of hard-wall geometries, including cylindrical (tube), slit, spherical, and cubic confinements. By varying confinement size and reduced volume, the solver captures a rich spectrum of deformations, including biaxial squeezed states, bent prolates, squeezed stomatocytes, and cubic and clam-like morphologies. Stability diagrams, bending-energy curves, and phase diagrams are constructed for each confinement, revealing both discontinuous (first-order) and continuous (second-order) shape transitions, as well as hysteresis and metastable branches. These results extend existing confinement studies by providing fully three-dimensional, non-axisymmetric solutions across multiple geometries and different regimes of confinement (free space to weak to strong) within a unified computational framework.

Overall, this work establishes a versatile and scalable neural-network-based phase-field approach for vesicle shape modeling. By unifying classical membrane elasticity theory with modern machine-learning optimization, the solver facilitates a structured exploration of equilibrium morphologies, phase transitions, and confinement effects beyond the reach of traditional axisymmetric or surface-discretization methods. The framework provides a foundation for future extensions to more complex membrane models, dynamic processes, and biologically relevant geometries in soft-matter and biophysical systems.